# Problem 1 : F
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# Testing accuracy of FIXED POINT CALCULATIONS
# C1A
# c = 1 , Initial Value: 4500
print " n x(n)"
P = vector (RDF,1001)
P[0] = 4500;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 4500.0 1 2249.00066667 2 1123.50166726 3 560.753503853 4 279.382101871 5 138.701788917 6 68.3725235957 7 33.2301390726 8 15.705349026 9 7.04369223539 10 2.94775910477 11 1.49160179269 12 1.7570615523 13 1.58592685233 14 1.68460174202 15 1.62313721065 16 1.65984118531 17 1.63732242516 18 1.65092097639 19 1.6426280225 20 1.64765567772 21 1.64459660783 22 1.64645383587 23 1.6453247713 24 1.64601060986 25 1.64559379982 26 1.64584703557 27 1.64569315261 28 1.64578665189 29 1.64572983793 30 1.64576435899 31 1.64574338293 32 1.64575612844 33 1.64574838392 34 1.64575308968 35 1.64575023034 36 1.64575196774 37 1.64575091205 38 1.64575155352 39 1.64575116375 40 1.64575140058 41 1.64575125667 42 1.64575134411 43 1.64575129098 44 1.64575132327 45 1.64575130365 46 1.64575131557 47 1.64575130833 48 1.64575131273 49 1.64575131005 50 1.64575131168 51 1.64575131069 52 1.64575131129 53 1.64575131093 54 1.64575131115 55 1.64575131101 56 1.6457513111 57 1.64575131105 ... 941 1.64575131106 942 1.64575131106 943 1.64575131106 944 1.64575131106 945 1.64575131106 946 1.64575131106 947 1.64575131106 948 1.64575131106 949 1.64575131106 950 1.64575131106 951 1.64575131106 952 1.64575131106 953 1.64575131106 954 1.64575131106 955 1.64575131106 956 1.64575131106 957 1.64575131106 958 1.64575131106 959 1.64575131106 960 1.64575131106 961 1.64575131106 962 1.64575131106 963 1.64575131106 964 1.64575131106 965 1.64575131106 966 1.64575131106 967 1.64575131106 968 1.64575131106 969 1.64575131106 970 1.64575131106 971 1.64575131106 972 1.64575131106 973 1.64575131106 974 1.64575131106 975 1.64575131106 976 1.64575131106 977 1.64575131106 978 1.64575131106 979 1.64575131106 980 1.64575131106 981 1.64575131106 982 1.64575131106 983 1.64575131106 984 1.64575131106 985 1.64575131106 986 1.64575131106 987 1.64575131106 988 1.64575131106 989 1.64575131106 990 1.64575131106 991 1.64575131106 992 1.64575131106 993 1.64575131106 994 1.64575131106 995 1.64575131106 996 1.64575131106 997 1.64575131106 998 1.64575131106 999 1.64575131106 1000 1.64575131106 WARNING: Output truncated! full_output.txt n x(n) 0 4500.0 1 2249.00066667 2 1123.50166726 3 560.753503853 4 279.382101871 5 138.701788917 6 68.3725235957 7 33.2301390726 8 15.705349026 9 7.04369223539 10 2.94775910477 11 1.49160179269 12 1.7570615523 13 1.58592685233 14 1.68460174202 15 1.62313721065 16 1.65984118531 17 1.63732242516 18 1.65092097639 19 1.6426280225 20 1.64765567772 21 1.64459660783 22 1.64645383587 23 1.6453247713 24 1.64601060986 25 1.64559379982 26 1.64584703557 27 1.64569315261 28 1.64578665189 29 1.64572983793 30 1.64576435899 31 1.64574338293 32 1.64575612844 33 1.64574838392 34 1.64575308968 35 1.64575023034 36 1.64575196774 37 1.64575091205 38 1.64575155352 39 1.64575116375 40 1.64575140058 41 1.64575125667 42 1.64575134411 43 1.64575129098 44 1.64575132327 45 1.64575130365 46 1.64575131557 47 1.64575130833 48 1.64575131273 49 1.64575131005 50 1.64575131168 51 1.64575131069 52 1.64575131129 53 1.64575131093 54 1.64575131115 55 1.64575131101 56 1.6457513111 57 1.64575131105 ... 941 1.64575131106 942 1.64575131106 943 1.64575131106 944 1.64575131106 945 1.64575131106 946 1.64575131106 947 1.64575131106 948 1.64575131106 949 1.64575131106 950 1.64575131106 951 1.64575131106 952 1.64575131106 953 1.64575131106 954 1.64575131106 955 1.64575131106 956 1.64575131106 957 1.64575131106 958 1.64575131106 959 1.64575131106 960 1.64575131106 961 1.64575131106 962 1.64575131106 963 1.64575131106 964 1.64575131106 965 1.64575131106 966 1.64575131106 967 1.64575131106 968 1.64575131106 969 1.64575131106 970 1.64575131106 971 1.64575131106 972 1.64575131106 973 1.64575131106 974 1.64575131106 975 1.64575131106 976 1.64575131106 977 1.64575131106 978 1.64575131106 979 1.64575131106 980 1.64575131106 981 1.64575131106 982 1.64575131106 983 1.64575131106 984 1.64575131106 985 1.64575131106 986 1.64575131106 987 1.64575131106 988 1.64575131106 989 1.64575131106 990 1.64575131106 991 1.64575131106 992 1.64575131106 993 1.64575131106 994 1.64575131106 995 1.64575131106 996 1.64575131106 997 1.64575131106 998 1.64575131106 999 1.64575131106 1000 1.64575131106 |
# C1B
# c = 1 , Initial Value: 2000
print " n x(n)"
P = vector (RDF,1001)
P[0] = 2000;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 2000.0 1 999.0015 2 498.503752998 3 248.257894508 4 123.141031462 5 60.5948780411 6 29.3469481556 7 13.7756993581 8 6.1056244665 9 2.54416246556 10 1.45125121135 11 1.79280734274 12 1.56975692494 13 1.69600237748 14 1.61686663365 15 1.66387391877 16 1.63495818963 17 1.6523884026 18 1.64174797502 19 1.64819465161 20 1.64427068765 21 1.64665245194 22 1.64520430137 23 1.6460838894 24 1.64554930251 25 1.64587408402 26 1.64567672126 27 1.64579663745 28 1.64572377099 29 1.64576804562 30 1.64574114292 31 1.64575748956 32 1.64574755688 33 1.64575359221 34 1.64574992499 35 1.64575215328 36 1.64575079931 37 1.64575162202 38 1.64575112212 39 1.64575142587 40 1.64575124131 41 1.64575135345 42 1.64575128531 43 1.64575132671 44 1.64575130156 45 1.64575131684 46 1.64575130755 47 1.6457513132 48 1.64575130977 49 1.64575131185 50 1.64575131059 51 1.64575131136 52 1.64575131089 53 1.64575131117 54 1.645751311 55 1.6457513111 56 1.64575131104 57 1.64575131108 ... 941 1.64575131106 942 1.64575131106 943 1.64575131106 944 1.64575131106 945 1.64575131106 946 1.64575131106 947 1.64575131106 948 1.64575131106 949 1.64575131106 950 1.64575131106 951 1.64575131106 952 1.64575131106 953 1.64575131106 954 1.64575131106 955 1.64575131106 956 1.64575131106 957 1.64575131106 958 1.64575131106 959 1.64575131106 960 1.64575131106 961 1.64575131106 962 1.64575131106 963 1.64575131106 964 1.64575131106 965 1.64575131106 966 1.64575131106 967 1.64575131106 968 1.64575131106 969 1.64575131106 970 1.64575131106 971 1.64575131106 972 1.64575131106 973 1.64575131106 974 1.64575131106 975 1.64575131106 976 1.64575131106 977 1.64575131106 978 1.64575131106 979 1.64575131106 980 1.64575131106 981 1.64575131106 982 1.64575131106 983 1.64575131106 984 1.64575131106 985 1.64575131106 986 1.64575131106 987 1.64575131106 988 1.64575131106 989 1.64575131106 990 1.64575131106 991 1.64575131106 992 1.64575131106 993 1.64575131106 994 1.64575131106 995 1.64575131106 996 1.64575131106 997 1.64575131106 998 1.64575131106 999 1.64575131106 1000 1.64575131106 WARNING: Output truncated! full_output.txt n x(n) 0 2000.0 1 999.0015 2 498.503752998 3 248.257894508 4 123.141031462 5 60.5948780411 6 29.3469481556 7 13.7756993581 8 6.1056244665 9 2.54416246556 10 1.45125121135 11 1.79280734274 12 1.56975692494 13 1.69600237748 14 1.61686663365 15 1.66387391877 16 1.63495818963 17 1.6523884026 18 1.64174797502 19 1.64819465161 20 1.64427068765 21 1.64665245194 22 1.64520430137 23 1.6460838894 24 1.64554930251 25 1.64587408402 26 1.64567672126 27 1.64579663745 28 1.64572377099 29 1.64576804562 30 1.64574114292 31 1.64575748956 32 1.64574755688 33 1.64575359221 34 1.64574992499 35 1.64575215328 36 1.64575079931 37 1.64575162202 38 1.64575112212 39 1.64575142587 40 1.64575124131 41 1.64575135345 42 1.64575128531 43 1.64575132671 44 1.64575130156 45 1.64575131684 46 1.64575130755 47 1.6457513132 48 1.64575130977 49 1.64575131185 50 1.64575131059 51 1.64575131136 52 1.64575131089 53 1.64575131117 54 1.645751311 55 1.6457513111 56 1.64575131104 57 1.64575131108 ... 941 1.64575131106 942 1.64575131106 943 1.64575131106 944 1.64575131106 945 1.64575131106 946 1.64575131106 947 1.64575131106 948 1.64575131106 949 1.64575131106 950 1.64575131106 951 1.64575131106 952 1.64575131106 953 1.64575131106 954 1.64575131106 955 1.64575131106 956 1.64575131106 957 1.64575131106 958 1.64575131106 959 1.64575131106 960 1.64575131106 961 1.64575131106 962 1.64575131106 963 1.64575131106 964 1.64575131106 965 1.64575131106 966 1.64575131106 967 1.64575131106 968 1.64575131106 969 1.64575131106 970 1.64575131106 971 1.64575131106 972 1.64575131106 973 1.64575131106 974 1.64575131106 975 1.64575131106 976 1.64575131106 977 1.64575131106 978 1.64575131106 979 1.64575131106 980 1.64575131106 981 1.64575131106 982 1.64575131106 983 1.64575131106 984 1.64575131106 985 1.64575131106 986 1.64575131106 987 1.64575131106 988 1.64575131106 989 1.64575131106 990 1.64575131106 991 1.64575131106 992 1.64575131106 993 1.64575131106 994 1.64575131106 995 1.64575131106 996 1.64575131106 997 1.64575131106 998 1.64575131106 999 1.64575131106 1000 1.64575131106 |
# C1C
# c = 1 , Initial Value: 300
print " n x(n)"
P = vector (RDF,1001)
P[0] = 300;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 300.0 1 149.01 2 73.525132877 3 35.8033688129 4 16.9854754043 5 7.66935919368 6 3.22584656715 7 1.54291171841 8 1.71583152517 9 1.60633916896 10 1.67077018712 11 1.63096417626 12 1.65488484367 13 1.64025738082 14 1.64910994365 15 1.6437180977 16 1.64698952846 17 1.64499997004 18 1.64620822492 19 1.64547381915 20 1.64591997398 21 1.64564884637 22 1.64581357827 23 1.64571347855 24 1.64577430002 25 1.64573734275 26 1.64575979869 27 1.64574615382 28 1.64575444476 29 1.64574940696 30 1.64575246805 31 1.64575060805 32 1.64575173823 33 1.64575105151 34 1.64575146878 35 1.64575121523 36 1.64575136929 37 1.64575127568 38 1.64575133256 39 1.645751298 40 1.645751319 41 1.64575130624 42 1.645751314 43 1.64575130928 44 1.64575131215 45 1.64575131041 46 1.64575131146 47 1.64575131082 48 1.64575131121 49 1.64575131097 50 1.64575131112 51 1.64575131103 52 1.64575131108 53 1.64575131105 54 1.64575131107 55 1.64575131106 56 1.64575131107 57 1.64575131106 ... 941 1.64575131106 942 1.64575131106 943 1.64575131106 944 1.64575131106 945 1.64575131106 946 1.64575131106 947 1.64575131106 948 1.64575131106 949 1.64575131106 950 1.64575131106 951 1.64575131106 952 1.64575131106 953 1.64575131106 954 1.64575131106 955 1.64575131106 956 1.64575131106 957 1.64575131106 958 1.64575131106 959 1.64575131106 960 1.64575131106 961 1.64575131106 962 1.64575131106 963 1.64575131106 964 1.64575131106 965 1.64575131106 966 1.64575131106 967 1.64575131106 968 1.64575131106 969 1.64575131106 970 1.64575131106 971 1.64575131106 972 1.64575131106 973 1.64575131106 974 1.64575131106 975 1.64575131106 976 1.64575131106 977 1.64575131106 978 1.64575131106 979 1.64575131106 980 1.64575131106 981 1.64575131106 982 1.64575131106 983 1.64575131106 984 1.64575131106 985 1.64575131106 986 1.64575131106 987 1.64575131106 988 1.64575131106 989 1.64575131106 990 1.64575131106 991 1.64575131106 992 1.64575131106 993 1.64575131106 994 1.64575131106 995 1.64575131106 996 1.64575131106 997 1.64575131106 998 1.64575131106 999 1.64575131106 1000 1.64575131106 WARNING: Output truncated! full_output.txt n x(n) 0 300.0 1 149.01 2 73.525132877 3 35.8033688129 4 16.9854754043 5 7.66935919368 6 3.22584656715 7 1.54291171841 8 1.71583152517 9 1.60633916896 10 1.67077018712 11 1.63096417626 12 1.65488484367 13 1.64025738082 14 1.64910994365 15 1.6437180977 16 1.64698952846 17 1.64499997004 18 1.64620822492 19 1.64547381915 20 1.64591997398 21 1.64564884637 22 1.64581357827 23 1.64571347855 24 1.64577430002 25 1.64573734275 26 1.64575979869 27 1.64574615382 28 1.64575444476 29 1.64574940696 30 1.64575246805 31 1.64575060805 32 1.64575173823 33 1.64575105151 34 1.64575146878 35 1.64575121523 36 1.64575136929 37 1.64575127568 38 1.64575133256 39 1.645751298 40 1.645751319 41 1.64575130624 42 1.645751314 43 1.64575130928 44 1.64575131215 45 1.64575131041 46 1.64575131146 47 1.64575131082 48 1.64575131121 49 1.64575131097 50 1.64575131112 51 1.64575131103 52 1.64575131108 53 1.64575131105 54 1.64575131107 55 1.64575131106 56 1.64575131107 57 1.64575131106 ... 941 1.64575131106 942 1.64575131106 943 1.64575131106 944 1.64575131106 945 1.64575131106 946 1.64575131106 947 1.64575131106 948 1.64575131106 949 1.64575131106 950 1.64575131106 951 1.64575131106 952 1.64575131106 953 1.64575131106 954 1.64575131106 955 1.64575131106 956 1.64575131106 957 1.64575131106 958 1.64575131106 959 1.64575131106 960 1.64575131106 961 1.64575131106 962 1.64575131106 963 1.64575131106 964 1.64575131106 965 1.64575131106 966 1.64575131106 967 1.64575131106 968 1.64575131106 969 1.64575131106 970 1.64575131106 971 1.64575131106 972 1.64575131106 973 1.64575131106 974 1.64575131106 975 1.64575131106 976 1.64575131106 977 1.64575131106 978 1.64575131106 979 1.64575131106 980 1.64575131106 981 1.64575131106 982 1.64575131106 983 1.64575131106 984 1.64575131106 985 1.64575131106 986 1.64575131106 987 1.64575131106 988 1.64575131106 989 1.64575131106 990 1.64575131106 991 1.64575131106 992 1.64575131106 993 1.64575131106 994 1.64575131106 995 1.64575131106 996 1.64575131106 997 1.64575131106 998 1.64575131106 999 1.64575131106 1000 1.64575131106 |
# In the above tables, it appears that the fixed point is 1.645751311...
# With a value of c=1, our calculations predicted the fixed point to likewise be 1.645751311...
# It appears that our calculations are correct.
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# C2A
# c = 0.5 , Initial Value: 300
print " n x(n)"
P = vector (RDF,1001)
P[0] = 300;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-0.5;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 300.0 1 149.51 2 74.2750655475 3 36.677923179 4 17.9207546612 5 8.62778099505 6 4.1616044696 7 2.3016780465 8 1.95423590995 9 2.01224479648 10 1.99699468438 11 2.00075472097 12 1.99981153328 13 2.00004713 14 1.99998821833 15 2.00000294547 16 1.99999926364 17 2.00000018409 18 1.99999995398 19 2.00000001151 20 1.99999999712 21 2.00000000072 22 1.99999999982 23 2.00000000004 24 1.99999999999 25 2.0 26 2.0 27 2.0 28 2.0 29 2.0 30 2.0 31 2.0 32 2.0 33 2.0 34 2.0 35 2.0 36 2.0 37 2.0 38 2.0 39 2.0 40 2.0 41 2.0 42 2.0 43 2.0 44 2.0 45 2.0 46 2.0 47 2.0 48 2.0 49 2.0 50 2.0 51 2.0 52 2.0 53 2.0 54 2.0 55 2.0 56 2.0 57 2.0 ... 941 2.0 942 2.0 943 2.0 944 2.0 945 2.0 946 2.0 947 2.0 948 2.0 949 2.0 950 2.0 951 2.0 952 2.0 953 2.0 954 2.0 955 2.0 956 2.0 957 2.0 958 2.0 959 2.0 960 2.0 961 2.0 962 2.0 963 2.0 964 2.0 965 2.0 966 2.0 967 2.0 968 2.0 969 2.0 970 2.0 971 2.0 972 2.0 973 2.0 974 2.0 975 2.0 976 2.0 977 2.0 978 2.0 979 2.0 980 2.0 981 2.0 982 2.0 983 2.0 984 2.0 985 2.0 986 2.0 987 2.0 988 2.0 989 2.0 990 2.0 991 2.0 992 2.0 993 2.0 994 2.0 995 2.0 996 2.0 997 2.0 998 2.0 999 2.0 1000 2.0 WARNING: Output truncated! full_output.txt n x(n) 0 300.0 1 149.51 2 74.2750655475 3 36.677923179 4 17.9207546612 5 8.62778099505 6 4.1616044696 7 2.3016780465 8 1.95423590995 9 2.01224479648 10 1.99699468438 11 2.00075472097 12 1.99981153328 13 2.00004713 14 1.99998821833 15 2.00000294547 16 1.99999926364 17 2.00000018409 18 1.99999995398 19 2.00000001151 20 1.99999999712 21 2.00000000072 22 1.99999999982 23 2.00000000004 24 1.99999999999 25 2.0 26 2.0 27 2.0 28 2.0 29 2.0 30 2.0 31 2.0 32 2.0 33 2.0 34 2.0 35 2.0 36 2.0 37 2.0 38 2.0 39 2.0 40 2.0 41 2.0 42 2.0 43 2.0 44 2.0 45 2.0 46 2.0 47 2.0 48 2.0 49 2.0 50 2.0 51 2.0 52 2.0 53 2.0 54 2.0 55 2.0 56 2.0 57 2.0 ... 941 2.0 942 2.0 943 2.0 944 2.0 945 2.0 946 2.0 947 2.0 948 2.0 949 2.0 950 2.0 951 2.0 952 2.0 953 2.0 954 2.0 955 2.0 956 2.0 957 2.0 958 2.0 959 2.0 960 2.0 961 2.0 962 2.0 963 2.0 964 2.0 965 2.0 966 2.0 967 2.0 968 2.0 969 2.0 970 2.0 971 2.0 972 2.0 973 2.0 974 2.0 975 2.0 976 2.0 977 2.0 978 2.0 979 2.0 980 2.0 981 2.0 982 2.0 983 2.0 984 2.0 985 2.0 986 2.0 987 2.0 988 2.0 989 2.0 990 2.0 991 2.0 992 2.0 993 2.0 994 2.0 995 2.0 996 2.0 997 2.0 998 2.0 999 2.0 1000 2.0 |
# C2B
# c = 0.5 , Initial Value: 5000
print " n x(n)"
P = vector (RDF,1001)
P[0] = 5000;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-0.5;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 5000.0 1 2499.5006 2 1249.25150024 3 624.128151558 4 311.568882484 5 155.294069931 6 77.1663531535 7 38.1220536245 8 18.6397214185 9 8.98080732068 10 4.32444935277 11 2.35595461869 12 1.95134648896 13 2.01307319727 14 1.99679537515 15 2.0008050135 16 1.99979898954 17 2.00005026777 18 1.99998743401 19 2.00000314156 20 1.99999921461 21 2.00000019635 22 1.99999995091 23 2.00000001227 24 1.99999999693 25 2.00000000077 26 1.99999999981 27 2.00000000005 28 1.99999999999 29 2.0 30 2.0 31 2.0 32 2.0 33 2.0 34 2.0 35 2.0 36 2.0 37 2.0 38 2.0 39 2.0 40 2.0 41 2.0 42 2.0 43 2.0 44 2.0 45 2.0 46 2.0 47 2.0 48 2.0 49 2.0 50 2.0 51 2.0 52 2.0 53 2.0 54 2.0 55 2.0 56 2.0 57 2.0 ... 941 2.0 942 2.0 943 2.0 944 2.0 945 2.0 946 2.0 947 2.0 948 2.0 949 2.0 950 2.0 951 2.0 952 2.0 953 2.0 954 2.0 955 2.0 956 2.0 957 2.0 958 2.0 959 2.0 960 2.0 961 2.0 962 2.0 963 2.0 964 2.0 965 2.0 966 2.0 967 2.0 968 2.0 969 2.0 970 2.0 971 2.0 972 2.0 973 2.0 974 2.0 975 2.0 976 2.0 977 2.0 978 2.0 979 2.0 980 2.0 981 2.0 982 2.0 983 2.0 984 2.0 985 2.0 986 2.0 987 2.0 988 2.0 989 2.0 990 2.0 991 2.0 992 2.0 993 2.0 994 2.0 995 2.0 996 2.0 997 2.0 998 2.0 999 2.0 1000 2.0 WARNING: Output truncated! full_output.txt n x(n) 0 5000.0 1 2499.5006 2 1249.25150024 3 624.128151558 4 311.568882484 5 155.294069931 6 77.1663531535 7 38.1220536245 8 18.6397214185 9 8.98080732068 10 4.32444935277 11 2.35595461869 12 1.95134648896 13 2.01307319727 14 1.99679537515 15 2.0008050135 16 1.99979898954 17 2.00005026777 18 1.99998743401 19 2.00000314156 20 1.99999921461 21 2.00000019635 22 1.99999995091 23 2.00000001227 24 1.99999999693 25 2.00000000077 26 1.99999999981 27 2.00000000005 28 1.99999999999 29 2.0 30 2.0 31 2.0 32 2.0 33 2.0 34 2.0 35 2.0 36 2.0 37 2.0 38 2.0 39 2.0 40 2.0 41 2.0 42 2.0 43 2.0 44 2.0 45 2.0 46 2.0 47 2.0 48 2.0 49 2.0 50 2.0 51 2.0 52 2.0 53 2.0 54 2.0 55 2.0 56 2.0 57 2.0 ... 941 2.0 942 2.0 943 2.0 944 2.0 945 2.0 946 2.0 947 2.0 948 2.0 949 2.0 950 2.0 951 2.0 952 2.0 953 2.0 954 2.0 955 2.0 956 2.0 957 2.0 958 2.0 959 2.0 960 2.0 961 2.0 962 2.0 963 2.0 964 2.0 965 2.0 966 2.0 967 2.0 968 2.0 969 2.0 970 2.0 971 2.0 972 2.0 973 2.0 974 2.0 975 2.0 976 2.0 977 2.0 978 2.0 979 2.0 980 2.0 981 2.0 982 2.0 983 2.0 984 2.0 985 2.0 986 2.0 987 2.0 988 2.0 989 2.0 990 2.0 991 2.0 992 2.0 993 2.0 994 2.0 995 2.0 996 2.0 997 2.0 998 2.0 999 2.0 1000 2.0 |
# C2C
# c = 0.5 , Initial Value: 2350
print " n x(n)"
P = vector (RDF,1001)
P[0] = 2350;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-0.5;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 2350.0 1 1174.5012766 2 586.753192574 3 292.881709169 4 145.951097628 5 72.4961036439 6 35.7894333562 7 17.478540302 8 8.41090919797 9 4.06213423128 10 2.26959514775 11 1.95661922254 12 2.0115665503 13 1.99715824336 14 2.00071347181 15 1.99982182287 16 2.00004455619 17 1.9999888617 18 2.00000278462 19 1.99999930385 20 2.00000017404 21 1.99999995649 22 2.00000001088 23 1.99999999728 24 2.00000000068 25 1.99999999983 26 2.00000000004 27 1.99999999999 28 2.0 29 2.0 30 2.0 31 2.0 32 2.0 33 2.0 34 2.0 35 2.0 36 2.0 37 2.0 38 2.0 39 2.0 40 2.0 41 2.0 42 2.0 43 2.0 44 2.0 45 2.0 46 2.0 47 2.0 48 2.0 49 2.0 50 2.0 51 2.0 52 2.0 53 2.0 54 2.0 55 2.0 56 2.0 57 2.0 ... 941 2.0 942 2.0 943 2.0 944 2.0 945 2.0 946 2.0 947 2.0 948 2.0 949 2.0 950 2.0 951 2.0 952 2.0 953 2.0 954 2.0 955 2.0 956 2.0 957 2.0 958 2.0 959 2.0 960 2.0 961 2.0 962 2.0 963 2.0 964 2.0 965 2.0 966 2.0 967 2.0 968 2.0 969 2.0 970 2.0 971 2.0 972 2.0 973 2.0 974 2.0 975 2.0 976 2.0 977 2.0 978 2.0 979 2.0 980 2.0 981 2.0 982 2.0 983 2.0 984 2.0 985 2.0 986 2.0 987 2.0 988 2.0 989 2.0 990 2.0 991 2.0 992 2.0 993 2.0 994 2.0 995 2.0 996 2.0 997 2.0 998 2.0 999 2.0 1000 2.0 WARNING: Output truncated! full_output.txt n x(n) 0 2350.0 1 1174.5012766 2 586.753192574 3 292.881709169 4 145.951097628 5 72.4961036439 6 35.7894333562 7 17.478540302 8 8.41090919797 9 4.06213423128 10 2.26959514775 11 1.95661922254 12 2.0115665503 13 1.99715824336 14 2.00071347181 15 1.99982182287 16 2.00004455619 17 1.9999888617 18 2.00000278462 19 1.99999930385 20 2.00000017404 21 1.99999995649 22 2.00000001088 23 1.99999999728 24 2.00000000068 25 1.99999999983 26 2.00000000004 27 1.99999999999 28 2.0 29 2.0 30 2.0 31 2.0 32 2.0 33 2.0 34 2.0 35 2.0 36 2.0 37 2.0 38 2.0 39 2.0 40 2.0 41 2.0 42 2.0 43 2.0 44 2.0 45 2.0 46 2.0 47 2.0 48 2.0 49 2.0 50 2.0 51 2.0 52 2.0 53 2.0 54 2.0 55 2.0 56 2.0 57 2.0 ... 941 2.0 942 2.0 943 2.0 944 2.0 945 2.0 946 2.0 947 2.0 948 2.0 949 2.0 950 2.0 951 2.0 952 2.0 953 2.0 954 2.0 955 2.0 956 2.0 957 2.0 958 2.0 959 2.0 960 2.0 961 2.0 962 2.0 963 2.0 964 2.0 965 2.0 966 2.0 967 2.0 968 2.0 969 2.0 970 2.0 971 2.0 972 2.0 973 2.0 974 2.0 975 2.0 976 2.0 977 2.0 978 2.0 979 2.0 980 2.0 981 2.0 982 2.0 983 2.0 984 2.0 985 2.0 986 2.0 987 2.0 988 2.0 989 2.0 990 2.0 991 2.0 992 2.0 993 2.0 994 2.0 995 2.0 996 2.0 997 2.0 998 2.0 999 2.0 1000 2.0 |
# In the above tables, it appears that the fixed point is 2.
# With a value of c=0.5, our calculations predicted the fixed point to likewise be 2.
# It appears that our calculations are correct.
|
|
# C3A
# c = 0.8 , Initial Value: 2350
print " n x(n)"
P = vector (RDF,1001)
P[0] = 2350;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-0.8;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 2350.0 1 1174.2012766 2 586.303193226 3 292.35671342 4 145.388618147 5 71.9149434259 6 35.1991876604 7 16.8848230698 8 7.82008588706 9 3.4936704437 10 1.80553098275 11 1.76432656602 12 1.78252876964 13 1.77426667409 14 1.7779727378 15 1.77630132948 16 1.77705330204 17 1.77671461733 18 1.77686708437 19 1.7767984325 20 1.77682934154 21 1.77681542478 22 1.77682169066 23 1.77681886948 24 1.7768201397 25 1.77681956779 26 1.77681982529 27 1.77681970935 28 1.77681976155 29 1.77681973805 30 1.77681974863 31 1.77681974387 32 1.77681974601 33 1.77681974505 34 1.77681974548 35 1.77681974528 36 1.77681974537 37 1.77681974533 38 1.77681974535 39 1.77681974534 40 1.77681974535 41 1.77681974534 42 1.77681974535 43 1.77681974534 44 1.77681974535 45 1.77681974535 46 1.77681974535 47 1.77681974535 48 1.77681974535 49 1.77681974535 50 1.77681974535 51 1.77681974535 52 1.77681974535 53 1.77681974535 54 1.77681974535 55 1.77681974535 56 1.77681974535 57 1.77681974535 ... 941 1.77681974535 942 1.77681974535 943 1.77681974535 944 1.77681974535 945 1.77681974535 946 1.77681974535 947 1.77681974535 948 1.77681974535 949 1.77681974535 950 1.77681974535 951 1.77681974535 952 1.77681974535 953 1.77681974535 954 1.77681974535 955 1.77681974535 956 1.77681974535 957 1.77681974535 958 1.77681974535 959 1.77681974535 960 1.77681974535 961 1.77681974535 962 1.77681974535 963 1.77681974535 964 1.77681974535 965 1.77681974535 966 1.77681974535 967 1.77681974535 968 1.77681974535 969 1.77681974535 970 1.77681974535 971 1.77681974535 972 1.77681974535 973 1.77681974535 974 1.77681974535 975 1.77681974535 976 1.77681974535 977 1.77681974535 978 1.77681974535 979 1.77681974535 980 1.77681974535 981 1.77681974535 982 1.77681974535 983 1.77681974535 984 1.77681974535 985 1.77681974535 986 1.77681974535 987 1.77681974535 988 1.77681974535 989 1.77681974535 990 1.77681974535 991 1.77681974535 992 1.77681974535 993 1.77681974535 994 1.77681974535 995 1.77681974535 996 1.77681974535 997 1.77681974535 998 1.77681974535 999 1.77681974535 1000 1.77681974535 WARNING: Output truncated! full_output.txt n x(n) 0 2350.0 1 1174.2012766 2 586.303193226 3 292.35671342 4 145.388618147 5 71.9149434259 6 35.1991876604 7 16.8848230698 8 7.82008588706 9 3.4936704437 10 1.80553098275 11 1.76432656602 12 1.78252876964 13 1.77426667409 14 1.7779727378 15 1.77630132948 16 1.77705330204 17 1.77671461733 18 1.77686708437 19 1.7767984325 20 1.77682934154 21 1.77681542478 22 1.77682169066 23 1.77681886948 24 1.7768201397 25 1.77681956779 26 1.77681982529 27 1.77681970935 28 1.77681976155 29 1.77681973805 30 1.77681974863 31 1.77681974387 32 1.77681974601 33 1.77681974505 34 1.77681974548 35 1.77681974528 36 1.77681974537 37 1.77681974533 38 1.77681974535 39 1.77681974534 40 1.77681974535 41 1.77681974534 42 1.77681974535 43 1.77681974534 44 1.77681974535 45 1.77681974535 46 1.77681974535 47 1.77681974535 48 1.77681974535 49 1.77681974535 50 1.77681974535 51 1.77681974535 52 1.77681974535 53 1.77681974535 54 1.77681974535 55 1.77681974535 56 1.77681974535 57 1.77681974535 ... 941 1.77681974535 942 1.77681974535 943 1.77681974535 944 1.77681974535 945 1.77681974535 946 1.77681974535 947 1.77681974535 948 1.77681974535 949 1.77681974535 950 1.77681974535 951 1.77681974535 952 1.77681974535 953 1.77681974535 954 1.77681974535 955 1.77681974535 956 1.77681974535 957 1.77681974535 958 1.77681974535 959 1.77681974535 960 1.77681974535 961 1.77681974535 962 1.77681974535 963 1.77681974535 964 1.77681974535 965 1.77681974535 966 1.77681974535 967 1.77681974535 968 1.77681974535 969 1.77681974535 970 1.77681974535 971 1.77681974535 972 1.77681974535 973 1.77681974535 974 1.77681974535 975 1.77681974535 976 1.77681974535 977 1.77681974535 978 1.77681974535 979 1.77681974535 980 1.77681974535 981 1.77681974535 982 1.77681974535 983 1.77681974535 984 1.77681974535 985 1.77681974535 986 1.77681974535 987 1.77681974535 988 1.77681974535 989 1.77681974535 990 1.77681974535 991 1.77681974535 992 1.77681974535 993 1.77681974535 994 1.77681974535 995 1.77681974535 996 1.77681974535 997 1.77681974535 998 1.77681974535 999 1.77681974535 1000 1.77681974535 |
# C3B
# c = 0.8 , Initial Value: 50
print " n x(n)"
P = vector (RDF,1001)
P[0] = 50;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-0.8;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 50.0 1 24.26 2 11.4536603462 3 5.18875517433 4 2.37255094459 5 1.65073725628 6 1.84273840554 7 1.74938107411 8 1.78958276059 9 1.77115978642 10 1.77938528728 11 1.77566814397 12 1.77733895508 13 1.77658611911 14 1.77692496303 15 1.77677237778 16 1.77684107344 17 1.77681014277 18 1.77682406888 19 1.77681779871 20 1.7768206218 21 1.77681935073 22 1.77681992302 23 1.77681966535 24 1.77681978136 25 1.77681972913 26 1.77681975265 27 1.77681974206 28 1.77681974683 29 1.77681974468 30 1.77681974565 31 1.77681974521 32 1.77681974541 33 1.77681974532 34 1.77681974536 35 1.77681974534 36 1.77681974535 37 1.77681974534 38 1.77681974535 39 1.77681974534 40 1.77681974535 41 1.77681974534 42 1.77681974535 43 1.77681974535 44 1.77681974535 45 1.77681974535 46 1.77681974535 47 1.77681974535 48 1.77681974535 49 1.77681974535 50 1.77681974535 51 1.77681974535 52 1.77681974535 53 1.77681974535 54 1.77681974535 55 1.77681974535 56 1.77681974535 57 1.77681974535 ... 941 1.77681974535 942 1.77681974535 943 1.77681974535 944 1.77681974535 945 1.77681974535 946 1.77681974535 947 1.77681974535 948 1.77681974535 949 1.77681974535 950 1.77681974535 951 1.77681974535 952 1.77681974535 953 1.77681974535 954 1.77681974535 955 1.77681974535 956 1.77681974535 957 1.77681974535 958 1.77681974535 959 1.77681974535 960 1.77681974535 961 1.77681974535 962 1.77681974535 963 1.77681974535 964 1.77681974535 965 1.77681974535 966 1.77681974535 967 1.77681974535 968 1.77681974535 969 1.77681974535 970 1.77681974535 971 1.77681974535 972 1.77681974535 973 1.77681974535 974 1.77681974535 975 1.77681974535 976 1.77681974535 977 1.77681974535 978 1.77681974535 979 1.77681974535 980 1.77681974535 981 1.77681974535 982 1.77681974535 983 1.77681974535 984 1.77681974535 985 1.77681974535 986 1.77681974535 987 1.77681974535 988 1.77681974535 989 1.77681974535 990 1.77681974535 991 1.77681974535 992 1.77681974535 993 1.77681974535 994 1.77681974535 995 1.77681974535 996 1.77681974535 997 1.77681974535 998 1.77681974535 999 1.77681974535 1000 1.77681974535 WARNING: Output truncated! full_output.txt n x(n) 0 50.0 1 24.26 2 11.4536603462 3 5.18875517433 4 2.37255094459 5 1.65073725628 6 1.84273840554 7 1.74938107411 8 1.78958276059 9 1.77115978642 10 1.77938528728 11 1.77566814397 12 1.77733895508 13 1.77658611911 14 1.77692496303 15 1.77677237778 16 1.77684107344 17 1.77681014277 18 1.77682406888 19 1.77681779871 20 1.7768206218 21 1.77681935073 22 1.77681992302 23 1.77681966535 24 1.77681978136 25 1.77681972913 26 1.77681975265 27 1.77681974206 28 1.77681974683 29 1.77681974468 30 1.77681974565 31 1.77681974521 32 1.77681974541 33 1.77681974532 34 1.77681974536 35 1.77681974534 36 1.77681974535 37 1.77681974534 38 1.77681974535 39 1.77681974534 40 1.77681974535 41 1.77681974534 42 1.77681974535 43 1.77681974535 44 1.77681974535 45 1.77681974535 46 1.77681974535 47 1.77681974535 48 1.77681974535 49 1.77681974535 50 1.77681974535 51 1.77681974535 52 1.77681974535 53 1.77681974535 54 1.77681974535 55 1.77681974535 56 1.77681974535 57 1.77681974535 ... 941 1.77681974535 942 1.77681974535 943 1.77681974535 944 1.77681974535 945 1.77681974535 946 1.77681974535 947 1.77681974535 948 1.77681974535 949 1.77681974535 950 1.77681974535 951 1.77681974535 952 1.77681974535 953 1.77681974535 954 1.77681974535 955 1.77681974535 956 1.77681974535 957 1.77681974535 958 1.77681974535 959 1.77681974535 960 1.77681974535 961 1.77681974535 962 1.77681974535 963 1.77681974535 964 1.77681974535 965 1.77681974535 966 1.77681974535 967 1.77681974535 968 1.77681974535 969 1.77681974535 970 1.77681974535 971 1.77681974535 972 1.77681974535 973 1.77681974535 974 1.77681974535 975 1.77681974535 976 1.77681974535 977 1.77681974535 978 1.77681974535 979 1.77681974535 980 1.77681974535 981 1.77681974535 982 1.77681974535 983 1.77681974535 984 1.77681974535 985 1.77681974535 986 1.77681974535 987 1.77681974535 988 1.77681974535 989 1.77681974535 990 1.77681974535 991 1.77681974535 992 1.77681974535 993 1.77681974535 994 1.77681974535 995 1.77681974535 996 1.77681974535 997 1.77681974535 998 1.77681974535 999 1.77681974535 1000 1.77681974535 |
# C3C
# c = 0.8 , Initial Value: 45000
print " n x(n)"
P = vector (RDF,1001)
P[0] = 45000;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-0.8;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 45000.0 1 22499.2000667 2 11248.8001667 3 5623.60035003 4 2811.00070848 5 1404.70142148 6 701.552846423 7 349.98069944 8 174.198921621 9 86.3166825016 10 42.3930969888 11 20.4673147326 12 9.58023252963 13 4.30326106487 14 2.04877624467 15 1.68867686924 16 1.82087724713 17 1.75799614274 18 1.78548645725 19 1.772957597 20 1.77856664374 21 1.77603484763 22 1.77717346937 23 1.77666055061 24 1.77689143516 25 1.77678747028 26 1.77683427751 27 1.77681320246 28 1.77682269125 29 1.77681841898 30 1.77682034253 31 1.77681947647 32 1.77681986641 33 1.77681969084 34 1.77681976989 35 1.7768197343 36 1.77681975032 37 1.77681974311 38 1.77681974635 39 1.77681974489 40 1.77681974555 41 1.77681974525 42 1.77681974539 43 1.77681974533 44 1.77681974535 45 1.77681974534 46 1.77681974535 47 1.77681974534 48 1.77681974535 49 1.77681974534 50 1.77681974535 51 1.77681974534 52 1.77681974535 53 1.77681974535 54 1.77681974535 55 1.77681974535 56 1.77681974535 57 1.77681974535 ... 941 1.77681974535 942 1.77681974535 943 1.77681974535 944 1.77681974535 945 1.77681974535 946 1.77681974535 947 1.77681974535 948 1.77681974535 949 1.77681974535 950 1.77681974535 951 1.77681974535 952 1.77681974535 953 1.77681974535 954 1.77681974535 955 1.77681974535 956 1.77681974535 957 1.77681974535 958 1.77681974535 959 1.77681974535 960 1.77681974535 961 1.77681974535 962 1.77681974535 963 1.77681974535 964 1.77681974535 965 1.77681974535 966 1.77681974535 967 1.77681974535 968 1.77681974535 969 1.77681974535 970 1.77681974535 971 1.77681974535 972 1.77681974535 973 1.77681974535 974 1.77681974535 975 1.77681974535 976 1.77681974535 977 1.77681974535 978 1.77681974535 979 1.77681974535 980 1.77681974535 981 1.77681974535 982 1.77681974535 983 1.77681974535 984 1.77681974535 985 1.77681974535 986 1.77681974535 987 1.77681974535 988 1.77681974535 989 1.77681974535 990 1.77681974535 991 1.77681974535 992 1.77681974535 993 1.77681974535 994 1.77681974535 995 1.77681974535 996 1.77681974535 997 1.77681974535 998 1.77681974535 999 1.77681974535 1000 1.77681974535 WARNING: Output truncated! full_output.txt n x(n) 0 45000.0 1 22499.2000667 2 11248.8001667 3 5623.60035003 4 2811.00070848 5 1404.70142148 6 701.552846423 7 349.98069944 8 174.198921621 9 86.3166825016 10 42.3930969888 11 20.4673147326 12 9.58023252963 13 4.30326106487 14 2.04877624467 15 1.68867686924 16 1.82087724713 17 1.75799614274 18 1.78548645725 19 1.772957597 20 1.77856664374 21 1.77603484763 22 1.77717346937 23 1.77666055061 24 1.77689143516 25 1.77678747028 26 1.77683427751 27 1.77681320246 28 1.77682269125 29 1.77681841898 30 1.77682034253 31 1.77681947647 32 1.77681986641 33 1.77681969084 34 1.77681976989 35 1.7768197343 36 1.77681975032 37 1.77681974311 38 1.77681974635 39 1.77681974489 40 1.77681974555 41 1.77681974525 42 1.77681974539 43 1.77681974533 44 1.77681974535 45 1.77681974534 46 1.77681974535 47 1.77681974534 48 1.77681974535 49 1.77681974534 50 1.77681974535 51 1.77681974534 52 1.77681974535 53 1.77681974535 54 1.77681974535 55 1.77681974535 56 1.77681974535 57 1.77681974535 ... 941 1.77681974535 942 1.77681974535 943 1.77681974535 944 1.77681974535 945 1.77681974535 946 1.77681974535 947 1.77681974535 948 1.77681974535 949 1.77681974535 950 1.77681974535 951 1.77681974535 952 1.77681974535 953 1.77681974535 954 1.77681974535 955 1.77681974535 956 1.77681974535 957 1.77681974535 958 1.77681974535 959 1.77681974535 960 1.77681974535 961 1.77681974535 962 1.77681974535 963 1.77681974535 964 1.77681974535 965 1.77681974535 966 1.77681974535 967 1.77681974535 968 1.77681974535 969 1.77681974535 970 1.77681974535 971 1.77681974535 972 1.77681974535 973 1.77681974535 974 1.77681974535 975 1.77681974535 976 1.77681974535 977 1.77681974535 978 1.77681974535 979 1.77681974535 980 1.77681974535 981 1.77681974535 982 1.77681974535 983 1.77681974535 984 1.77681974535 985 1.77681974535 986 1.77681974535 987 1.77681974535 988 1.77681974535 989 1.77681974535 990 1.77681974535 991 1.77681974535 992 1.77681974535 993 1.77681974535 994 1.77681974535 995 1.77681974535 996 1.77681974535 997 1.77681974535 998 1.77681974535 999 1.77681974535 1000 1.77681974535 |
# In the above tables, it appears that the fixed point is 1.77681974535...
# With a value of c=0.8, our calculations predicted the fixed point to likewise be 1.77681974535...
# It appears that our calculations are correct.
|
|
# Testing accuracy of FIXED POINT CALCULATIONS
# C1A
# c = 1.6 , Initial Value: 4500
print " n x(n)"
P = vector (RDF,1001)
P[0] = 4500;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.6;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 4500.0 1 2248.40066667 2 1122.60166762 3 559.703506171 4 278.257113066 5 137.539337929 6 67.1914809059 7 32.0403889703 8 14.5138263072 9 5.86361260947 10 1.84343628438 11 0.949113688986 12 2.03540051877 13 0.89161164554 14 2.21049942565 15 0.862409079251 16 2.30983210997 17 0.853711749724 18 2.34092253843 19 0.85200730534 20 2.34710022215 21 0.851723054731 22 2.34813321338 23 0.85167725625 24 2.34829972238 25 0.851669920239 26 2.34832639572 27 0.851668746266 28 2.34833066427 29 0.851668558425 30 2.34833134726 31 0.85166852837 32 2.34833145654 33 0.851668523561 34 2.34833147402 35 0.851668522792 36 2.34833147682 37 0.851668522669 38 2.34833147727 39 0.851668522649 40 2.34833147734 41 0.851668522646 42 2.34833147735 43 0.851668522645 44 2.34833147735 45 0.851668522645 46 2.34833147735 47 0.851668522645 48 2.34833147735 49 0.851668522645 50 2.34833147735 51 0.851668522645 52 2.34833147735 53 0.851668522645 54 2.34833147735 55 0.851668522645 56 2.34833147735 57 0.851668522645 ... 941 0.851668522645 942 2.34833147735 943 0.851668522645 944 2.34833147735 945 0.851668522645 946 2.34833147735 947 0.851668522645 948 2.34833147735 949 0.851668522645 950 2.34833147735 951 0.851668522645 952 2.34833147735 953 0.851668522645 954 2.34833147735 955 0.851668522645 956 2.34833147735 957 0.851668522645 958 2.34833147735 959 0.851668522645 960 2.34833147735 961 0.851668522645 962 2.34833147735 963 0.851668522645 964 2.34833147735 965 0.851668522645 966 2.34833147735 967 0.851668522645 968 2.34833147735 969 0.851668522645 970 2.34833147735 971 0.851668522645 972 2.34833147735 973 0.851668522645 974 2.34833147735 975 0.851668522645 976 2.34833147735 977 0.851668522645 978 2.34833147735 979 0.851668522645 980 2.34833147735 981 0.851668522645 982 2.34833147735 983 0.851668522645 984 2.34833147735 985 0.851668522645 986 2.34833147735 987 0.851668522645 988 2.34833147735 989 0.851668522645 990 2.34833147735 991 0.851668522645 992 2.34833147735 993 0.851668522645 994 2.34833147735 995 0.851668522645 996 2.34833147735 997 0.851668522645 998 2.34833147735 999 0.851668522645 1000 2.34833147735 WARNING: Output truncated! full_output.txt n x(n) 0 4500.0 1 2248.40066667 2 1122.60166762 3 559.703506171 4 278.257113066 5 137.539337929 6 67.1914809059 7 32.0403889703 8 14.5138263072 9 5.86361260947 10 1.84343628438 11 0.949113688986 12 2.03540051877 13 0.89161164554 14 2.21049942565 15 0.862409079251 16 2.30983210997 17 0.853711749724 18 2.34092253843 19 0.85200730534 20 2.34710022215 21 0.851723054731 22 2.34813321338 23 0.85167725625 24 2.34829972238 25 0.851669920239 26 2.34832639572 27 0.851668746266 28 2.34833066427 29 0.851668558425 30 2.34833134726 31 0.85166852837 32 2.34833145654 33 0.851668523561 34 2.34833147402 35 0.851668522792 36 2.34833147682 37 0.851668522669 38 2.34833147727 39 0.851668522649 40 2.34833147734 41 0.851668522646 42 2.34833147735 43 0.851668522645 44 2.34833147735 45 0.851668522645 46 2.34833147735 47 0.851668522645 48 2.34833147735 49 0.851668522645 50 2.34833147735 51 0.851668522645 52 2.34833147735 53 0.851668522645 54 2.34833147735 55 0.851668522645 56 2.34833147735 57 0.851668522645 ... 941 0.851668522645 942 2.34833147735 943 0.851668522645 944 2.34833147735 945 0.851668522645 946 2.34833147735 947 0.851668522645 948 2.34833147735 949 0.851668522645 950 2.34833147735 951 0.851668522645 952 2.34833147735 953 0.851668522645 954 2.34833147735 955 0.851668522645 956 2.34833147735 957 0.851668522645 958 2.34833147735 959 0.851668522645 960 2.34833147735 961 0.851668522645 962 2.34833147735 963 0.851668522645 964 2.34833147735 965 0.851668522645 966 2.34833147735 967 0.851668522645 968 2.34833147735 969 0.851668522645 970 2.34833147735 971 0.851668522645 972 2.34833147735 973 0.851668522645 974 2.34833147735 975 0.851668522645 976 2.34833147735 977 0.851668522645 978 2.34833147735 979 0.851668522645 980 2.34833147735 981 0.851668522645 982 2.34833147735 983 0.851668522645 984 2.34833147735 985 0.851668522645 986 2.34833147735 987 0.851668522645 988 2.34833147735 989 0.851668522645 990 2.34833147735 991 0.851668522645 992 2.34833147735 993 0.851668522645 994 2.34833147735 995 0.851668522645 996 2.34833147735 997 0.851668522645 998 2.34833147735 999 0.851668522645 1000 2.34833147735 |
# C1B
# c = 1.6 , Initial Value: 2222
print " n x(n)"
P = vector (RDF,1001)
P[0] = 2222;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.6;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 2222.0 1 1109.40135014 2 553.103379229 3 274.957113555 4 135.88946757 5 66.3668105512 6 31.628608593 7 14.3091551335 8 5.76423355079 9 1.80256758183 10 0.965576453359 11 1.98974054471 12 0.902604538501 13 2.17501699927 14 0.866808064193 15 2.29437783233 16 0.85473292994 17 2.33723474653 18 0.852185489157 19 2.34645308651 20 0.851751998227 21 2.34802799435 22 0.851681898588 23 2.34828284336 24 0.851670663306 25 2.34832369394 26 0.851668865164 27 2.34833023196 28 0.851668577449 29 2.34833127809 30 0.851668531414 31 2.34833144547 32 0.851668524048 33 2.34833147225 34 0.85166852287 35 2.34833147654 36 0.851668522681 37 2.34833147722 38 0.851668522651 39 2.34833147733 40 0.851668522646 41 2.34833147735 42 0.851668522645 43 2.34833147735 44 0.851668522645 45 2.34833147735 46 0.851668522645 47 2.34833147735 48 0.851668522645 49 2.34833147735 50 0.851668522645 51 2.34833147735 52 0.851668522645 53 2.34833147735 54 0.851668522645 55 2.34833147735 56 0.851668522645 57 2.34833147735 ... 941 2.34833147735 942 0.851668522645 943 2.34833147735 944 0.851668522645 945 2.34833147735 946 0.851668522645 947 2.34833147735 948 0.851668522645 949 2.34833147735 950 0.851668522645 951 2.34833147735 952 0.851668522645 953 2.34833147735 954 0.851668522645 955 2.34833147735 956 0.851668522645 957 2.34833147735 958 0.851668522645 959 2.34833147735 960 0.851668522645 961 2.34833147735 962 0.851668522645 963 2.34833147735 964 0.851668522645 965 2.34833147735 966 0.851668522645 967 2.34833147735 968 0.851668522645 969 2.34833147735 970 0.851668522645 971 2.34833147735 972 0.851668522645 973 2.34833147735 974 0.851668522645 975 2.34833147735 976 0.851668522645 977 2.34833147735 978 0.851668522645 979 2.34833147735 980 0.851668522645 981 2.34833147735 982 0.851668522645 983 2.34833147735 984 0.851668522645 985 2.34833147735 986 0.851668522645 987 2.34833147735 988 0.851668522645 989 2.34833147735 990 0.851668522645 991 2.34833147735 992 0.851668522645 993 2.34833147735 994 0.851668522645 995 2.34833147735 996 0.851668522645 997 2.34833147735 998 0.851668522645 999 2.34833147735 1000 0.851668522645 WARNING: Output truncated! full_output.txt n x(n) 0 2222.0 1 1109.40135014 2 553.103379229 3 274.957113555 4 135.88946757 5 66.3668105512 6 31.628608593 7 14.3091551335 8 5.76423355079 9 1.80256758183 10 0.965576453359 11 1.98974054471 12 0.902604538501 13 2.17501699927 14 0.866808064193 15 2.29437783233 16 0.85473292994 17 2.33723474653 18 0.852185489157 19 2.34645308651 20 0.851751998227 21 2.34802799435 22 0.851681898588 23 2.34828284336 24 0.851670663306 25 2.34832369394 26 0.851668865164 27 2.34833023196 28 0.851668577449 29 2.34833127809 30 0.851668531414 31 2.34833144547 32 0.851668524048 33 2.34833147225 34 0.85166852287 35 2.34833147654 36 0.851668522681 37 2.34833147722 38 0.851668522651 39 2.34833147733 40 0.851668522646 41 2.34833147735 42 0.851668522645 43 2.34833147735 44 0.851668522645 45 2.34833147735 46 0.851668522645 47 2.34833147735 48 0.851668522645 49 2.34833147735 50 0.851668522645 51 2.34833147735 52 0.851668522645 53 2.34833147735 54 0.851668522645 55 2.34833147735 56 0.851668522645 57 2.34833147735 ... 941 2.34833147735 942 0.851668522645 943 2.34833147735 944 0.851668522645 945 2.34833147735 946 0.851668522645 947 2.34833147735 948 0.851668522645 949 2.34833147735 950 0.851668522645 951 2.34833147735 952 0.851668522645 953 2.34833147735 954 0.851668522645 955 2.34833147735 956 0.851668522645 957 2.34833147735 958 0.851668522645 959 2.34833147735 960 0.851668522645 961 2.34833147735 962 0.851668522645 963 2.34833147735 964 0.851668522645 965 2.34833147735 966 0.851668522645 967 2.34833147735 968 0.851668522645 969 2.34833147735 970 0.851668522645 971 2.34833147735 972 0.851668522645 973 2.34833147735 974 0.851668522645 975 2.34833147735 976 0.851668522645 977 2.34833147735 978 0.851668522645 979 2.34833147735 980 0.851668522645 981 2.34833147735 982 0.851668522645 983 2.34833147735 984 0.851668522645 985 2.34833147735 986 0.851668522645 987 2.34833147735 988 0.851668522645 989 2.34833147735 990 0.851668522645 991 2.34833147735 992 0.851668522645 993 2.34833147735 994 0.851668522645 995 2.34833147735 996 0.851668522645 997 2.34833147735 998 0.851668522645 999 2.34833147735 1000 0.851668522645 |
# C1C
# c = 1.6 , Initial Value: 3409
print " n x(n)"
P = vector (RDF,1001)
P[0] = 3409;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.6;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 3409.0 1 1702.90088002 2 849.852201711 3 423.329630881 4 210.071902117 5 103.450231883 6 50.1541153932 7 23.5368733264 8 10.2958962437 9 3.83932634954 10 1.10105025335 11 1.67519640382 12 1.02843299189 13 1.83127577318 14 0.953840086354 15 2.02210134025 16 0.89465583881 17 2.2005726755 18 0.863567829609 19 2.30574380208 20 0.853970486795 21 2.33998721055 22 0.852051894515 23 2.34693825238 24 0.851730280678 25 2.34810694394 26 0.851678414796 27 2.34829551002 28 0.851670105668 29 2.3483257215 30 0.851668775936 31 2.34833055639 32 0.851668563172 33 2.34833133 34 0.851668529129 35 2.34833145378 36 0.851668523683 37 2.34833147358 38 0.851668522811 39 2.34833147675 40 0.851668522672 41 2.34833147726 42 0.851668522649 43 2.34833147734 44 0.851668522646 45 2.34833147735 46 0.851668522645 47 2.34833147735 48 0.851668522645 49 2.34833147735 50 0.851668522645 51 2.34833147735 52 0.851668522645 53 2.34833147735 54 0.851668522645 55 2.34833147735 56 0.851668522645 57 2.34833147735 ... 941 2.34833147735 942 0.851668522645 943 2.34833147735 944 0.851668522645 945 2.34833147735 946 0.851668522645 947 2.34833147735 948 0.851668522645 949 2.34833147735 950 0.851668522645 951 2.34833147735 952 0.851668522645 953 2.34833147735 954 0.851668522645 955 2.34833147735 956 0.851668522645 957 2.34833147735 958 0.851668522645 959 2.34833147735 960 0.851668522645 961 2.34833147735 962 0.851668522645 963 2.34833147735 964 0.851668522645 965 2.34833147735 966 0.851668522645 967 2.34833147735 968 0.851668522645 969 2.34833147735 970 0.851668522645 971 2.34833147735 972 0.851668522645 973 2.34833147735 974 0.851668522645 975 2.34833147735 976 0.851668522645 977 2.34833147735 978 0.851668522645 979 2.34833147735 980 0.851668522645 981 2.34833147735 982 0.851668522645 983 2.34833147735 984 0.851668522645 985 2.34833147735 986 0.851668522645 987 2.34833147735 988 0.851668522645 989 2.34833147735 990 0.851668522645 991 2.34833147735 992 0.851668522645 993 2.34833147735 994 0.851668522645 995 2.34833147735 996 0.851668522645 997 2.34833147735 998 0.851668522645 999 2.34833147735 1000 0.851668522645 WARNING: Output truncated! full_output.txt n x(n) 0 3409.0 1 1702.90088002 2 849.852201711 3 423.329630881 4 210.071902117 5 103.450231883 6 50.1541153932 7 23.5368733264 8 10.2958962437 9 3.83932634954 10 1.10105025335 11 1.67519640382 12 1.02843299189 13 1.83127577318 14 0.953840086354 15 2.02210134025 16 0.89465583881 17 2.2005726755 18 0.863567829609 19 2.30574380208 20 0.853970486795 21 2.33998721055 22 0.852051894515 23 2.34693825238 24 0.851730280678 25 2.34810694394 26 0.851678414796 27 2.34829551002 28 0.851670105668 29 2.3483257215 30 0.851668775936 31 2.34833055639 32 0.851668563172 33 2.34833133 34 0.851668529129 35 2.34833145378 36 0.851668523683 37 2.34833147358 38 0.851668522811 39 2.34833147675 40 0.851668522672 41 2.34833147726 42 0.851668522649 43 2.34833147734 44 0.851668522646 45 2.34833147735 46 0.851668522645 47 2.34833147735 48 0.851668522645 49 2.34833147735 50 0.851668522645 51 2.34833147735 52 0.851668522645 53 2.34833147735 54 0.851668522645 55 2.34833147735 56 0.851668522645 57 2.34833147735 ... 941 2.34833147735 942 0.851668522645 943 2.34833147735 944 0.851668522645 945 2.34833147735 946 0.851668522645 947 2.34833147735 948 0.851668522645 949 2.34833147735 950 0.851668522645 951 2.34833147735 952 0.851668522645 953 2.34833147735 954 0.851668522645 955 2.34833147735 956 0.851668522645 957 2.34833147735 958 0.851668522645 959 2.34833147735 960 0.851668522645 961 2.34833147735 962 0.851668522645 963 2.34833147735 964 0.851668522645 965 2.34833147735 966 0.851668522645 967 2.34833147735 968 0.851668522645 969 2.34833147735 970 0.851668522645 971 2.34833147735 972 0.851668522645 973 2.34833147735 974 0.851668522645 975 2.34833147735 976 0.851668522645 977 2.34833147735 978 0.851668522645 979 2.34833147735 980 0.851668522645 981 2.34833147735 982 0.851668522645 983 2.34833147735 984 0.851668522645 985 2.34833147735 986 0.851668522645 987 2.34833147735 988 0.851668522645 989 2.34833147735 990 0.851668522645 991 2.34833147735 992 0.851668522645 993 2.34833147735 994 0.851668522645 995 2.34833147735 996 0.851668522645 997 2.34833147735 998 0.851668522645 999 2.34833147735 1000 0.851668522645 |
# In the above tables, it appears that the 2-cycle periodic points are 2.34833147735... and 0.851668522645...
# With a value of c=1.6, our calculations predicted the periodic points to likewise be 2.34833147735... and 0.851668522645...
# It appears that our calculations are correct.
|
|
# C2A
# c = 1.5 , Initial Value: 409
print " n x(n)"
P = vector (RDF,1001)
P[0] = 409;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.5;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 409.0 1 203.007334963 2 100.018445273 3 48.5392171038 4 22.8314142459 5 10.047105028 6 3.82214598859 7 1.19597236992 8 1.60640533866 9 1.17072634334 10 1.64787491241 11 1.14446398853 12 1.69354645244 13 1.11820382128 14 1.74197594748 15 1.0931701912 16 1.79089702812 17 1.07058670062 18 1.83749516943 19 1.051404938 20 1.87902747402 21 1.03608432551 22 1.91355937708 23 1.02453872227 24 1.94041637479 25 1.01626811503 26 1.96011095773 27 1.01058107905 28 1.97387966335 29 1.00678932185 30 1.98316404753 31 1.00431618398 32 1.98926518821 33 1.00272714971 34 1.99320437707 35 1.00171628246 36 1.99571811559 37 1.00107736131 38 1.99731007511 39 1.00067519826 40 1.9983133711 41 1.00042272489 42 1.99894372363 43 1.00026448771 44 1.99933899053 45 1.00016541627 46 1.9995865414 47 1.00010342877 48 1.99974146016 49 1.00006466003 50 1.99983836247 51 1.00004041918 52 1.99989895695 53 1.00002526459 54 1.99993684044 55 1.00001579139 56 1.99996052228 57 1.00000987001 ... 941 1.0 942 2.0 943 1.0 944 2.0 945 1.0 946 2.0 947 1.0 948 2.0 949 1.0 950 2.0 951 1.0 952 2.0 953 1.0 954 2.0 955 1.0 956 2.0 957 1.0 958 2.0 959 1.0 960 2.0 961 1.0 962 2.0 963 1.0 964 2.0 965 1.0 966 2.0 967 1.0 968 2.0 969 1.0 970 2.0 971 1.0 972 2.0 973 1.0 974 2.0 975 1.0 976 2.0 977 1.0 978 2.0 979 1.0 980 2.0 981 1.0 982 2.0 983 1.0 984 2.0 985 1.0 986 2.0 987 1.0 988 2.0 989 1.0 990 2.0 991 1.0 992 2.0 993 1.0 994 2.0 995 1.0 996 2.0 997 1.0 998 2.0 999 1.0 1000 2.0 WARNING: Output truncated! full_output.txt n x(n) 0 409.0 1 203.007334963 2 100.018445273 3 48.5392171038 4 22.8314142459 5 10.047105028 6 3.82214598859 7 1.19597236992 8 1.60640533866 9 1.17072634334 10 1.64787491241 11 1.14446398853 12 1.69354645244 13 1.11820382128 14 1.74197594748 15 1.0931701912 16 1.79089702812 17 1.07058670062 18 1.83749516943 19 1.051404938 20 1.87902747402 21 1.03608432551 22 1.91355937708 23 1.02453872227 24 1.94041637479 25 1.01626811503 26 1.96011095773 27 1.01058107905 28 1.97387966335 29 1.00678932185 30 1.98316404753 31 1.00431618398 32 1.98926518821 33 1.00272714971 34 1.99320437707 35 1.00171628246 36 1.99571811559 37 1.00107736131 38 1.99731007511 39 1.00067519826 40 1.9983133711 41 1.00042272489 42 1.99894372363 43 1.00026448771 44 1.99933899053 45 1.00016541627 46 1.9995865414 47 1.00010342877 48 1.99974146016 49 1.00006466003 50 1.99983836247 51 1.00004041918 52 1.99989895695 53 1.00002526459 54 1.99993684044 55 1.00001579139 56 1.99996052228 57 1.00000987001 ... 941 1.0 942 2.0 943 1.0 944 2.0 945 1.0 946 2.0 947 1.0 948 2.0 949 1.0 950 2.0 951 1.0 952 2.0 953 1.0 954 2.0 955 1.0 956 2.0 957 1.0 958 2.0 959 1.0 960 2.0 961 1.0 962 2.0 963 1.0 964 2.0 965 1.0 966 2.0 967 1.0 968 2.0 969 1.0 970 2.0 971 1.0 972 2.0 973 1.0 974 2.0 975 1.0 976 2.0 977 1.0 978 2.0 979 1.0 980 2.0 981 1.0 982 2.0 983 1.0 984 2.0 985 1.0 986 2.0 987 1.0 988 2.0 989 1.0 990 2.0 991 1.0 992 2.0 993 1.0 994 2.0 995 1.0 996 2.0 997 1.0 998 2.0 999 1.0 1000 2.0 |
# C2B
# c = 1.5 , Initial Value: 5100
print " n x(n)"
P = vector (RDF,1001)
P[0] = 5100;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.5;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 5100.0 1 2548.50058824 2 1272.75147128 3 634.878092738 4 315.943771686 5 156.481381203 6 76.7598622116 7 36.9190140316 8 17.0407659573 9 7.19643140319 10 2.51508898669 11 0.95034523156 12 2.13192018537 13 0.973142228577 14 2.06936817304 15 0.984401946824 16 2.03973659611 17 0.990646439567 18 2.02364884655 19 0.99429506294 20 2.01436054179 21 0.996486647664 22 2.00882054234 23 0.997823912046 24 2.00545445694 25 0.998647512083 26 2.0033867149 27 0.999157615194 28 2.00210809265 29 0.999474641604 30 2.00131422443 31 0.999672091162 32 2.00082009477 33 0.999795228412 34 2.00051205479 35 0.999872084602 36 2.00031983759 37 0.999920078958 38 2.00019982177 39 0.99995005953 40 2.00012485866 41 0.999968791181 42 2.00007802497 43 0.999980496041 44 2.00004876104 45 0.999987810632 46 2.00003047387 47 0.999992381882 48 2.00001904547 49 0.999995238768 50 2.00001190315 51 0.999997024266 52 2.00000743936 53 0.999998140181 54 2.00000464956 55 0.999998837618 56 2.00000290596 57 0.999999273514 ... 941 1.0 942 2.0 943 1.0 944 2.0 945 1.0 946 2.0 947 1.0 948 2.0 949 1.0 950 2.0 951 1.0 952 2.0 953 1.0 954 2.0 955 1.0 956 2.0 957 1.0 958 2.0 959 1.0 960 2.0 961 1.0 962 2.0 963 1.0 964 2.0 965 1.0 966 2.0 967 1.0 968 2.0 969 1.0 970 2.0 971 1.0 972 2.0 973 1.0 974 2.0 975 1.0 976 2.0 977 1.0 978 2.0 979 1.0 980 2.0 981 1.0 982 2.0 983 1.0 984 2.0 985 1.0 986 2.0 987 1.0 988 2.0 989 1.0 990 2.0 991 1.0 992 2.0 993 1.0 994 2.0 995 1.0 996 2.0 997 1.0 998 2.0 999 1.0 1000 2.0 WARNING: Output truncated! full_output.txt n x(n) 0 5100.0 1 2548.50058824 2 1272.75147128 3 634.878092738 4 315.943771686 5 156.481381203 6 76.7598622116 7 36.9190140316 8 17.0407659573 9 7.19643140319 10 2.51508898669 11 0.95034523156 12 2.13192018537 13 0.973142228577 14 2.06936817304 15 0.984401946824 16 2.03973659611 17 0.990646439567 18 2.02364884655 19 0.99429506294 20 2.01436054179 21 0.996486647664 22 2.00882054234 23 0.997823912046 24 2.00545445694 25 0.998647512083 26 2.0033867149 27 0.999157615194 28 2.00210809265 29 0.999474641604 30 2.00131422443 31 0.999672091162 32 2.00082009477 33 0.999795228412 34 2.00051205479 35 0.999872084602 36 2.00031983759 37 0.999920078958 38 2.00019982177 39 0.99995005953 40 2.00012485866 41 0.999968791181 42 2.00007802497 43 0.999980496041 44 2.00004876104 45 0.999987810632 46 2.00003047387 47 0.999992381882 48 2.00001904547 49 0.999995238768 50 2.00001190315 51 0.999997024266 52 2.00000743936 53 0.999998140181 54 2.00000464956 55 0.999998837618 56 2.00000290596 57 0.999999273514 ... 941 1.0 942 2.0 943 1.0 944 2.0 945 1.0 946 2.0 947 1.0 948 2.0 949 1.0 950 2.0 951 1.0 952 2.0 953 1.0 954 2.0 955 1.0 956 2.0 957 1.0 958 2.0 959 1.0 960 2.0 961 1.0 962 2.0 963 1.0 964 2.0 965 1.0 966 2.0 967 1.0 968 2.0 969 1.0 970 2.0 971 1.0 972 2.0 973 1.0 974 2.0 975 1.0 976 2.0 977 1.0 978 2.0 979 1.0 980 2.0 981 1.0 982 2.0 983 1.0 984 2.0 985 1.0 986 2.0 987 1.0 988 2.0 989 1.0 990 2.0 991 1.0 992 2.0 993 1.0 994 2.0 995 1.0 996 2.0 997 1.0 998 2.0 999 1.0 1000 2.0 |
# C2C
# c = 1.5 , Initial Value: 69
print " n x(n)"
P = vector (RDF,1001)
P[0] = 69;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.5;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 69.0 1 33.0434782609 2 15.1125286041 3 6.25477509258 4 2.10702109915 5 0.977321635857 6 2.05827463792 7 0.986668760456 8 2.0338684685 9 0.991955872307 10 2.02030601742 11 0.995076566889 12 2.01238166316 13 0.996961720084 14 2.00762347762 15 0.998115841866 16 2.0047210656 17 0.998828072091 18 2.00293394485 19 0.99926973707 20 2.00182725834 21 0.999544436348 22 2.00113953203 23 0.999715603666 24 2.00071123355 25 0.99982238124 26 2.00044414156 27 0.999889038566 28 2.00027744053 29 0.99993066873 30 2.0001733426 31 0.999956675618 32 2.00010831659 33 0.999972925253 34 2.00006768907 35 0.999983079451 36 2.00004230223 37 0.999989425113 38 2.00002643755 39 0.999993390874 40 2.00001652295 41 0.999995869366 42 2.00001032664 43 0.999997418381 44 2.00000645407 45 0.999998386499 46 2.00000403376 47 0.999998991566 48 2.00000252109 49 0.99999936973 50 2.00000157568 51 0.999999606082 52 2.0000009848 53 0.999999753802 54 2.0000006155 55 0.999999846126 56 2.00000038468 57 0.999999903829 ... 941 1.0 942 2.0 943 1.0 944 2.0 945 1.0 946 2.0 947 1.0 948 2.0 949 1.0 950 2.0 951 1.0 952 2.0 953 1.0 954 2.0 955 1.0 956 2.0 957 1.0 958 2.0 959 1.0 960 2.0 961 1.0 962 2.0 963 1.0 964 2.0 965 1.0 966 2.0 967 1.0 968 2.0 969 1.0 970 2.0 971 1.0 972 2.0 973 1.0 974 2.0 975 1.0 976 2.0 977 1.0 978 2.0 979 1.0 980 2.0 981 1.0 982 2.0 983 1.0 984 2.0 985 1.0 986 2.0 987 1.0 988 2.0 989 1.0 990 2.0 991 1.0 992 2.0 993 1.0 994 2.0 995 1.0 996 2.0 997 1.0 998 2.0 999 1.0 1000 2.0 WARNING: Output truncated! full_output.txt n x(n) 0 69.0 1 33.0434782609 2 15.1125286041 3 6.25477509258 4 2.10702109915 5 0.977321635857 6 2.05827463792 7 0.986668760456 8 2.0338684685 9 0.991955872307 10 2.02030601742 11 0.995076566889 12 2.01238166316 13 0.996961720084 14 2.00762347762 15 0.998115841866 16 2.0047210656 17 0.998828072091 18 2.00293394485 19 0.99926973707 20 2.00182725834 21 0.999544436348 22 2.00113953203 23 0.999715603666 24 2.00071123355 25 0.99982238124 26 2.00044414156 27 0.999889038566 28 2.00027744053 29 0.99993066873 30 2.0001733426 31 0.999956675618 32 2.00010831659 33 0.999972925253 34 2.00006768907 35 0.999983079451 36 2.00004230223 37 0.999989425113 38 2.00002643755 39 0.999993390874 40 2.00001652295 41 0.999995869366 42 2.00001032664 43 0.999997418381 44 2.00000645407 45 0.999998386499 46 2.00000403376 47 0.999998991566 48 2.00000252109 49 0.99999936973 50 2.00000157568 51 0.999999606082 52 2.0000009848 53 0.999999753802 54 2.0000006155 55 0.999999846126 56 2.00000038468 57 0.999999903829 ... 941 1.0 942 2.0 943 1.0 944 2.0 945 1.0 946 2.0 947 1.0 948 2.0 949 1.0 950 2.0 951 1.0 952 2.0 953 1.0 954 2.0 955 1.0 956 2.0 957 1.0 958 2.0 959 1.0 960 2.0 961 1.0 962 2.0 963 1.0 964 2.0 965 1.0 966 2.0 967 1.0 968 2.0 969 1.0 970 2.0 971 1.0 972 2.0 973 1.0 974 2.0 975 1.0 976 2.0 977 1.0 978 2.0 979 1.0 980 2.0 981 1.0 982 2.0 983 1.0 984 2.0 985 1.0 986 2.0 987 1.0 988 2.0 989 1.0 990 2.0 991 1.0 992 2.0 993 1.0 994 2.0 995 1.0 996 2.0 997 1.0 998 2.0 999 1.0 1000 2.0 |
# In the above tables, it appears that the 2-cycle periodic points are 1 and 2.
# With a value of c=1.5, our calculations predicted the periodic points to likewise be 1 and 2.
# It appears that our calculations are correct.
|
|
# C3A
# c = 1.8 , Initial Value: 6879
print " n x(n)"
P = vector (RDF,1001)
P[0] = 6879;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.8;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 6879.0 1 3437.70043611 2 1717.05109073 3 856.727292547 4 426.567147971 5 211.490606876 6 103.959488465 7 50.2086016274 8 23.3640515317 9 10.0104281534 10 3.50490155802 11 0.808394933314 12 2.3152548674 13 0.653381107406 14 3.1181915414 15 0.721191896025 16 2.72037646764 17 0.662976776394 18 3.0565337696 19 0.709770845214 20 2.78160172158 21 0.669316155314 22 3.01684482331 23 0.702838822078 24 2.81982335482 25 0.67380810726 26 2.9892102335 27 0.698214687726 28 2.84578006178 29 0.67708253167 30 2.96931574262 31 0.694991651897 32 2.86409443805 33 0.679498713691 34 2.95476875211 35 0.692692256869 36 2.87727370883 37 0.681290527159 38 2.94405303223 39 0.69102989926 40 2.88686111021 41 0.682621526015 42 2.93613261621 43 0.689818521694 44 2.89387917925 45 0.68361037447 46 2.93026989175 47 0.688931425662 48 2.89903554119 49 0.684344683676 50 2.92592817652 51 0.688279755294 52 2.90283265762 53 0.684889612957 54 2.92271272593 55 0.687800040916 56 2.90563282292 57 0.685293734932 ... 941 0.686447127434 942 2.91355287257 943 0.686447127434 944 2.91355287257 945 0.686447127434 946 2.91355287257 947 0.686447127434 948 2.91355287257 949 0.686447127434 950 2.91355287257 951 0.686447127434 952 2.91355287257 953 0.686447127434 954 2.91355287257 955 0.686447127434 956 2.91355287257 957 0.686447127434 958 2.91355287257 959 0.686447127434 960 2.91355287257 961 0.686447127434 962 2.91355287257 963 0.686447127434 964 2.91355287257 965 0.686447127434 966 2.91355287257 967 0.686447127434 968 2.91355287257 969 0.686447127434 970 2.91355287257 971 0.686447127434 972 2.91355287257 973 0.686447127434 974 2.91355287257 975 0.686447127434 976 2.91355287257 977 0.686447127434 978 2.91355287257 979 0.686447127434 980 2.91355287257 981 0.686447127434 982 2.91355287257 983 0.686447127434 984 2.91355287257 985 0.686447127434 986 2.91355287257 987 0.686447127434 988 2.91355287257 989 0.686447127434 990 2.91355287257 991 0.686447127434 992 2.91355287257 993 0.686447127434 994 2.91355287257 995 0.686447127434 996 2.91355287257 997 0.686447127434 998 2.91355287257 999 0.686447127434 1000 2.91355287257 WARNING: Output truncated! full_output.txt n x(n) 0 6879.0 1 3437.70043611 2 1717.05109073 3 856.727292547 4 426.567147971 5 211.490606876 6 103.959488465 7 50.2086016274 8 23.3640515317 9 10.0104281534 10 3.50490155802 11 0.808394933314 12 2.3152548674 13 0.653381107406 14 3.1181915414 15 0.721191896025 16 2.72037646764 17 0.662976776394 18 3.0565337696 19 0.709770845214 20 2.78160172158 21 0.669316155314 22 3.01684482331 23 0.702838822078 24 2.81982335482 25 0.67380810726 26 2.9892102335 27 0.698214687726 28 2.84578006178 29 0.67708253167 30 2.96931574262 31 0.694991651897 32 2.86409443805 33 0.679498713691 34 2.95476875211 35 0.692692256869 36 2.87727370883 37 0.681290527159 38 2.94405303223 39 0.69102989926 40 2.88686111021 41 0.682621526015 42 2.93613261621 43 0.689818521694 44 2.89387917925 45 0.68361037447 46 2.93026989175 47 0.688931425662 48 2.89903554119 49 0.684344683676 50 2.92592817652 51 0.688279755294 52 2.90283265762 53 0.684889612957 54 2.92271272593 55 0.687800040916 56 2.90563282292 57 0.685293734932 ... 941 0.686447127434 942 2.91355287257 943 0.686447127434 944 2.91355287257 945 0.686447127434 946 2.91355287257 947 0.686447127434 948 2.91355287257 949 0.686447127434 950 2.91355287257 951 0.686447127434 952 2.91355287257 953 0.686447127434 954 2.91355287257 955 0.686447127434 956 2.91355287257 957 0.686447127434 958 2.91355287257 959 0.686447127434 960 2.91355287257 961 0.686447127434 962 2.91355287257 963 0.686447127434 964 2.91355287257 965 0.686447127434 966 2.91355287257 967 0.686447127434 968 2.91355287257 969 0.686447127434 970 2.91355287257 971 0.686447127434 972 2.91355287257 973 0.686447127434 974 2.91355287257 975 0.686447127434 976 2.91355287257 977 0.686447127434 978 2.91355287257 979 0.686447127434 980 2.91355287257 981 0.686447127434 982 2.91355287257 983 0.686447127434 984 2.91355287257 985 0.686447127434 986 2.91355287257 987 0.686447127434 988 2.91355287257 989 0.686447127434 990 2.91355287257 991 0.686447127434 992 2.91355287257 993 0.686447127434 994 2.91355287257 995 0.686447127434 996 2.91355287257 997 0.686447127434 998 2.91355287257 999 0.686447127434 1000 2.91355287257 |
# C3B
# c = 1.8 , Initial Value: 111
print " n x(n)"
P = vector (RDF,1001)
P[0] = 111;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.8;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 111.0 1 53.727027027 2 25.1193513323 3 10.8791055018 4 3.91531071641 5 0.923878071368 6 1.90912077208 7 0.725964376754 8 2.69541636292 9 0.660708770634 10 3.07093282674 11 0.712368276503 12 2.76748901938 13 0.667759650849 14 3.02651425786 15 0.70449647043 16 2.81060881744 17 0.672688806501 18 2.99605886707 19 0.699344872618 20 2.83940146338 21 0.676261432489 22 2.97428492507 23 0.695788263316 24 2.85955064869 25 0.678891206022 26 2.95841579056 27 0.693264137669 28 2.87398702201 29 0.680839560769 30 2.94674422577 31 0.691444863741 32 2.88446317219 33 0.682286466645 34 2.93812330938 35 0.69012159129 36 2.89212084689 37 0.683361476486 38 2.93174383549 39 0.689153680525 40 2.89774230216 41 0.684159899069 42 2.92701979193 43 0.688443177343 44 2.9018797059 45 0.684752520615 46 2.92352114066 47 0.68792041514 48 2.90492978192 49 0.685192091002 50 2.9209302957 51 0.687535189342 52 2.9071806192 53 0.685517937415 54 2.91901207058 55 0.687251014571 56 2.90884278073 57 0.685759357434 ... 941 0.686447127434 942 2.91355287257 943 0.686447127434 944 2.91355287257 945 0.686447127434 946 2.91355287257 947 0.686447127434 948 2.91355287257 949 0.686447127434 950 2.91355287257 951 0.686447127434 952 2.91355287257 953 0.686447127434 954 2.91355287257 955 0.686447127434 956 2.91355287257 957 0.686447127434 958 2.91355287257 959 0.686447127434 960 2.91355287257 961 0.686447127434 962 2.91355287257 963 0.686447127434 964 2.91355287257 965 0.686447127434 966 2.91355287257 967 0.686447127434 968 2.91355287257 969 0.686447127434 970 2.91355287257 971 0.686447127434 972 2.91355287257 973 0.686447127434 974 2.91355287257 975 0.686447127434 976 2.91355287257 977 0.686447127434 978 2.91355287257 979 0.686447127434 980 2.91355287257 981 0.686447127434 982 2.91355287257 983 0.686447127434 984 2.91355287257 985 0.686447127434 986 2.91355287257 987 0.686447127434 988 2.91355287257 989 0.686447127434 990 2.91355287257 991 0.686447127434 992 2.91355287257 993 0.686447127434 994 2.91355287257 995 0.686447127434 996 2.91355287257 997 0.686447127434 998 2.91355287257 999 0.686447127434 1000 2.91355287257 WARNING: Output truncated! full_output.txt n x(n) 0 111.0 1 53.727027027 2 25.1193513323 3 10.8791055018 4 3.91531071641 5 0.923878071368 6 1.90912077208 7 0.725964376754 8 2.69541636292 9 0.660708770634 10 3.07093282674 11 0.712368276503 12 2.76748901938 13 0.667759650849 14 3.02651425786 15 0.70449647043 16 2.81060881744 17 0.672688806501 18 2.99605886707 19 0.699344872618 20 2.83940146338 21 0.676261432489 22 2.97428492507 23 0.695788263316 24 2.85955064869 25 0.678891206022 26 2.95841579056 27 0.693264137669 28 2.87398702201 29 0.680839560769 30 2.94674422577 31 0.691444863741 32 2.88446317219 33 0.682286466645 34 2.93812330938 35 0.69012159129 36 2.89212084689 37 0.683361476486 38 2.93174383549 39 0.689153680525 40 2.89774230216 41 0.684159899069 42 2.92701979193 43 0.688443177343 44 2.9018797059 45 0.684752520615 46 2.92352114066 47 0.68792041514 48 2.90492978192 49 0.685192091002 50 2.9209302957 51 0.687535189342 52 2.9071806192 53 0.685517937415 54 2.91901207058 55 0.687251014571 56 2.90884278073 57 0.685759357434 ... 941 0.686447127434 942 2.91355287257 943 0.686447127434 944 2.91355287257 945 0.686447127434 946 2.91355287257 947 0.686447127434 948 2.91355287257 949 0.686447127434 950 2.91355287257 951 0.686447127434 952 2.91355287257 953 0.686447127434 954 2.91355287257 955 0.686447127434 956 2.91355287257 957 0.686447127434 958 2.91355287257 959 0.686447127434 960 2.91355287257 961 0.686447127434 962 2.91355287257 963 0.686447127434 964 2.91355287257 965 0.686447127434 966 2.91355287257 967 0.686447127434 968 2.91355287257 969 0.686447127434 970 2.91355287257 971 0.686447127434 972 2.91355287257 973 0.686447127434 974 2.91355287257 975 0.686447127434 976 2.91355287257 977 0.686447127434 978 2.91355287257 979 0.686447127434 980 2.91355287257 981 0.686447127434 982 2.91355287257 983 0.686447127434 984 2.91355287257 985 0.686447127434 986 2.91355287257 987 0.686447127434 988 2.91355287257 989 0.686447127434 990 2.91355287257 991 0.686447127434 992 2.91355287257 993 0.686447127434 994 2.91355287257 995 0.686447127434 996 2.91355287257 997 0.686447127434 998 2.91355287257 999 0.686447127434 1000 2.91355287257 |
# C3C
# c = 1.8 , Initial Value: 4590
print " n x(n)"
P = vector (RDF,1001)
P[0] = 4590;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.8;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 4590.0 1 2293.20065359 2 1144.80163501 3 570.603438048 4 283.506976616 5 139.964070058 6 68.2034691013 7 32.345720583 8 14.4656082639 9 5.6401925767 10 1.55199302046 11 0.908994893893 12 1.95484601893 13 0.712070736677 14 2.76909994954 15 0.667934487668 16 3.0254256958 17 0.704308841868 18 2.81164943458 19 0.672814066354 20 2.99529121831 21 0.699217670549 22 2.84011825224 23 0.676353172198 24 2.97372908044 25 0.695698875443 26 2.86005994415 27 0.678959036005 28 2.95800823757 29 0.693200077369 30 2.87435489322 31 0.680889900865 32 2.94644362392 33 0.691398428558 34 2.88473134986 35 0.682323867274 36 2.93790099538 37 0.690087698949 38 2.89231739828 39 0.683389260968 40 2.93157924142 41 0.689128835838 42 2.89788682149 43 0.684180528273 44 2.92689789316 45 0.688424914143 46 2.90198617846 47 0.684767826776 48 2.92343086492 49 0.687906965154 50 2.90500832279 51 0.685203440245 52 2.92086345058 53 0.687525271687 54 2.90723860326 55 0.685526347937 56 2.91896258487 57 0.687243695266 ... 941 0.686447127434 942 2.91355287257 943 0.686447127434 944 2.91355287257 945 0.686447127434 946 2.91355287257 947 0.686447127434 948 2.91355287257 949 0.686447127434 950 2.91355287257 951 0.686447127434 952 2.91355287257 953 0.686447127434 954 2.91355287257 955 0.686447127434 956 2.91355287257 957 0.686447127434 958 2.91355287257 959 0.686447127434 960 2.91355287257 961 0.686447127434 962 2.91355287257 963 0.686447127434 964 2.91355287257 965 0.686447127434 966 2.91355287257 967 0.686447127434 968 2.91355287257 969 0.686447127434 970 2.91355287257 971 0.686447127434 972 2.91355287257 973 0.686447127434 974 2.91355287257 975 0.686447127434 976 2.91355287257 977 0.686447127434 978 2.91355287257 979 0.686447127434 980 2.91355287257 981 0.686447127434 982 2.91355287257 983 0.686447127434 984 2.91355287257 985 0.686447127434 986 2.91355287257 987 0.686447127434 988 2.91355287257 989 0.686447127434 990 2.91355287257 991 0.686447127434 992 2.91355287257 993 0.686447127434 994 2.91355287257 995 0.686447127434 996 2.91355287257 997 0.686447127434 998 2.91355287257 999 0.686447127434 1000 2.91355287257 WARNING: Output truncated! full_output.txt n x(n) 0 4590.0 1 2293.20065359 2 1144.80163501 3 570.603438048 4 283.506976616 5 139.964070058 6 68.2034691013 7 32.345720583 8 14.4656082639 9 5.6401925767 10 1.55199302046 11 0.908994893893 12 1.95484601893 13 0.712070736677 14 2.76909994954 15 0.667934487668 16 3.0254256958 17 0.704308841868 18 2.81164943458 19 0.672814066354 20 2.99529121831 21 0.699217670549 22 2.84011825224 23 0.676353172198 24 2.97372908044 25 0.695698875443 26 2.86005994415 27 0.678959036005 28 2.95800823757 29 0.693200077369 30 2.87435489322 31 0.680889900865 32 2.94644362392 33 0.691398428558 34 2.88473134986 35 0.682323867274 36 2.93790099538 37 0.690087698949 38 2.89231739828 39 0.683389260968 40 2.93157924142 41 0.689128835838 42 2.89788682149 43 0.684180528273 44 2.92689789316 45 0.688424914143 46 2.90198617846 47 0.684767826776 48 2.92343086492 49 0.687906965154 50 2.90500832279 51 0.685203440245 52 2.92086345058 53 0.687525271687 54 2.90723860326 55 0.685526347937 56 2.91896258487 57 0.687243695266 ... 941 0.686447127434 942 2.91355287257 943 0.686447127434 944 2.91355287257 945 0.686447127434 946 2.91355287257 947 0.686447127434 948 2.91355287257 949 0.686447127434 950 2.91355287257 951 0.686447127434 952 2.91355287257 953 0.686447127434 954 2.91355287257 955 0.686447127434 956 2.91355287257 957 0.686447127434 958 2.91355287257 959 0.686447127434 960 2.91355287257 961 0.686447127434 962 2.91355287257 963 0.686447127434 964 2.91355287257 965 0.686447127434 966 2.91355287257 967 0.686447127434 968 2.91355287257 969 0.686447127434 970 2.91355287257 971 0.686447127434 972 2.91355287257 973 0.686447127434 974 2.91355287257 975 0.686447127434 976 2.91355287257 977 0.686447127434 978 2.91355287257 979 0.686447127434 980 2.91355287257 981 0.686447127434 982 2.91355287257 983 0.686447127434 984 2.91355287257 985 0.686447127434 986 2.91355287257 987 0.686447127434 988 2.91355287257 989 0.686447127434 990 2.91355287257 991 0.686447127434 992 2.91355287257 993 0.686447127434 994 2.91355287257 995 0.686447127434 996 2.91355287257 997 0.686447127434 998 2.91355287257 999 0.686447127434 1000 2.91355287257 |
# In the above tables, it appears that the 2-cycle periodic points are 2.91355287257... and 0.686447127434...
# With a value of c=1.8, our calculations predicted the periodic points to likewise be 91355287257... and 0.686447127434...
# It appears that our calculations are correct.
|
|
# Problem 1 : G
|
|
# i : Finding a stable 4-cycle
# c = 1.89 , Initial Value: 100
print " n x(n)"
P = vector (RDF,1001)
P[0] = 100;
for n in range(1,1001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.89;
for n in range(0,1001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.14 2 22.2423182385 3 9.36603714624 4 3.11332479796 5 0.630262471149 6 3.18505291476 7 0.644426036536 8 3.08751840618 9 0.625413329587 10 3.21953439147 11 0.651578953345 12 3.03998975754 13 0.616840308943 14 3.28191554624 15 0.665058077571 16 2.95341327759 17 0.60248049705 18 3.39065454441 19 0.690112189307 20 2.80217536958 21 0.581684490603 22 3.5582773299 23 0.732243269188 24 2.57312071432 25 0.562459797207 26 3.72494444122 27 0.777853360473 28 2.35569474533 29 0.561357026642 30 3.73487099266 31 0.780676091759 32 2.34316099341 33 0.561902253698 34 3.72995800973 35 0.779277603448 36 2.34935806316 37 0.561623592327 38 3.73246773973 39 0.779991654041 40 2.34619082726 41 0.561763783289 42 3.73120479825 43 0.779632239945 44 2.34778423403 45 0.56169267318 46 3.73184532633 47 0.779814501562 48 2.34697600109 49 0.561728595509 50 3.73152173265 51 0.779722417387 52 2.34738429268 53 0.561710410906 54 3.7316855368 55 0.779769029179 56 2.34717760747 57 0.561719606642 ... 941 0.561716516768 942 3.73163053487 943 0.77975337759 944 2.34724700641 945 0.561716516768 946 3.73163053487 947 0.77975337759 948 2.34724700641 949 0.561716516768 950 3.73163053487 951 0.77975337759 952 2.34724700641 953 0.561716516768 954 3.73163053487 955 0.77975337759 956 2.34724700641 957 0.561716516768 958 3.73163053487 959 0.77975337759 960 2.34724700641 961 0.561716516768 962 3.73163053487 963 0.77975337759 964 2.34724700641 965 0.561716516768 966 3.73163053487 967 0.77975337759 968 2.34724700641 969 0.561716516768 970 3.73163053487 971 0.77975337759 972 2.34724700641 973 0.561716516768 974 3.73163053487 975 0.77975337759 976 2.34724700641 977 0.561716516768 978 3.73163053487 979 0.77975337759 980 2.34724700641 981 0.561716516768 982 3.73163053487 983 0.77975337759 984 2.34724700641 985 0.561716516768 986 3.73163053487 987 0.77975337759 988 2.34724700641 989 0.561716516768 990 3.73163053487 991 0.77975337759 992 2.34724700641 993 0.561716516768 994 3.73163053487 995 0.77975337759 996 2.34724700641 997 0.561716516768 998 3.73163053487 999 0.77975337759 1000 2.34724700641 WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.14 2 22.2423182385 3 9.36603714624 4 3.11332479796 5 0.630262471149 6 3.18505291476 7 0.644426036536 8 3.08751840618 9 0.625413329587 10 3.21953439147 11 0.651578953345 12 3.03998975754 13 0.616840308943 14 3.28191554624 15 0.665058077571 16 2.95341327759 17 0.60248049705 18 3.39065454441 19 0.690112189307 20 2.80217536958 21 0.581684490603 22 3.5582773299 23 0.732243269188 24 2.57312071432 25 0.562459797207 26 3.72494444122 27 0.777853360473 28 2.35569474533 29 0.561357026642 30 3.73487099266 31 0.780676091759 32 2.34316099341 33 0.561902253698 34 3.72995800973 35 0.779277603448 36 2.34935806316 37 0.561623592327 38 3.73246773973 39 0.779991654041 40 2.34619082726 41 0.561763783289 42 3.73120479825 43 0.779632239945 44 2.34778423403 45 0.56169267318 46 3.73184532633 47 0.779814501562 48 2.34697600109 49 0.561728595509 50 3.73152173265 51 0.779722417387 52 2.34738429268 53 0.561710410906 54 3.7316855368 55 0.779769029179 56 2.34717760747 57 0.561719606642 ... 941 0.561716516768 942 3.73163053487 943 0.77975337759 944 2.34724700641 945 0.561716516768 946 3.73163053487 947 0.77975337759 948 2.34724700641 949 0.561716516768 950 3.73163053487 951 0.77975337759 952 2.34724700641 953 0.561716516768 954 3.73163053487 955 0.77975337759 956 2.34724700641 957 0.561716516768 958 3.73163053487 959 0.77975337759 960 2.34724700641 961 0.561716516768 962 3.73163053487 963 0.77975337759 964 2.34724700641 965 0.561716516768 966 3.73163053487 967 0.77975337759 968 2.34724700641 969 0.561716516768 970 3.73163053487 971 0.77975337759 972 2.34724700641 973 0.561716516768 974 3.73163053487 975 0.77975337759 976 2.34724700641 977 0.561716516768 978 3.73163053487 979 0.77975337759 980 2.34724700641 981 0.561716516768 982 3.73163053487 983 0.77975337759 984 2.34724700641 985 0.561716516768 986 3.73163053487 987 0.77975337759 988 2.34724700641 989 0.561716516768 990 3.73163053487 991 0.77975337759 992 2.34724700641 993 0.561716516768 994 3.73163053487 995 0.77975337759 996 2.34724700641 997 0.561716516768 998 3.73163053487 999 0.77975337759 1000 2.34724700641 |
# With a value of c = 1.89, the equation seems to have a stable four-cycle that oscillates between 0.561716516768, 3.73163053487, 0.77975337759, and 2.34724700641
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# ii : Finding a stable 8-cycle
# c = 1.91 , Initial Value: 100
print " n x(n)"
P = vector (RDF,10001)
P[0] = 100;
for n in range(1,10001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.91;
for n in range(0,10001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.12 2 22.2123441397 3 9.33123210586 4 3.07711700436 5 0.603497055296 6 3.36277526867 7 0.663507910093 8 2.94317706752 9 0.580895198352 10 3.54489035668 11 0.708733694527 12 2.67726846203 13 0.549179309907 14 3.82728634779 15 0.78748836534 16 2.29332427558 17 0.544806839324 18 3.86894222105 19 0.799876848992 20 2.24051578372 21 0.54923529242 22 3.82675753671 23 0.787332277618 24 2.29400147661 25 0.544759268798 26 3.86939928846 27 0.800013788972 28 2.2399422598 29 0.549291367677 30 3.82622796337 31 0.787175994904 32 2.2946798237 33 0.544711846276 34 3.8698550174 35 0.800150349481 36 2.23937054397 37 0.549347440927 38 3.8256985228 39 0.787019781403 40 2.29535817062 41 0.544664652268 42 3.87030863322 43 0.800286298139 44 2.23880160664 45 0.54940341503 46 3.82517013186 47 0.786863907546 48 2.29603534202 49 0.544617767749 50 3.87075935621 51 0.80042140091 52 2.23823642364 53 0.549459191135 54 3.82464372277 55 0.786708648093 56 2.2967101417 57 0.544571273553 ... 9941 0.550995280309 9942 3.81019037566 9943 0.782457420218 9944 2.31530315618 9945 0.5433782271 9946 3.88270532359 9947 0.804009804166 9948 2.22330268534 9949 0.550995280309 9950 3.81019037566 9951 0.782457420218 9952 2.31530315618 9953 0.5433782271 9954 3.88270532359 9955 0.804009804166 9956 2.22330268534 9957 0.550995280309 9958 3.81019037566 9959 0.782457420218 9960 2.31530315618 9961 0.5433782271 9962 3.88270532359 9963 0.804009804166 9964 2.22330268534 9965 0.550995280309 9966 3.81019037566 9967 0.782457420218 9968 2.31530315618 9969 0.5433782271 9970 3.88270532359 9971 0.804009804166 9972 2.22330268534 9973 0.550995280309 9974 3.81019037566 9975 0.782457420218 9976 2.31530315618 9977 0.5433782271 9978 3.88270532359 9979 0.804009804166 9980 2.22330268534 9981 0.550995280309 9982 3.81019037566 9983 0.782457420218 9984 2.31530315618 9985 0.5433782271 9986 3.88270532359 9987 0.804009804166 9988 2.22330268534 9989 0.550995280309 9990 3.81019037566 9991 0.782457420218 9992 2.31530315618 9993 0.5433782271 9994 3.88270532359 9995 0.804009804166 9996 2.22330268534 9997 0.550995280309 9998 3.81019037566 9999 0.782457420218 10000 2.31530315618 WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.12 2 22.2123441397 3 9.33123210586 4 3.07711700436 5 0.603497055296 6 3.36277526867 7 0.663507910093 8 2.94317706752 9 0.580895198352 10 3.54489035668 11 0.708733694527 12 2.67726846203 13 0.549179309907 14 3.82728634779 15 0.78748836534 16 2.29332427558 17 0.544806839324 18 3.86894222105 19 0.799876848992 20 2.24051578372 21 0.54923529242 22 3.82675753671 23 0.787332277618 24 2.29400147661 25 0.544759268798 26 3.86939928846 27 0.800013788972 28 2.2399422598 29 0.549291367677 30 3.82622796337 31 0.787175994904 32 2.2946798237 33 0.544711846276 34 3.8698550174 35 0.800150349481 36 2.23937054397 37 0.549347440927 38 3.8256985228 39 0.787019781403 40 2.29535817062 41 0.544664652268 42 3.87030863322 43 0.800286298139 44 2.23880160664 45 0.54940341503 46 3.82517013186 47 0.786863907546 48 2.29603534202 49 0.544617767749 50 3.87075935621 51 0.80042140091 52 2.23823642364 53 0.549459191135 54 3.82464372277 55 0.786708648093 56 2.2967101417 57 0.544571273553 ... 9941 0.550995280309 9942 3.81019037566 9943 0.782457420218 9944 2.31530315618 9945 0.5433782271 9946 3.88270532359 9947 0.804009804166 9948 2.22330268534 9949 0.550995280309 9950 3.81019037566 9951 0.782457420218 9952 2.31530315618 9953 0.5433782271 9954 3.88270532359 9955 0.804009804166 9956 2.22330268534 9957 0.550995280309 9958 3.81019037566 9959 0.782457420218 9960 2.31530315618 9961 0.5433782271 9962 3.88270532359 9963 0.804009804166 9964 2.22330268534 9965 0.550995280309 9966 3.81019037566 9967 0.782457420218 9968 2.31530315618 9969 0.5433782271 9970 3.88270532359 9971 0.804009804166 9972 2.22330268534 9973 0.550995280309 9974 3.81019037566 9975 0.782457420218 9976 2.31530315618 9977 0.5433782271 9978 3.88270532359 9979 0.804009804166 9980 2.22330268534 9981 0.550995280309 9982 3.81019037566 9983 0.782457420218 9984 2.31530315618 9985 0.5433782271 9986 3.88270532359 9987 0.804009804166 9988 2.22330268534 9989 0.550995280309 9990 3.81019037566 9991 0.782457420218 9992 2.31530315618 9993 0.5433782271 9994 3.88270532359 9995 0.804009804166 9996 2.22330268534 9997 0.550995280309 9998 3.81019037566 9999 0.782457420218 10000 2.31530315618 |
# With a value of c = 1.91, the equation seems to have a stable eight-cycle that oscillates between 0.5433782271, 3.88270532359, 0.804009804166, 2.22330268534, 0.550995280309, 3.81019037566, 0.782457420218, and 2.31530315618
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# iii : Finding a stable 16-cycle
# c = 1.93 , Initial Value: 100
print " n x(n)"
P = vector (RDF,10001)
P[0] = 100;
for n in range(1,10001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.93;
for n in range(0,10001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.1 2 22.1823700624 3 9.29642756801 4 3.04091839048 5 0.577003263438 6 3.55777898437 7 0.692112191847 8 2.75061346298 9 0.535972519459 10 3.9352881569 11 0.799977071713 12 2.22009601528 13 0.531340915402 14 3.98176303788 15 0.814316610748 16 2.1612290469 17 0.538713579541 18 3.90817872768 19 0.79171034769 20 2.25511978148 21 0.527866167434 22 4.01719180549 23 0.825386234216 24 2.11735514034 25 0.545539551821 26 3.84191276194 27 0.771817422304 28 2.34283833117 29 0.521917252067 30 4.07899624717 31 0.844973181224 32 2.04289513217 33 0.559951726069 34 3.70758056418 35 0.732943299285 36 2.52955770818 37 0.520756936463 38 4.09122348044 39 0.848888720651 40 2.02847647433 41 0.563180703574 42 3.67847740768 43 0.724793529252 44 2.57150656476 45 0.522384564256 46 4.07408785231 47 0.843405072196 48 2.04871220903 49 0.558690620011 50 3.71904348024 51 0.736180768426 52 2.5131764618 53 0.520296689339 54 4.09608931925 55 0.850450563448 56 2.02276718591 57 0.564500394977 ... 9941 0.519554224551 9942 4.10395785846 9943 0.852980587498 9944 2.0135691628 9945 0.566676289827 9946 3.64736588854 9947 0.716194337884 9948 2.61690437016 9949 0.524844859649 9950 4.0483972457 9951 0.835232621723 9952 2.07943004272 9953 0.552418040152 9954 3.77687887867 9955 0.752746086801 9956 2.43178067634 9957 0.519554224551 9958 4.10395785846 9959 0.852980587498 9960 2.0135691628 9961 0.566676289827 9962 3.64736588854 9963 0.716194337884 9964 2.61690437016 9965 0.524844859649 9966 4.0483972457 9967 0.835232621723 9968 2.07943004272 9969 0.552418040152 9970 3.77687887867 9971 0.752746086801 9972 2.43178067634 9973 0.519554224551 9974 4.10395785846 9975 0.852980587498 9976 2.0135691628 9977 0.566676289827 9978 3.64736588854 9979 0.716194337884 9980 2.61690437016 9981 0.524844859649 9982 4.0483972457 9983 0.835232621723 9984 2.07943004272 9985 0.552418040152 9986 3.77687887867 9987 0.752746086801 9988 2.43178067634 9989 0.519554224551 9990 4.10395785846 9991 0.852980587498 9992 2.0135691628 9993 0.566676289827 9994 3.64736588854 9995 0.716194337884 9996 2.61690437016 9997 0.524844859649 9998 4.0483972457 9999 0.835232621723 10000 2.07943004272 WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.1 2 22.1823700624 3 9.29642756801 4 3.04091839048 5 0.577003263438 6 3.55777898437 7 0.692112191847 8 2.75061346298 9 0.535972519459 10 3.9352881569 11 0.799977071713 12 2.22009601528 13 0.531340915402 14 3.98176303788 15 0.814316610748 16 2.1612290469 17 0.538713579541 18 3.90817872768 19 0.79171034769 20 2.25511978148 21 0.527866167434 22 4.01719180549 23 0.825386234216 24 2.11735514034 25 0.545539551821 26 3.84191276194 27 0.771817422304 28 2.34283833117 29 0.521917252067 30 4.07899624717 31 0.844973181224 32 2.04289513217 33 0.559951726069 34 3.70758056418 35 0.732943299285 36 2.52955770818 37 0.520756936463 38 4.09122348044 39 0.848888720651 40 2.02847647433 41 0.563180703574 42 3.67847740768 43 0.724793529252 44 2.57150656476 45 0.522384564256 46 4.07408785231 47 0.843405072196 48 2.04871220903 49 0.558690620011 50 3.71904348024 51 0.736180768426 52 2.5131764618 53 0.520296689339 54 4.09608931925 55 0.850450563448 56 2.02276718591 57 0.564500394977 ... 9941 0.519554224551 9942 4.10395785846 9943 0.852980587498 9944 2.0135691628 9945 0.566676289827 9946 3.64736588854 9947 0.716194337884 9948 2.61690437016 9949 0.524844859649 9950 4.0483972457 9951 0.835232621723 9952 2.07943004272 9953 0.552418040152 9954 3.77687887867 9955 0.752746086801 9956 2.43178067634 9957 0.519554224551 9958 4.10395785846 9959 0.852980587498 9960 2.0135691628 9961 0.566676289827 9962 3.64736588854 9963 0.716194337884 9964 2.61690437016 9965 0.524844859649 9966 4.0483972457 9967 0.835232621723 9968 2.07943004272 9969 0.552418040152 9970 3.77687887867 9971 0.752746086801 9972 2.43178067634 9973 0.519554224551 9974 4.10395785846 9975 0.852980587498 9976 2.0135691628 9977 0.566676289827 9978 3.64736588854 9979 0.716194337884 9980 2.61690437016 9981 0.524844859649 9982 4.0483972457 9983 0.835232621723 9984 2.07943004272 9985 0.552418040152 9986 3.77687887867 9987 0.752746086801 9988 2.43178067634 9989 0.519554224551 9990 4.10395785846 9991 0.852980587498 9992 2.0135691628 9993 0.566676289827 9994 3.64736588854 9995 0.716194337884 9996 2.61690437016 9997 0.524844859649 9998 4.0483972457 9999 0.835232621723 10000 2.07943004272 |
# With a value of c = 1.93, the equation seems to have a stable sixteen-cycle.
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# iv : Finding a stable 32-cycle
# c = 1.931 , Initial Value: 100
print " n x(n)"
P = vector (RDF,100001)
P[0] = 100;
for n in range(1,100001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.931;
for n in range(0,100001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.099 2 22.1808713591 3 9.29468735434 4 3.03910870257 5 0.575685873584 6 3.56801822814 7 0.693811993478 8 2.73984393411 9 0.533874844717 10 3.95523196243 11 0.805104997408 12 2.19777455627 13 0.532904445815 14 3.96497928645 15 0.808114024657 16 2.18540441438 17 0.534445842308 18 3.94951385264 19 0.80334408361 20 2.2050619019 21 0.532036974575 22 3.9737243355 23 0.810821431833 24 2.17436224928 25 0.535896027707 26 3.93504884852 27 0.798903778489 28 2.22359747846 29 0.529963787788 30 3.99474602369 31 0.817359428054 32 2.14803561503 33 0.539642695393 34 3.89805531049 35 0.787642183782 36 2.27165721287 37 0.525450389952 38 4.04111290763 39 0.831926209022 40 2.09105216321 41 0.549210520751 42 3.80599151601 43 0.760226626987 44 2.39530500621 45 0.519102605377 46 4.10775558509 47 0.853203620813 48 2.01176129077 49 0.56611124703 50 3.65136740321 51 0.716293709231 52 2.61537294307 53 0.523750414347 54 4.05879430341 55 0.837532908717 56 2.06971525754 57 0.553332376116 ... 99941 0.525278850907 99942 4.04289163905 99943 0.832488957912 99944 2.08889588056 99945 0.54961334609 99946 3.80218940946 99947 0.759113787401 99948 2.4005335931 99949 0.518988945252 99950 4.10896441798 99951 0.853593179743 99952 2.01035138154 99953 0.566452129381 99954 3.64834879845 99955 0.715464198182 99956 2.61981402775 99957 0.524026464425 99958 4.05591493788 99959 0.836617951456 99960 2.07317513456 99961 0.552643317655 99962 3.77377777901 99963 0.750848257989 99964 2.43990519735 99965 0.518508568005 99966 4.11407960131 99967 0.855242998143 99968 2.00439648994 99969 0.567908110038 99970 3.63549879593 99971 0.711945655314 99972 2.63877759999 99973 0.525278850907 99974 4.04289163905 99975 0.832488957912 99976 2.08889588056 99977 0.54961334609 99978 3.80218940946 99979 0.759113787401 99980 2.4005335931 99981 0.518988945252 99982 4.10896441798 99983 0.853593179743 99984 2.01035138154 99985 0.566452129381 99986 3.64834879845 99987 0.715464198182 99988 2.61981402775 99989 0.524026464425 99990 4.05591493788 99991 0.836617951456 99992 2.07317513456 99993 0.552643317655 99994 3.77377777901 99995 0.750848257989 99996 2.43990519735 99997 0.518508568005 99998 4.11407960131 99999 0.855242998143 100000 2.00439648994 WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.099 2 22.1808713591 3 9.29468735434 4 3.03910870257 5 0.575685873584 6 3.56801822814 7 0.693811993478 8 2.73984393411 9 0.533874844717 10 3.95523196243 11 0.805104997408 12 2.19777455627 13 0.532904445815 14 3.96497928645 15 0.808114024657 16 2.18540441438 17 0.534445842308 18 3.94951385264 19 0.80334408361 20 2.2050619019 21 0.532036974575 22 3.9737243355 23 0.810821431833 24 2.17436224928 25 0.535896027707 26 3.93504884852 27 0.798903778489 28 2.22359747846 29 0.529963787788 30 3.99474602369 31 0.817359428054 32 2.14803561503 33 0.539642695393 34 3.89805531049 35 0.787642183782 36 2.27165721287 37 0.525450389952 38 4.04111290763 39 0.831926209022 40 2.09105216321 41 0.549210520751 42 3.80599151601 43 0.760226626987 44 2.39530500621 45 0.519102605377 46 4.10775558509 47 0.853203620813 48 2.01176129077 49 0.56611124703 50 3.65136740321 51 0.716293709231 52 2.61537294307 53 0.523750414347 54 4.05879430341 55 0.837532908717 56 2.06971525754 57 0.553332376116 ... 99941 0.525278850907 99942 4.04289163905 99943 0.832488957912 99944 2.08889588056 99945 0.54961334609 99946 3.80218940946 99947 0.759113787401 99948 2.4005335931 99949 0.518988945252 99950 4.10896441798 99951 0.853593179743 99952 2.01035138154 99953 0.566452129381 99954 3.64834879845 99955 0.715464198182 99956 2.61981402775 99957 0.524026464425 99958 4.05591493788 99959 0.836617951456 99960 2.07317513456 99961 0.552643317655 99962 3.77377777901 99963 0.750848257989 99964 2.43990519735 99965 0.518508568005 99966 4.11407960131 99967 0.855242998143 99968 2.00439648994 99969 0.567908110038 99970 3.63549879593 99971 0.711945655314 99972 2.63877759999 99973 0.525278850907 99974 4.04289163905 99975 0.832488957912 99976 2.08889588056 99977 0.54961334609 99978 3.80218940946 99979 0.759113787401 99980 2.4005335931 99981 0.518988945252 99982 4.10896441798 99983 0.853593179743 99984 2.01035138154 99985 0.566452129381 99986 3.64834879845 99987 0.715464198182 99988 2.61981402775 99989 0.524026464425 99990 4.05591493788 99991 0.836617951456 99992 2.07317513456 99993 0.552643317655 99994 3.77377777901 99995 0.750848257989 99996 2.43990519735 99997 0.518508568005 99998 4.11407960131 99999 0.855242998143 100000 2.00439648994 |
# With a value of c = 1.93, the equation seems to have a stable 32-cycle.
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# v : Finding chaos
# c = 1.95 , Initial Value: 100
print " n x(n)"
P = vector (RDF,100001)
P[0] = 100;
for n in range(1,100001): P[n]=(3/P[n-1])+(P[n-1]/2)-1.95;
for n in range(0,100001): print n, " ", P[n]
WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.08 2 22.1523960067 3 9.26162353468 4 3.00472905774 5 0.550790657266 6 3.77211081014 7 0.731366081506 8 2.5175959895 9 0.500410951123 10 4.2952781119 11 0.896080442479 12 1.84595393276 13 0.598152950872 14 3.36451608102 15 0.623916730127 16 3.17029232237 17 0.581431139018 18 3.50039794405 19 0.657244384562 20 2.94313437446 21 0.540888637869 22 3.86687252978 23 0.759257028801 24 2.38085963404 25 0.500478900598 26 4.29449814215 27 0.895817309016 28 1.84680576757 29 0.597829259681 30 3.36706981616 31 0.624517324186 32 3.16596847736 33 0.580561582376 34 3.50769126248 35 0.659109043983 36 2.93115342264 37 0.539064590469 38 3.88472810486 39 0.764618871565 40 2.35583279045 41 0.501351425148 42 4.28450232504 43 0.892449172093 44 1.85776056346 45 0.593727780923 46 3.39968455406 47 0.632277088935 48 3.11089373566 49 0.569799962123 50 3.59990559605 51 0.683307984698 52 2.7820607594 53 0.519367720056 54 4.08593810947 55 0.827194566632 56 2.09031356147 57 0.530348158383 ... 99941 0.5928748014 99942 3.40652764727 99943 0.633925984825 99944 3.09937666817 99945 0.567624897203 99946 3.61899288163 99947 0.6884563471 99948 2.75180293139 99949 0.516095812747 99950 4.12092203925 99951 0.838453437687 99952 2.04724293913 99953 0.539006911937 99954 3.88529479104 99955 0.76478957812 99956 2.35504238499 99957 0.501383616 99958 4.28413423465 99959 0.892325287513 99960 1.85816531474 99961 0.593578405518 99962 3.40088142059 99963 0.632564968388 99964 3.10887834379 99965 0.569417427152 99966 3.60325136251 99967 0.684207063508 99968 2.77674110714 99969 0.518773761591 99970 4.0922545239 99971 0.829219493258 99972 2.08246970336 99973 0.531832040277 99974 3.95679466753 99975 0.786586790657 99976 2.2572399752 99977 0.507676726343 99978 4.213110631 99979 0.868618269159 99980 1.93807024751 99981 0.566966631323 99982 3.62480000744 99983 0.690031871365 99984 2.74264120416 99985 0.515156717227 99986 4.13104896412 99987 0.84173228692 99988 2.03494470694 99989 0.541713884343 99990 3.85883572896 99991 0.75685443103 99992 2.39220132716 99993 0.500175714011 99994 4.29798002962 99995 0.896992327367 99996 1.84300637684 99997 0.599278348453 99998 3.3556601789 99999 0.621841950026 100000 3.18529797286 WARNING: Output truncated! full_output.txt n x(n) 0 100.0 1 48.08 2 22.1523960067 3 9.26162353468 4 3.00472905774 5 0.550790657266 6 3.77211081014 7 0.731366081506 8 2.5175959895 9 0.500410951123 10 4.2952781119 11 0.896080442479 12 1.84595393276 13 0.598152950872 14 3.36451608102 15 0.623916730127 16 3.17029232237 17 0.581431139018 18 3.50039794405 19 0.657244384562 20 2.94313437446 21 0.540888637869 22 3.86687252978 23 0.759257028801 24 2.38085963404 25 0.500478900598 26 4.29449814215 27 0.895817309016 28 1.84680576757 29 0.597829259681 30 3.36706981616 31 0.624517324186 32 3.16596847736 33 0.580561582376 34 3.50769126248 35 0.659109043983 36 2.93115342264 37 0.539064590469 38 3.88472810486 39 0.764618871565 40 2.35583279045 41 0.501351425148 42 4.28450232504 43 0.892449172093 44 1.85776056346 45 0.593727780923 46 3.39968455406 47 0.632277088935 48 3.11089373566 49 0.569799962123 50 3.59990559605 51 0.683307984698 52 2.7820607594 53 0.519367720056 54 4.08593810947 55 0.827194566632 56 2.09031356147 57 0.530348158383 ... 99941 0.5928748014 99942 3.40652764727 99943 0.633925984825 99944 3.09937666817 99945 0.567624897203 99946 3.61899288163 99947 0.6884563471 99948 2.75180293139 99949 0.516095812747 99950 4.12092203925 99951 0.838453437687 99952 2.04724293913 99953 0.539006911937 99954 3.88529479104 99955 0.76478957812 99956 2.35504238499 99957 0.501383616 99958 4.28413423465 99959 0.892325287513 99960 1.85816531474 99961 0.593578405518 99962 3.40088142059 99963 0.632564968388 99964 3.10887834379 99965 0.569417427152 99966 3.60325136251 99967 0.684207063508 99968 2.77674110714 99969 0.518773761591 99970 4.0922545239 99971 0.829219493258 99972 2.08246970336 99973 0.531832040277 99974 3.95679466753 99975 0.786586790657 99976 2.2572399752 99977 0.507676726343 99978 4.213110631 99979 0.868618269159 99980 1.93807024751 99981 0.566966631323 99982 3.62480000744 99983 0.690031871365 99984 2.74264120416 99985 0.515156717227 99986 4.13104896412 99987 0.84173228692 99988 2.03494470694 99989 0.541713884343 99990 3.85883572896 99991 0.75685443103 99992 2.39220132716 99993 0.500175714011 99994 4.29798002962 99995 0.896992327367 99996 1.84300637684 99997 0.599278348453 99998 3.3556601789 99999 0.621841950026 100000 3.18529797286 |
# At this point (c=1.95) there appears to be no real pattern. We have reached chaos.
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|
# Problem 2
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|
#DEFINING KEY FUNCTIONS USED IN CALCULATIONS;
#CompIterate calculates N iterations of the equation Pn+1=(3/Pn)+(Pn/2)+c for a given c and N returned from the function as a vector with values from all iterations;
def CompIterate(c,N):
P= vector(RDF,N);
P[0]=1;
for i in range(1,N):
P[i]=(3/P[i-1])+(P[i-1]/2)-c;
return P;
#LyapunovEst is a function that estimates the Lyapunov exponent for the equation Pn+1=(3/Pn)+(Pn/2)+c for a given c and N;
def LyapunovEst(c,N):
Pprime= vector(RDF,N);
Pprime[0]=1;
LogPprimeSum=ln(Pprime[0])
for i in range(1,N):
Pprime[i]=(1/2)-(3/((Pprime[i-1])^2));
Pprime[i]=((Pprime[i])^2)^(1/2);
LogPprimeSum= LogPprimeSum+ln(Pprime[i]);
lyapunov_estimate=(1/N)*LogPprimeSum;
return lyapunov_estimate;
#VectorMax takes a vector P with N values and determines the maximum value which it returns from the function;
def VectorMax(P,N):
vmaximum=P[0];
for i in range(N):
if P[i]>vmaximum:
vmaximum=P[i];
return vmaximum;
#VectorMin takes a vector P with N values and determines the minimum value which it returns from the function;
def VectorMin(P,N):
vminimum=P[0];
for i in range(N):
if P[i]<vminimum:
vminimum=P[i];
return vminimum;
#BinIt takes a vector P with N values finds the max and min values of that vector calculates the range then divides that range into 100 equal intervals;
def BinIt(P,N,max_value,min_value,num_intervals):
if num_intervals==0:
num_intervals=100;
val_range=max_value-min_value;
adding_interval=val_range/num_intervals;
#The values that define the maximum of each of these intervals is represented in the vector BinMax so that a set of data can be iteratively compared to these different maxima to determine which bin it belongs to;
BinMax= vector(RDF,num_intervals);
BinMax[0]= min_value+adding_interval;
for i in range(1,num_intervals):
BinMax[i]=BinMax[i-1]+adding_interval;
#BinHisto counts how many values of the vector P are in each interval defined by BinMax and then returns a vector that contains the number of values in each of these 100 intervals;
BinHisto= vector(RDF,num_intervals);
for j in range(N):
if P[j]<BinMax[0]:
BinHisto[0]=BinHisto[0]+1;
for i in range(1,num_intervals):
for j in range(N):
if P[j]>BinMax[i-1] and (P[j]<BinMax[i] or P[j]==BinMax[i]):
BinHisto[i]=BinHisto[i]+1;
return BinHisto;
#PART A - INTERVAL OF STABILITY OF THE FIXED POINT;
c=1.3;
print "c value chosen from interval of stability of the fixed point:";
print c;
print "";
N=2000;
print "Iterations computed:";
print N;
print "";
fixedpt=CompIterate(c,N);
print "Values for all iterations:";
print fixedpt;
print "";
fixedpt_lyapunov=LyapunovEst(c,N);
print "Estimate of Lyapunov exponent:";
print fixedpt_lyapunov;
print "";
num_intervals=100;
max_value=VectorMax(fixedpt,N);
min_value=VectorMin(fixedpt,N);
fixedpt_bars=BinIt(fixedpt,N,max_value,min_value,num_intervals);
print "Graphical approximation of frequency distribution across 100 equal intervals of value range:";
bar_chart(fixedpt_bars);
c value chosen from interval of stability of the fixed point: 1.30000000000000 Iterations computed: 2000 Values for all iterations: (1.0, 2.2, 1.16363636364, 1.85994318182, 1.24292408824, 1.73512513778, 1.29654449341, 1.66211493799, 1.33598679815, 1.61352457061, 1.3660460264, 1.57914228152, 1.38933662428, 1.55397221846, 1.40752255278, 1.53516584542, 1.42176918145, 1.5209317342, 1.43294098386, 1.51006683246, 1.44170045342, 1.501726315, 1.44856405016, 1.49529849113, 1.453937634, 1.49033104784, 1.45814110028, 1.48648460933, 1.46142667144, 1.48350192828, 1.4639930272, 1.48118660144, 1.46599644512, 1.47938790514, 1.46755965944, 1.47798974016, 1.46877891964, 1.47690243429, 1.46972960788, 1.47605658773, 1.47047069813, 1.47539840965, 1.47104828557, 1.47488616062, 1.47149837224, 1.47448742505, 1.47184906014, 1.47417701209, 1.4721222743, 1.47393533576, 1.47233511394, 1.47374716205, 1.47250091063, 1.47360063849, 1.47263005604, 1.47348654146, 1.47273064869, 1.47339769185, 1.47280899907, 1.47332850115, 1.47287002387, 1.47327461854, 1.47291755346, 1.47323265654, 1.47295457171, 1.47319997752, 1.47298340294, 1.47317452764, 1.47300585762, 1.47315470754, 1.47302334593, 1.47313927176, 1.47303696623, 1.47312725043, 1.47304757399, 1.47311788821, 1.4730558355, 1.4731105969, 1.47306226969, 1.4731049184, 1.47306728074, 1.47310049596, 1.4730711834, 1.47309705173, 1.47307422285, 1.47309436935, 1.47307659, 1.47309228029, 1.47307843357, 1.47309065331, 1.47307986936, 1.47308938621, 1.47308098757, 1.47308839938, 1.47308185844, 1.47308763083, 1.47308253668, 1.47308703227, 1.4730830649, 1.47308656611, 1.47308347629, 1.47308620306, 1.47308379668, 1.47308592032, 1.4730840462, 1.47308570011, 1.47308424053, 1.47308552861, 1.47308439188, 1.47308539505, 1.47308450975, 1.47308529103, 1.47308460155, 1.47308521002, 1.47308467304, 1.47308514692, 1.47308472872, 1.47308509779, 1.47308477209, 1.47308505952, 1.47308480586, 1.47308502971, 1.47308483216, 1.4730850065, 1.47308485265, 1.47308498842, 1.4730848686, 1.47308497434, 1.47308488103, 1.47308496338, 1.4730848907, 1.47308495484, 1.47308489824, 1.47308494819, 1.47308490411, 1.47308494301, 1.47308490868, 1.47308493898, 1.47308491224, 1.47308493583, 1.47308491501, 1.47308493339, 1.47308491717, 1.47308493148, 1.47308491885, 1.47308493, 1.47308492016, 1.47308492884, 1.47308492118, 1.47308492794, 1.47308492198, 1.47308492724, 1.47308492259, 1.47308492669, 1.47308492308, 1.47308492627, 1.47308492345, 1.47308492594, 1.47308492374, 1.47308492568, 1.47308492397, 1.47308492548, 1.47308492415, 1.47308492532, 1.47308492429, 1.4730849252, 1.47308492439, 1.47308492511, 1.47308492448, 1.47308492503, 1.47308492454, 1.47308492498, 1.47308492459, 1.47308492493, 1.47308492463, 1.4730849249, 1.47308492466, 1.47308492487, 1.47308492469, 1.47308492485, 1.47308492471, 1.47308492483, 1.47308492472, 1.47308492482, 1.47308492473, 1.47308492481, 1.47308492474, 1.4730849248, 1.47308492475, 1.47308492479, 1.47308492475, 1.47308492479, 1.47308492476, 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1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: c value chosen from interval of stability of the fixed point: 1.30000000000000 Iterations computed: 2000 Values for all iterations: (1.0, 2.2, 1.16363636364, 1.85994318182, 1.24292408824, 1.73512513778, 1.29654449341, 1.66211493799, 1.33598679815, 1.61352457061, 1.3660460264, 1.57914228152, 1.38933662428, 1.55397221846, 1.40752255278, 1.53516584542, 1.42176918145, 1.5209317342, 1.43294098386, 1.51006683246, 1.44170045342, 1.501726315, 1.44856405016, 1.49529849113, 1.453937634, 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1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477, 1.47308492477) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: |
#PART B - INTERVAL OF STABILITY OF THE TWO CYCLE;
c=1.55;
print "c value chosen from interval of stability of the two cycle:";
print c;
print "";
N=2000;
print "Iterations computed:";
print N;
print "";
twocycle=CompIterate(c,N);
print "Values for all iterations:";
print twocycle;
print "";
twocycle_lyapunov=LyapunovEst(c,N);
print "Estimate of Lyapunov exponent:";
print twocycle_lyapunov;
print "";
num_intervals=100;
max_value=VectorMax(twocycle,N);
min_value=VectorMin(twocycle,N);
twocycle_bars=BinIt(twocycle,N,max_value,min_value,num_intervals);
print "Graphical approximation of frequency distribution across 100 equal intervals of value range:";
bar_chart(twocycle_bars);
|
|
#PART C - INTERVAL OF STABILITY OF THE FOUR CYCLE;
c=1.89;
print "c value chosen from interval of stability of the four cycle:";
print c;
print "";
N=2000;
print "Iterations computed:";
print N;
print "";
fourcycle=CompIterate(c,N);
print "Values for all iterations:";
print fourcycle;
print "";
fourcycle_lyapunov=LyapunovEst(c,N);
print "Estimate of Lyapunov exponent:";
print fourcycle_lyapunov;
print "";
num_intervals=100;
max_value=VectorMax(fourcycle,N);
min_value=VectorMin(fourcycle,N);
fourcycle_bars=BinIt(fourcycle,N,max_value,min_value,num_intervals);
print "Graphical approximation of frequency distribution across 100 equal intervals of value range:";
bar_chart(fourcycle_bars);
c value chosen from interval of stability of the four cycle: 1.89000000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.61, 0.778354037267, 2.3534642149, 0.561448748989, 3.73404378771, 0.780440431654, 2.34420353386, 0.561854124892, 3.73039128829, 0.779400824822, 2.3488110426, 0.561647472904, 3.73225255921, 0.779930403899, 2.34646225477, 0.561751586814, 3.73131464649, 0.779663493784, 2.34764561008, 0.561698812866, 3.73179001603, 0.779798761204, 2.34704578459, 0.561725481992, 3.73154977795, 0.779730397683, 2.34734890464, 0.561711983967, 3.73167136646, 0.779764996773, 2.34719548681, 0.56171881038, 3.73160987422, 0.779747498396, 2.34727307548, 0.561715356663, 3.73164098506, 0.779756351314, 2.34723382072, 0.561717103668, 3.73162524812, 0.779751873187, 2.34725367708, 0.561716219886, 3.73163320918, 0.779754138594, 2.34724363206, 0.561716666954, 3.731629182, 0.779752992617, 2.34724871342, 0.561716440795, 3.73163121923, 0.779753572334, 2.3472461429, 0.561716555201, 3.73163018867, 0.779753279075, 2.34724744323, 0.561716497327, 3.73163071, 0.779753427425, 2.34724678544, 0.561716526603, 3.73163044628, 0.77975335238, 2.34724711819, 0.561716511793, 3.73163057969, 0.779753390343, 2.34724694986, 0.561716519285, 3.7316305122, 0.779753371139, 2.34724703502, 0.561716515495, 3.73163054634, 0.779753380853, 2.34724699194, 0.561716517412, 3.73163052907, 0.779753375939, 2.34724701373, 0.561716516442, 3.7316305378, 0.779753378425, 2.34724700271, 0.561716516933, 3.73163053339, 0.779753377167, 2.34724700828, 0.561716516685, 3.73163053562, 0.779753377804, 2.34724700546, 0.56171651681, 3.73163053449, 0.779753377482, 2.34724700689, 0.561716516747, 3.73163053506, 0.779753377645, 2.34724700617, 0.561716516779, 3.73163053477, 0.779753377562, 2.34724700653, 0.561716516763, 3.73163053492, 0.779753377604, 2.34724700635, 0.561716516771, 3.73163053484, 0.779753377583, 2.34724700644, 0.561716516767, 3.73163053488, 0.779753377593, 2.34724700639, 0.561716516769, 3.73163053486, 0.779753377588, 2.34724700642, 0.561716516768, 3.73163053487, 0.779753377591, 2.34724700641, 0.561716516768, 3.73163053487, 0.779753377589, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 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3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: c value chosen from interval of stability of the four cycle: 1.89000000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.61, 0.778354037267, 2.3534642149, 0.561448748989, 3.73404378771, 0.780440431654, 2.34420353386, 0.561854124892, 3.73039128829, 0.779400824822, 2.3488110426, 0.561647472904, 3.73225255921, 0.779930403899, 2.34646225477, 0.561751586814, 3.73131464649, 0.779663493784, 2.34764561008, 0.561698812866, 3.73179001603, 0.779798761204, 2.34704578459, 0.561725481992, 3.73154977795, 0.779730397683, 2.34734890464, 0.561711983967, 3.73167136646, 0.779764996773, 2.34719548681, 0.56171881038, 3.73160987422, 0.779747498396, 2.34727307548, 0.561715356663, 3.73164098506, 0.779756351314, 2.34723382072, 0.561717103668, 3.73162524812, 0.779751873187, 2.34725367708, 0.561716219886, 3.73163320918, 0.779754138594, 2.34724363206, 0.561716666954, 3.731629182, 0.779752992617, 2.34724871342, 0.561716440795, 3.73163121923, 0.779753572334, 2.3472461429, 0.561716555201, 3.73163018867, 0.779753279075, 2.34724744323, 0.561716497327, 3.73163071, 0.779753427425, 2.34724678544, 0.561716526603, 3.73163044628, 0.77975335238, 2.34724711819, 0.561716511793, 3.73163057969, 0.779753390343, 2.34724694986, 0.561716519285, 3.7316305122, 0.779753371139, 2.34724703502, 0.561716515495, 3.73163054634, 0.779753380853, 2.34724699194, 0.561716517412, 3.73163052907, 0.779753375939, 2.34724701373, 0.561716516442, 3.7316305378, 0.779753378425, 2.34724700271, 0.561716516933, 3.73163053339, 0.779753377167, 2.34724700828, 0.561716516685, 3.73163053562, 0.779753377804, 2.34724700546, 0.56171651681, 3.73163053449, 0.779753377482, 2.34724700689, 0.561716516747, 3.73163053506, 0.779753377645, 2.34724700617, 0.561716516779, 3.73163053477, 0.779753377562, 2.34724700653, 0.561716516763, 3.73163053492, 0.779753377604, 2.34724700635, 0.561716516771, 3.73163053484, 0.779753377583, 2.34724700644, 0.561716516767, 3.73163053488, 0.779753377593, 2.34724700639, 0.561716516769, 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3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641, 0.561716516768, 3.73163053487, 0.77975337759, 2.34724700641) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: |
#PART D - INTERVAL OF STABILITY OF THE EIGHT CYCLE;
c=1.91;
print "c value chosen from interval of stability of the eight cycle:";
print c;
print "";
N=2000;
print "Iterations computed:";
print N;
print "";
eightcycle=CompIterate(c,N);
print "Values for all iterations:";
print eightcycle;
print "";
eightcycle_lyapunov=LyapunovEst(c,N);
print "Estimate of Lyapunov exponent:";
print eightcycle_lyapunov;
print "";
num_intervals=100;
max_value=VectorMax(eightcycle,N);
min_value=VectorMin(eightcycle,N);
eightcycle_bars=BinIt(eightcycle,N,max_value,min_value,num_intervals);
print "Graphical approximation of frequency distribution across 100 equal intervals of value range:";
bar_chart(eightcycle_bars);
c value chosen from interval of stability of the eight cycle: 1.91000000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.59, 0.77179245283, 2.36295159862, 0.541074381758, 3.90506138412, 0.810764454583, 2.19559377831, 0.554169862898, 3.78058754293, 0.773821223463, 2.3537750722, 0.541435828939, 3.90154074084, 0.809697366003, 2.19993668991, 0.553643951512, 3.78546692222, 0.775238074579, 2.34739801019, 0.541709801294, 3.89887542904, 0.808890356596, 2.2032296576, 0.553252273018, 3.7891072474, 0.776296850831, 2.34264948166, 0.541926052928, 3.89677364821, 0.808254481514, 2.20582951846, 0.552947334229, 3.79194516194, 0.777123262714, 2.33895308267, 0.542101670598, 3.89506809205, 0.807738809869, 2.20794128479, 0.552702426011, 3.79422679093, 0.777788324852, 2.3359847154, 0.542247336006, 3.89365430632, 0.807311578326, 2.20969316623, 0.552501141606, 3.79610360473, 0.778335817865, 2.33354532693, 0.542370146993, 3.89246295933, 0.806951723209, 2.2111703825, 0.552332741662, 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2.31530315641, 0.543378227087, 3.88270532371, 0.804009804205, 2.22330268518) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: c value chosen from interval of stability of the eight cycle: 1.91000000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.59, 0.77179245283, 2.36295159862, 0.541074381758, 3.90506138412, 0.810764454583, 2.19559377831, 0.554169862898, 3.78058754293, 0.773821223463, 2.3537750722, 0.541435828939, 3.90154074084, 0.809697366003, 2.19993668991, 0.553643951512, 3.78546692222, 0.775238074579, 2.34739801019, 0.541709801294, 3.89887542904, 0.808890356596, 2.2032296576, 0.553252273018, 3.7891072474, 0.776296850831, 2.34264948166, 0.541926052928, 3.89677364821, 0.808254481514, 2.20582951846, 0.552947334229, 3.79194516194, 0.777123262714, 2.33895308267, 0.542101670598, 3.89506809205, 0.807738809869, 2.20794128479, 0.552702426011, 3.79422679093, 0.777788324852, 2.3359847154, 0.542247336006, 3.89365430632, 0.807311578326, 2.20969316623, 0.552501141606, 3.79610360473, 0.778335817865, 2.33354532693, 0.542370146993, 3.89246295933, 0.806951723209, 2.2111703825, 0.552332741662, 3.79767490493, 0.778794485413, 2.33150464227, 0.542475043282, 3.89144584579, 0.806644610844, 2.21243225752, 0.552189849448, 3.79900899042, 0.779184121546, 2.32977318876, 0.542565590118, 3.89056820246, 0.806379695469, 2.21352161806, 0.552067202029, 3.80015464456, 0.779518879427, 2.32828713952, 0.542644437663, 3.88980420773, 0.80614914865, 2.21447030491, 0.55196092744, 3.80114779325, 0.77980919164, 2.32699954182, 0.542713604469, 3.88913420692, 0.805947014871, 2.21530257454, 0.551868103735, 3.80201557152, 0.780062944417, 2.3258749666, 0.542774659881, 3.88854292828, 0.805768668963, 2.21603728837, 0.551786478212, 3.80277891583, 0.780286226935, 2.32488610193, 0.542828845568, 3.88801829524, 0.805610455072, 2.21668937168, 0.551714282107, 3.80345427655, 0.780483826768, 2.3240115049, 0.542877158927, 3.88755061175, 0.805469439175, 2.21727081505, 0.551650104529, 3.80405478407, 0.780659567512, 2.32323407245, 0.542920411798, 3.88713198911, 0.805343235051, 2.21779137942, 0.551592804433, 3.80459106359, 0.780816544922, 2.32253997358, 0.542959272713, 3.88675593381, 0.805229879164, 2.2182591036, 0.551541447746, 3.80507181767, 0.780957295802, 2.3219178868, 0.542994297904, 3.88641704513, 0.805127738945, 2.21868067664, 0.551495261553, 3.8055042509, 0.781083921242, 2.3213584445, 0.543025954444, 3.88611078963, 0.805035444382, 2.21906171592, 0.551453600116, 3.80589438509, 0.781198178109, 2.32085382091, 0.54305463779, 3.88583333018, 0.804951836207, 2.2194069779, 0.551415919285, 3.80624729679, 0.781301548062, 2.32039742171, 0.543080685266, 3.88558139471, 0.804875926114, 2.21972052011, 0.551381756937, 3.80656729895, 0.781395290329, 2.31998364692, 0.543104386552, 3.88535217406, 0.804806865813, 2.22000582726, 0.551350717839, 3.80685808152, 0.781480482594, 2.31960770737, 0.543125991936, 3.88514324169, 0.804743922708, 2.22026591041, 0.551322461802, 3.80712282163, 0.781558053044, 2.319265481, 0.543145718887, 3.88495248977, 0.804686460556, 2.22050338588, 0.551296694301, 3.80736427074, 0.781628805735, 2.31895339927, 0.543163757316, 3.88477807795, 0.80463392396, 2.22072053865, 0.551273158988, 3.80758482437, 0.781693440889, 2.31866835624, 0.543180273844, 3.88461839192, 0.804585825819, 2.22091937354, 0.551251631647, 3.8077865783, 0.781752571257, 2.31840763545, 0.543195415289, 3.88447200953, 0.804541737085, 2.2211016572, 0.551231915292, 3.80797137426, 0.781806735439, 2.31816885037, 0.543209311524, 3.88433767295, 0.804501278346, 2.22126895244, 0.551213836141, 3.80814083739, 0.78185640879, 2.31794989568, 0.543222077854, 3.88421426568, 0.804464112863, 2.22142264683, 0.551197240303, 3.80829640715, 0.781902012436, 2.31774890709, 0.543233816987, 3.88410079331, 0.804429940761, 2.22156397641, 0.551181991021, 3.80843936297, 0.781943920754, 2.3175642279, 0.543244620699, 3.88399636743, 0.804398494168, 2.22169404568, 0.551167966371, 3.80857084571, 0.781982467634, 2.31739438116, 0.543254571227, 3.88390019208, 0.80436953311, 2.22181384438, 0.551155057329, 3.80869187572, 0.78201795174, 2.31723804622, 0.543263742453, 3.88381155223, 0.804342842037, 2.22192426174, 0.551143166137, 3.80880336807, 0.782050640961, 2.31709403889, 0.543272200922, 3.88372980397, 0.804318226861, 2.22202609868, 0.551132204908, 3.80890614552, 0.782080776196, 2.31696129462, 0.543280006698, 3.88365436614, 0.804295512423, 2.22212007815, 0.551122094451, 3.80900094957, 0.782108574579, 2.31683885418, 0.543287214113, 3.88358471311, 0.804274540314, 2.22220685415, 0.551112763244, 3.80908844998, 0.782134232257, 2.31672585126, 0.543293872409, 3.88352036853, 0.804255166989, 2.22228701938, 0.551104146565, 3.8091692529, 0.782157926775, 2.31662150193, 0.543300026296, 3.88346089995, 0.804237262142, 2.22236111201, 0.551096185732, 3.80924390799, 0.782179819142, 2.31652509542, 0.543305716433, 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0.804009804632, 2.22330268341, 0.550995280515, 3.81019037373, 0.782457419651, 2.31530315868, 0.543378226951, 3.88270532502, 0.804009804599, 2.22330268354, 0.550995280501, 3.81019037387, 0.782457419691, 2.3153031585, 0.543378226962, 3.88270532492, 0.804009804569, 2.22330268367, 0.550995280487, 3.81019037399, 0.782457419728, 2.31530315834, 0.543378226971, 3.88270532483, 0.80400980454, 2.22330268379, 0.550995280475, 3.81019037411, 0.782457419763, 2.31530315819, 0.54337822698, 3.88270532474, 0.804009804514, 2.2233026839, 0.550995280463, 3.81019037422, 0.782457419795, 2.31530315805, 0.543378226989, 3.88270532466, 0.80400980449, 2.223302684, 0.550995280452, 3.81019037432, 0.782457419824, 2.31530315792, 0.543378226997, 3.88270532458, 0.804009804467, 2.22330268409, 0.550995280442, 3.81019037442, 0.782457419852, 2.31530315779, 0.543378227004, 3.88270532451, 0.804009804446, 2.22330268418, 0.550995280433, 3.8101903745, 0.782457419878, 2.31530315768, 0.543378227011, 3.88270532445, 0.804009804426, 2.22330268426, 0.550995280424, 3.81019037459, 0.782457419902, 2.31530315758, 0.543378227017, 3.88270532439, 0.804009804408, 2.22330268434, 0.550995280416, 3.81019037466, 0.782457419924, 2.31530315748, 0.543378227023, 3.88270532433, 0.804009804391, 2.22330268441, 0.550995280409, 3.81019037473, 0.782457419945, 2.31530315739, 0.543378227028, 3.88270532428, 0.804009804375, 2.22330268447, 0.550995280402, 3.8101903748, 0.782457419964, 2.3153031573, 0.543378227033, 3.88270532423, 0.80400980436, 2.22330268453, 0.550995280395, 3.81019037486, 0.782457419982, 2.31530315722, 0.543378227038, 3.88270532418, 0.804009804347, 2.22330268459, 0.550995280389, 3.81019037492, 0.782457419999, 2.31530315715, 0.543378227042, 3.88270532414, 0.804009804334, 2.22330268464, 0.550995280384, 3.81019037497, 0.782457420014, 2.31530315708, 0.543378227047, 3.8827053241, 0.804009804322, 2.22330268469, 0.550995280378, 3.81019037502, 0.782457420029, 2.31530315702, 0.54337822705, 3.88270532407, 0.804009804311, 2.22330268474, 0.550995280373, 3.81019037506, 0.782457420042, 2.31530315696, 0.543378227054, 3.88270532403, 0.804009804301, 2.22330268478, 0.550995280369, 3.8101903751, 0.782457420054, 2.3153031569, 0.543378227057, 3.882705324, 0.804009804291, 2.22330268482, 0.550995280365, 3.81019037514, 0.782457420066, 2.31530315685, 0.54337822706, 3.88270532397, 0.804009804283, 2.22330268486, 0.550995280361, 3.81019037518, 0.782457420077, 2.31530315681, 0.543378227063, 3.88270532394, 0.804009804274, 2.22330268489, 0.550995280357, 3.81019037521, 0.782457420087, 2.31530315676, 0.543378227066, 3.88270532392, 0.804009804267, 2.22330268492, 0.550995280354, 3.81019037525, 0.782457420096, 2.31530315672, 0.543378227068, 3.8827053239, 0.80400980426, 2.22330268495, 0.550995280351, 3.81019037528, 0.782457420105, 2.31530315668, 0.54337822707, 3.88270532387, 0.804009804253, 2.22330268498, 0.550995280348, 3.8101903753, 0.782457420113, 2.31530315665, 0.543378227072, 3.88270532385, 0.804009804247, 2.223302685, 0.550995280345, 3.81019037533, 0.78245742012, 2.31530315662, 0.543378227074, 3.88270532383, 0.804009804241, 2.22330268503, 0.550995280343, 3.81019037535, 0.782457420127, 2.31530315659, 0.543378227076, 3.88270532382, 0.804009804236, 2.22330268505, 0.55099528034, 3.81019037537, 0.782457420133, 2.31530315656, 0.543378227078, 3.8827053238, 0.804009804231, 2.22330268507, 0.550995280338, 3.81019037539, 0.782457420139, 2.31530315653, 0.543378227079, 3.88270532379, 0.804009804227, 2.22330268509, 0.550995280336, 3.81019037541, 0.782457420145, 2.31530315651, 0.543378227081, 3.88270532377, 0.804009804222, 2.2233026851, 0.550995280334, 3.81019037543, 0.78245742015, 2.31530315648, 0.543378227082, 3.88270532376, 0.804009804218, 2.22330268512, 0.550995280332, 3.81019037545, 0.782457420155, 2.31530315646, 0.543378227083, 3.88270532375, 0.804009804215, 2.22330268514, 0.550995280331, 3.81019037546, 0.782457420159, 2.31530315644, 0.543378227085, 3.88270532374, 0.804009804211, 2.22330268515, 0.550995280329, 3.81019037548, 0.782457420164, 2.31530315642, 0.543378227086, 3.88270532372, 0.804009804208, 2.22330268516, 0.550995280328, 3.81019037549, 0.782457420167, 2.31530315641, 0.543378227087, 3.88270532371, 0.804009804205, 2.22330268518) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: |
#PART E - INTERVAL OF CHAOS;
c=1.999;
print "c value chosen from interval of chaos:";
print c;
print "";
N=2000;
print "Iterations computed:";
print N;
print "";
chaos=CompIterate(c,N);
print "Values for all iterations:";
print chaos;
print "";
chaos_lyapunov=LyapunovEst(c,N);
print "Estimate of Lyapunov exponent:";
print chaos_lyapunov;
print "";
num_intervals=100;
max_value=VectorMax(chaos,N);
min_value=VectorMin(chaos,N);
chaos_bars=BinIt(chaos,N,max_value,min_value,num_intervals);
print "Graphical approximation of frequency distribution across 100 equal intervals of value range:";
bar_chart(chaos_bars);
c value chosen from interval of chaos: 1.99900000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.501, 0.750167554963, 2.37519035061, 0.45165183905, 4.86911039757, 1.05168417411, 1.37940949641, 0.865548409792, 1.89978484747, 0.530018609522, 3.92618792133, 0.728193912097, 2.48487872044, 0.450741742957, 4.8820668368, 1.05652725887, 1.3687549675, 0.877150351232, 1.85974139431, 0.543998204656, 3.78772318473, 0.686894128453, 2.7119323516, 0.463188459785, 4.70943953254, 0.992738243581, 1.51931374758, 0.735232569418, 2.44895783211, 0.450489800548, 4.88566315203, 1.05787309027, 1.36581546823, 0.88039772107, 1.84874970739, 0.548093163269, 3.74856860498, 0.675589783365, 2.779359379, 0.47006511287, 4.61812711709, 0.959677552257, 1.60688876338, 0.671406220223, 2.80493695966, 0.473011233605, 4.57984970266, 0.9459681813, 1.6453381164, 0.647002493495, 2.96126853924, 0.494713617287, 4.31247126834, 0.852892442837, 1.94488857326, 0.515949107342, 4.07350152408, 0.774217897813, 2.26298707879, 0.458174992956, 4.77780408102, 1.01780556044, 1.457420574, 0.788141500008, 2.20149383516, 0.464457978627, 4.69237090968, 0.986521103651, 1.53524972975, 0.722704349054, 2.51342723575, 0.451302975597, 4.87407054377, 1.05353723745, 1.37531865622, 0.86997199077, 1.8843728766, 0.535228033279, 3.87370159328, 0.712303843101, 2.56883774784, 0.453262192371, 4.84631681605, 1.04318521, 1.39840024802, 0.845508668005, 1.97191429205, 0.508321442685, 4.15693811668, 0.801154086067, 2.14617505754, 0.471923175709, 4.59392847234, 0.951000089484, 1.63107388041, 0.655815979637, 2.90336194207, 0.485965852444, 4.41725618176, 0.888782723657, 1.82079450609, 0.559029640974, 3.64695656188, 0.647081991485, 2.96073863295, 0.494629982787, 4.31345480176, 0.853225589194, 1.94368174283, 0.516303432129, 4.06968833719, 0.773001353972, 2.26847706155, 0.457711678443, 4.78420029075, 1.02016419322, 1.45178519122, 0.793314004331, 2.17926174332, 0.467243850434, 4.65525217474, 0.973059490199, 1.57058893132, 0.696405978718, 2.65703504663, 0.458595592423, 4.77200916363, 1.01567060007, 1.46254883647, 0.783487974453, 2.22177531613, 0.462159208127, 4.72334916496, 0.997817125457, 1.50647151245, 0.745644140122, 2.3971890337, 0.451060278551, 4.87752589895, 1.05482887883, 1.37247764265, 0.873066777112, 1.87369666798, 0.538961284269, 3.83674316889, 0.701284750883, 2.62950524957, 0.456651659115, 4.79888479232, 1.02558763924, 1.43894607889, 0.80533225968, 2.12883665907, 0.474638706879, 4.55891638128, 0.938509303909, 1.66681326624, 0.634248319504, 3.04813275062, 0.509275478208, 4.14635922163, 0.797705923814, 2.16063736934, 0.469797863346, 4.62162401486, 0.96093447831, 1.60342827575, 0.673705217037, 2.79083875756, 0.471365035125, 4.60117664938, 0.95359545865, 1.6237858659, 0.660427181773, 2.87372801609, 0.481804138497, 4.46849864719, 0.906615757124, 1.76331720961, 0.583997413075, 3.43000776314, 0.590637470521, 3.37557657062, 0.577525346147, 3.48433988084, 0.604165165511, 3.26861216184, 0.553126810777, 3.70127446323, 0.662168855195, 2.86265087823, 0.480305136121, 4.4871819656, 0.913162042097, 1.74286869366, 0.593734414306, 3.35063141909, 0.571669338388, 3.53462355439, 0.61705845526, 3.17130524646, 0.532635354899, 3.89968881543, 0.720136559471, 2.52694482154, 0.45167680654, 4.86875571353, 1.05155171648, 1.37970258832, 0.865232951044, 1.9008908048, 0.529652836738, 3.92991389589, 0.729332451702, 2.47901671917, 0.450665587167, 4.88315347266, 1.05693383504, 1.36786597388, 0.878130315691, 1.85641458389, 0.545225624291, 3.77592206971, 0.683468957586, 2.7321072026, 0.465107146617, 4.68368022008, 0.983362062658, 1.5434393572, 0.716430642044, 2.5466409055, 0.452342840406, 4.85930910874, 1.04802626975, 1.38753680112, 0.856873216501, 1.93053798734, 0.520239969466, 4.0276895989, 0.759688692292, 2.3298303327, 0.45356257908, 4.84208356749, 1.04160977719, 1.40196218521, 0.841839094229, 1.98554592247, 0.504692434842, 4.19756054062, 0.81448109975, 2.09156727277, 0.481114742565, 4.4770761251, 0.90961824796, 1.75389599364, 0.588425705234, 3.39356252244, 0.581807997184, 3.4482442595, 0.595130103151, 3.3394796081, 0.569083364143, 3.55717706807, 0.62295391692, 3.12824248865, 0.524126184282, 3.98687558227, 0.746906720275, 2.39101917561, 0.451204669552, 4.87546969464, 1.05406017757, 1.37416740033, 0.871223832353, 1.88004388922, 0.536729373146, 3.85877370739, 0.707835915882, 2.59318834333, 0.454471190535, 4.8293140849, 1.03686327324, 1.4127735819, 0.830869306783, 2.02711081571, 0.494494233474, 4.31505193855, 0.853766731779, 1.94172372571, 0.516880837633, 4.06348611287, 0.771025387583, 2.27743515667, 0.45698889173, 4.79420543619, 1.02385813053, 1.44302251332, 0.801480900101, 2.14481155707, 0.472130058116, 4.59124635777, 0.950040522722, 1.63378030238, 0.654122351829, 2.91435912402, 0.487565396883, 4.39780344023, 0.882060447938, 1.84315767166, 0.550220380626, 3.72847092728, 0.669854852796, 2.81450959607, 0.474159850975, 4.56506014921, 0.94069556683, 1.66047730181, 0.637948081218, 3.02255108147, 0.50481459769, 4.19618314908, 0.814027003577, 2.09339492872, 0.480776315772, 4.48129620158, 0.91109726573, 1.74928159656, 0.590630543727, 3.37563267571, 0.577538627336, 3.48422706594, 0.604136636006, 3.26883238726, 0.55317508873, 3.70082525178, 0.662042632955, 2.86345154569, 0.480412437742, 4.4858405462, 0.912691257957, 1.74432791773, 0.593024066308, 3.35632864861, 0.572998127273, 3.52311826937, 0.614077525675, 3.19341540433, 0.537140760556, 3.85469856324, 0.706620254249, 2.59987198579, 0.454838960578, 4.82416052276, 1.03495011539, 1.41716549253, 0.826484417819, 2.04407470825, 0.490694083055, 4.36013586352, 0.869119885621, 1.88732770193, 0.534212924467, 3.88384479914, 0.715352853193, 2.55241101032, 0.45256479795, 4.85616739751, 1.04685482451, 1.39015428265, 0.854110992271, 1.94047955711, 0.517249366239, 4.05953512441, 0.769768434979, 2.28316015617, 0.45654834785, 4.80031974692, 1.02611824252, 1.43769879132, 0.806517352896, 2.12395545573, 0.475436728315, 4.5487062581, 0.934881317895, 1.67740413118, 0.628179835635, 3.09079239952, 0.517021079149, 4.0619818913, 0.770546675409, 2.27961308693, 0.45681934283, 4.79655715999, 1.02472718822, 1.44097206833, 0.803413967355, 2.13677201354, 0.473372946411, 4.57518425956, 0.944303425719, 1.65009666328, 0.644123640053, 2.98055273241, 0.497801084718, 4.27640408383, 0.84072601888, 1.98970743156, 0.503613073973, 4.20976070486, 0.818509931716, 2.07545178724, 0.48419430606, 4.43895687804, 0.896312893472, 1.7962019058, 0.56929180968, 3.55535108765, 0.622474067808, 3.13171491282, 0.52479905822, 3.97987320224, 0.744729460227, 2.4016732102, 0.45096574859, 4.87887279388, 1.055332527, 1.37137216259, 0.874276060033, 1.8695484702, 0.540439772164, 3.8222547183, 0.69700440715, 2.65363567709, 0.45834228314, 4.77549787741, 1.01695568845, 1.45945888476, 0.786285807362, 2.20954942878, 0.463517592157, 4.70500505115, 0.991121394714, 1.52343512061, 0.731951405195, 2.46560845159, 0.450542430138, 4.88491155463, 1.05759176861, 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0.450535944207, 4.88500416968, 1.0576264327, 1.36635351758, 0.879801801216, 1.85075979821, 0.547335791331, 3.75576386548, 0.677654195175, 2.76686380947, 0.468691993624, 4.63613807522, 0.966159342369, 1.58915754855, 0.683371462129, 2.7326846793, 0.465163841971, 4.68292240989, 0.983086809518, 1.54415590831, 0.715886956702, 2.54954923239, 0.452453207802, 4.85774650799, 1.04744356053, 1.38883791315, 0.855498236954, 1.93547756093, 0.518743833769, 4.04357319025, 0.764704660867, 2.30643551996, 0.45492613619, 4.82294019602, 1.03449730093, 1.41820788409, 0.825449677785, 2.04810750417, 0.489820613147, 4.37060150878, 0.872705130073, 1.87493978278, 0.538521278401, 3.84107115371, 0.702567710846, 2.62233491767, 0.456186088868, 4.8053567079, 1.02798164346, 1.43333087098, 0.810692286957, 2.10688708641, 0.478345193821, 4.51179409275, 0.921821007516, 1.71633839761, 0.607076257329, 3.24625667519, 0.548269679508, 3.74689465455, 0.675110350014, 2.78227315376, 0.470391598658, 4.61386072481, 0.958145047877, 1.6111224695, 0.668617072707, 2.82218169957, 0.475098238876, 4.55303263993, 0.936417812058, 1.67290704309, 0.630739055486, 3.0726946259, 0.513689047243, 4.09795346911, 0.78204946857, 2.22809920344, 0.461488752799, 4.73244451656, 1.00114411188, 1.49814364279, 0.752550031303, 2.36372093315, 0.452045825103, 4.86351822653, 1.04959652607, 1.38403943045, 0.860588018552, 1.91728258375, 0.524355865213, 3.98448325029, 0.746162345738, 2.39465393044, 0.451117592653, 4.87670955272, 1.05452366587, 1.37314819927, 0.872334638665, 1.87621452327, 0.538071539328, 3.84550256307, 0.703883384112, 2.61501132989, 0.455728227887, 4.81173481719, 1.03034316401, 1.42782286946, 0.816012393521, 2.0854209471, 0.48226895939, 4.46272973383, 0.904599167205, 1.76968556842, 0.581059185096, 3.45451483906, 0.596686168505, 3.32711171431, 0.566238835546, 3.58223712301, 0.629584060572, 3.08084278026, 0.515180914381, 4.08178754438, 0.776865881193, 2.25110336869, 0.459231490046, 4.76326880785, 1.01245399233, 1.47032460132, 0.776528079278, 2.25261435504, 0.459093061458, 4.76516935964, 1.01315306944, 1.46862959817, 0.778035444058, 2.24588318794, 0.459718991525, 4.7565851136, 1.00999713337, 1.47630402685, 0.7712537658, 2.27639719331, 0.457070543087, 4.79307353795, 1.02343995537, 1.44401065504, 0.800552324507, 2.14868892737, 0.47154470551, 4.59884145235, 0.952758935934, 1.62612978334, 0.658936088749, 2.88326163274, 0.483119118206, 4.45220825017, 0.900927053266, 1.78136686245, 0.575783468806, 3.49918373223, 0.607934672201, 3.2397080857, 0.546863399446, 3.7602623533, 0.678947849566, 2.75907544464, 0.467858478927, 4.64712466848, 0.970122807439, 1.57845339471, 0.690821298409, 2.68906772551, 0.461162142229, 4.73688522617, 1.00277018105, 1.49409750549, 0.755949836157, 2.34749221448, 0.452705617178, 4.85417581711, 1.04611249461, 1.39181666071, 0.852364637706, 1.94680275744, 0.515389539197, 4.07953468162, 0.776145327304, 2.25432816989, 0.458937501198, 4.76730653824, 1.01393942388, 1.46672634902, 0.779734488662, 2.2383307638, 0.460449869125, 4.74659218985, 1.00632848364, 1.48529818444, 0.763445542138, 2.31227613724, 0.454560963612, 4.82805529784, 1.03639584272, 1.41384480852, 0.829796022866, 2.0312443343, 0.493549363618, 4.32619402777, 0.857547187581, 1.92812335628, 0.520978725785, 4.01988188742, 0.75723152737, 2.34141594179, 0.452983948587, 4.85024319839, 1.04464728488, 1.39510634214, 0.8489268805, 1.95933669708, 0.511798758374, 4.11857831239, 0.788695847159, 2.19909560503, 0.464744972087, 4.68852569949, 0.98512283821, 1.53886692177, 0.7199197729, 2.52809088206, 0.451711640924, 4.86826092985, 1.05136694921, 1.38011157813, 0.8647930778, 1.90243448221, 0.529144083029, 3.93510534297, 0.730921080594, 2.47087082432, 0.450582250785, 4.88434300109, 1.05737897929, 1.36689361302, 0.879204300019, 1.85277836427, 0.546579074382, 3.76297386597, 0.679728717504, 2.75438981787, 0.467365359878, 4.65364364184, 0.972477972292, 1.57214176593, 0.695295742208, 2.66335860192, 0.459076624189, 4.76539511362, 1.0132361215, 1.46842841517, 0.778214716845, 2.24508457109, 0.459794843079, 4.75554650179, 1.00961557341, 1.4772358025, 0.770437892837, 2.28010841875, 0.456781117504, 4.79708761329, 1.02492325389, 1.44051005579, 0.803850694573, 2.13496168221, 0.473658285271, 4.57150912521, 0.942992998937, 1.65385627985, 0.641870522071, 2.99577507272, 0.500297831584, 4.2475770644, 0.831073539985, 2.0263256222, 0.494675107413, 4.31292409756, 0.853045817969, 1.94433283503, 0.516112124111, 4.07174648309, 0.773657816043, 2.26551221683, 0.457959958526, 4.78077103457, 1.01889935936, 1.45480328284, 0.790536113746, 2.19116110203, 0.4657176707, 4.67552980628, 0.980403417769, 1.55116656175, 0.710611562543, 2.57802149508, 0.453693829983, 4.84023571497, 1.04092238286, 1.40352045868, 0.840242430931, 1.99151934053, 0.503147250312, 4.21504285884, 0.820257964604, 2.06851498702, 0.485573308878, 4.42205046167, 0.890443540622, 1.81532925792, 0.561257395211, 3.62676984959, 0.641567272839, 2.99783262926, 0.500639293865, 4.24365791675, 0.82976624193, 2.03135920134, 0.493523281502, 4.32650222391, 0.857651888087, 1.92774863467, 0.521093808841, 4.01866769361, 0.756849913504, 2.34322272758, 0.452899389631, 4.85143742516, 1.04509214237, 1.39410635805, 0.849969338563, 1.9555237525, 0.512877734553, 4.10678618319, 0.78489131239, 2.21563090993, 0.462831589401, 4.71425512611, 0.99449532843, 1.51485308632, 0.738816606784, 2.43095590404, 0.450560394657, 4.88465504691, 1.05749576485, 1.36663867642, 0.879486247915, 1.85182545425, 0.546935820075, 3.75957217618, 0.678749223099, 2.76026917335, 0.467985111604, 4.64545290083, 0.969519242647, 1.58007675093, 0.689680323231, 2.69568152862, 0.461731871123, 4.72914322488, 0.999935990372, 1.50116003636, 0.75003449827, 2.37583326682, 0.45163150566, 4.86939928034, 1.05179206293, 1.37917083534, 0.86580542895, 1.89888445251, 0.530317187071, 3.92315043378, 0.727266770436, 2.48966717193, 0.450813927845, 4.88103721091, 1.05614206976, 1.36959797334, 0.876222788693, 1.86289817405, 0.542843064427, 3.79888064461, 0.690146628096, 2.69297566475, 0.461497156436, 4.73233034381, 1.00110231958, 1.49824784236, 0.752462863493, 2.36413915245, 0.452030414208, 4.86373677651, 1.04967808373, 1.38385813028, 0.860781343025, 1.91659632137, 0.524573000541, 3.98222361084, 0.745459757011, 2.39809198764, 0.451040540928, 4.87780707903, 1.05493401354, 1.37224677072, 0.873319093021, 1.87283006319, 0.539268856436, 3.83372223534, 0.700390423613, 2.63452048134, 0.456987390923, 4.79422624517, 1.02386581897, 1.4430043548, 0.801497982246, 2.14474032292, 0.472140897451, 4.59110589916, 0.949990283828, 1.63392217661, 0.654033848039, 2.91493548947, 0.487650040822, 4.39677775057, 0.881706738547, 1.84434522835, 0.549766135759, 3.73274882382, 0.671071671381, 2.80699723041, 0.473256349448, 4.57668734107, 0.944839616932, 1.64856186034, 0.6450488577, 2.97433492822, 0.496796308003, 4.28809037388, 0.844657307587, 1.97506492962, 0.507469870365, 4.16641595349, 0.804251302588, 2.13330301047, 0.473921493735, 4.56812310426, 0.941786410792, 1.6573288473, 0.639806089189, 3.00982372024, 0.502647974539, 4.22071568874, 0.822137773945, 2.0610923172, 0.487085037167, 4.40363132462, 0.884071602885, 1.83642609948, 0.552820849612, 3.70412325972, 0.662969883353, 2.85757737096, 0.479629025933, 4.49564864058, 0.916136256754, 1.73369019238, 0.598258069156, 3.31468739103, 0.563406419715, 3.60745616797, 0.636339019868, 3.03363759429, 0.506730592558, 4.17467096572, 0.80695499195, 2.12215695756, 0.475734518363, 4.54490536279, 0.933532432352, 1.68136640406, 0.625946282795, 3.1067166762, 0.520008029616, 4.03014569944, 0.760462810711, 2.32619749409, 0.453757087586, 4.83934552045, 1.04059129824, 1.40427190029, 0.839474361093, 1.99440201064, 0.502411281882, 4.22340913136, 0.823031201506, 2.05757789549, 0.487813942364, 4.39479269495, 0.881022402498, 1.84664595321, 0.548889935305, 3.74102159247, 0.673430786764, 2.79251618207, 0.471558042907, 4.59866817832, 0.952696878486, 1.62630386016, 0.658825655262, 2.88396956307, 0.483217673871, 4.45099102621, 0.900502713374, 1.78272382806, 0.575180056011, 3.50434806524, 0.609253378664, 3.22968637461, 0.544725949292, 3.78071947914, 0.684859499749, 2.7238902693, 0.4643110868, 4.6943409109, 0.987237804189, 1.53340042694, 0.724136345411, 2.50593438707, 0.451125433383, 4.87659789095, 1.05448192079, 1.37323995074, 0.872234542094, 1.87655913584, 0.537950210747, 3.8466993822, 0.70423907208, 2.61303654283, 0.455607841089, 4.81341403794, 1.03096526693, 1.42637697833, 0.817419293907, 2.07979673979, 0.483347020095, 4.44939430334, 0.899946228166, 1.78450561426, 0.574390695422, 3.51112117835, 0.610988519062, 3.21657015523, 0.541955548093, 3.80748711712, 0.692664806281, 2.67843161996, 0.460274271672, 4.74899004363, 1.00720828557, 1.48313404864, 0.765310688486, 2.30363195653, 0.455107341039, 4.82040515143, 1.03355690191, 1.42037626097, 0.823304534032, 2.05650442134) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: c value chosen from interval of chaos: 1.99900000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.501, 0.750167554963, 2.37519035061, 0.45165183905, 4.86911039757, 1.05168417411, 1.37940949641, 0.865548409792, 1.89978484747, 0.530018609522, 3.92618792133, 0.728193912097, 2.48487872044, 0.450741742957, 4.8820668368, 1.05652725887, 1.3687549675, 0.877150351232, 1.85974139431, 0.543998204656, 3.78772318473, 0.686894128453, 2.7119323516, 0.463188459785, 4.70943953254, 0.992738243581, 1.51931374758, 0.735232569418, 2.44895783211, 0.450489800548, 4.88566315203, 1.05787309027, 1.36581546823, 0.88039772107, 1.84874970739, 0.548093163269, 3.74856860498, 0.675589783365, 2.779359379, 0.47006511287, 4.61812711709, 0.959677552257, 1.60688876338, 0.671406220223, 2.80493695966, 0.473011233605, 4.57984970266, 0.9459681813, 1.6453381164, 0.647002493495, 2.96126853924, 0.494713617287, 4.31247126834, 0.852892442837, 1.94488857326, 0.515949107342, 4.07350152408, 0.774217897813, 2.26298707879, 0.458174992956, 4.77780408102, 1.01780556044, 1.457420574, 0.788141500008, 2.20149383516, 0.464457978627, 4.69237090968, 0.986521103651, 1.53524972975, 0.722704349054, 2.51342723575, 0.451302975597, 4.87407054377, 1.05353723745, 1.37531865622, 0.86997199077, 1.8843728766, 0.535228033279, 3.87370159328, 0.712303843101, 2.56883774784, 0.453262192371, 4.84631681605, 1.04318521, 1.39840024802, 0.845508668005, 1.97191429205, 0.508321442685, 4.15693811668, 0.801154086067, 2.14617505754, 0.471923175709, 4.59392847234, 0.951000089484, 1.63107388041, 0.655815979637, 2.90336194207, 0.485965852444, 4.41725618176, 0.888782723657, 1.82079450609, 0.559029640974, 3.64695656188, 0.647081991485, 2.96073863295, 0.494629982787, 4.31345480176, 0.853225589194, 1.94368174283, 0.516303432129, 4.06968833719, 0.773001353972, 2.26847706155, 0.457711678443, 4.78420029075, 1.02016419322, 1.45178519122, 0.793314004331, 2.17926174332, 0.467243850434, 4.65525217474, 0.973059490199, 1.57058893132, 0.696405978718, 2.65703504663, 0.458595592423, 4.77200916363, 1.01567060007, 1.46254883647, 0.783487974453, 2.22177531613, 0.462159208127, 4.72334916496, 0.997817125457, 1.50647151245, 0.745644140122, 2.3971890337, 0.451060278551, 4.87752589895, 1.05482887883, 1.37247764265, 0.873066777112, 1.87369666798, 0.538961284269, 3.83674316889, 0.701284750883, 2.62950524957, 0.456651659115, 4.79888479232, 1.02558763924, 1.43894607889, 0.80533225968, 2.12883665907, 0.474638706879, 4.55891638128, 0.938509303909, 1.66681326624, 0.634248319504, 3.04813275062, 0.509275478208, 4.14635922163, 0.797705923814, 2.16063736934, 0.469797863346, 4.62162401486, 0.96093447831, 1.60342827575, 0.673705217037, 2.79083875756, 0.471365035125, 4.60117664938, 0.95359545865, 1.6237858659, 0.660427181773, 2.87372801609, 0.481804138497, 4.46849864719, 0.906615757124, 1.76331720961, 0.583997413075, 3.43000776314, 0.590637470521, 3.37557657062, 0.577525346147, 3.48433988084, 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0.452650695618, 4.85495241001, 1.04640193252, 1.39116814853, 0.853045177234, 1.94433515618, 0.516111442712, 4.07175381662, 0.773660155803, 2.26550165953, 0.457960850702, 4.78075871874, 1.018894818, 1.45481413557, 0.790526156846, 2.19120392133, 0.465712325427, 4.67560106875, 0.980429269566, 1.551098803, 0.710662169944, 2.57774616358, 0.453680458321, 4.84042392099, 1.04099238654, 1.40336165048, 0.840404910097, 1.99091029939, 0.503303550106, 4.21326937723, 0.819670813391, 2.07084129075, 0.485107231546, 4.42775332444, 0.89242118202, 1.80885200029, 0.563936480558, 3.60271630309, 0.635063185086, 3.04247099015, 0.50827612117, 4.15744170038, 0.801318461462, 2.14548911146, 0.472027112363, 4.5925806866, 0.950517843199, 1.63243323381, 0.654964059109, 2.90888603536, 0.486765648932, 4.40751287808, 0.885412419632, 1.83195775619, 0.55457122424, 3.68787029461, 0.658412780829, 2.88661854909, 0.483587568155, 4.44642720552, 0.898912605202, 1.78782189592, 0.572930445791, 3.52370292337, 0.614228568451, 3.19228958002, 0.536909158122, 3.85699197643, 0.70730419169, 2.59610865053, 0.454629997873, 4.8270876581, 1.03603658254, 1.41466893773, 0.828971969264, 2.03442620364, 0.492830364728, 4.33470244125, 0.860440248802, 1.9178073734, 0.524190090855, 3.98620971763, 0.746699481907, 2.39203031084, 0.451179864958, 4.87582282828, 1.05419217918, 1.37387701952, 0.871540067523, 1.87895256993, 0.537110520448, 3.85499789407, 0.706709488893, 2.59938052541, 0.454811396827, 4.82454647442, 1.03509334318, 1.41683600922, 0.826811958798, 2.04280052314, 0.490972432969, 4.35680891192, 0.867981820872, 1.89128449615, 0.532865794085, 3.89736830161, 0.719434343199, 2.5306598889, 0.451791496656, 4.86712696606, 1.0509435407, 1.38104947489, 0.863785800171, 1.90597615684, 0.527984683378, 3.94697535075, 0.734563362048, 2.45234053149, 0.450491399771, 4.88564031103, 1.05786454049, 1.36583411323, 0.880377059295, 1.84881934911, 0.548066859231, 3.74881815035, 0.675661282587, 2.77892522317, 0.470016669027, 4.61876068713, 0.959905227665, 1.60626114799, 0.671821891661, 2.80238019778, 0.47270865393, 4.58375813205, 0.947363860723, 1.64136384043, 0.649430239087, 2.94514892389, 0.492198673326, 4.34219906484, 0.862993698073, 1.90876788867, 0.527078448316, 3.95629158016, 0.737431659364, 2.43788941825, 0.45051734197, 4.88526981334, 1.05772586062, 1.36613659221, 0.880041976265, 1.84994929032, 0.547640717684, 3.7528644544, 0.676821610019, 2.77189339066, 0.469239401864, 4.62894470323, 0.96356823358, 1.59621178642, 0.678555741612, 2.76143270898, 0.468108932605, 4.64381915982, 0.968929568362, 1.58166506329, 0.688567857998, 2.70215299449, 0.462302308336, 4.72141141938, 0.997108925704, 1.50825283338, 0.744182849146, 2.40435874475, 0.450913308316, 4.87962023278, 1.05561205935, 1.37075916335, 0.874947843859, 1.8672497296, 0.541265879357, 3.8141955035, 0.69463320785, 2.66714266363, 0.459370556434, 4.76136069478, 1.0117523357, 1.4720286999, 0.775018092813, 2.25938642052, 0.458487328501, 4.77349974465, 1.01621958228, 1.46122767226, 0.784681990362, 2.21654585632, 0.462730151455, 4.71562533134, 0.994995523554, 1.51358670337, 0.739840361826, 2.42584899366, 0.450604936478, 4.88401914451, 1.05725777875, 1.36715826045, 0.878911773913, 1.85376776767, 0.546209572909, 3.7665021167, 0.680746030886, 2.74830285989, 0.466734182262, 4.66200859247, 0.97550375371, 1.56408601943, 0.701096085267, 2.6305620937, 0.456721718136, 4.79791208156, 1.02522802347, 1.43979231679, 0.80453000268, 2.13215017654, 0.474105433978, 4.56575914049, 0.940944454241, 1.65975819499, 0.638371302719, 3.0196450212, 0.504316771991, 4.20180052248, 0.815879895435, 2.08595174416, 0.482168298342, 4.46397806115, 0.905035344177, 1.76830534232, 0.581692249594, 3.44921241616, 0.595369979642, 3.33756854556, 0.568642216534, 3.56104618243, 0.623972148687, 3.12089297581, 0.522709826076, 4.00167690459, 0.751524164442, 2.36864969135, 0.451869244771, 4.86602332939, 1.05053152022, 1.38196303551, 0.862806585372, 1.90942821445, 0.526865081792, 3.95848990715, 0.738109712704, 2.4344912787, 0.450535944207, 4.88500416968, 1.0576264327, 1.36635351758, 0.879801801216, 1.85075979821, 0.547335791331, 3.75576386548, 0.677654195175, 2.76686380947, 0.468691993624, 4.63613807522, 0.966159342369, 1.58915754855, 0.683371462129, 2.7326846793, 0.465163841971, 4.68292240989, 0.983086809518, 1.54415590831, 0.715886956702, 2.54954923239, 0.452453207802, 4.85774650799, 1.04744356053, 1.38883791315, 0.855498236954, 1.93547756093, 0.518743833769, 4.04357319025, 0.764704660867, 2.30643551996, 0.45492613619, 4.82294019602, 1.03449730093, 1.41820788409, 0.825449677785, 2.04810750417, 0.489820613147, 4.37060150878, 0.872705130073, 1.87493978278, 0.538521278401, 3.84107115371, 0.702567710846, 2.62233491767, 0.456186088868, 4.8053567079, 1.02798164346, 1.43333087098, 0.810692286957, 2.10688708641, 0.478345193821, 4.51179409275, 0.921821007516, 1.71633839761, 0.607076257329, 3.24625667519, 0.548269679508, 3.74689465455, 0.675110350014, 2.78227315376, 0.470391598658, 4.61386072481, 0.958145047877, 1.6111224695, 0.668617072707, 2.82218169957, 0.475098238876, 4.55303263993, 0.936417812058, 1.67290704309, 0.630739055486, 3.0726946259, 0.513689047243, 4.09795346911, 0.78204946857, 2.22809920344, 0.461488752799, 4.73244451656, 1.00114411188, 1.49814364279, 0.752550031303, 2.36372093315, 0.452045825103, 4.86351822653, 1.04959652607, 1.38403943045, 0.860588018552, 1.91728258375, 0.524355865213, 3.98448325029, 0.746162345738, 2.39465393044, 0.451117592653, 4.87670955272, 1.05452366587, 1.37314819927, 0.872334638665, 1.87621452327, 0.538071539328, 3.84550256307, 0.703883384112, 2.61501132989, 0.455728227887, 4.81173481719, 1.03034316401, 1.42782286946, 0.816012393521, 2.0854209471, 0.48226895939, 4.46272973383, 0.904599167205, 1.76968556842, 0.581059185096, 3.45451483906, 0.596686168505, 3.32711171431, 0.566238835546, 3.58223712301, 0.629584060572, 3.08084278026, 0.515180914381, 4.08178754438, 0.776865881193, 2.25110336869, 0.459231490046, 4.76326880785, 1.01245399233, 1.47032460132, 0.776528079278, 2.25261435504, 0.459093061458, 4.76516935964, 1.01315306944, 1.46862959817, 0.778035444058, 2.24588318794, 0.459718991525, 4.7565851136, 1.00999713337, 1.47630402685, 0.7712537658, 2.27639719331, 0.457070543087, 4.79307353795, 1.02343995537, 1.44401065504, 0.800552324507, 2.14868892737, 0.47154470551, 4.59884145235, 0.952758935934, 1.62612978334, 0.658936088749, 2.88326163274, 0.483119118206, 4.45220825017, 0.900927053266, 1.78136686245, 0.575783468806, 3.49918373223, 0.607934672201, 3.2397080857, 0.546863399446, 3.7602623533, 0.678947849566, 2.75907544464, 0.467858478927, 4.64712466848, 0.970122807439, 1.57845339471, 0.690821298409, 2.68906772551, 0.461162142229, 4.73688522617, 1.00277018105, 1.49409750549, 0.755949836157, 2.34749221448, 0.452705617178, 4.85417581711, 1.04611249461, 1.39181666071, 0.852364637706, 1.94680275744, 0.515389539197, 4.07953468162, 0.776145327304, 2.25432816989, 0.458937501198, 4.76730653824, 1.01393942388, 1.46672634902, 0.779734488662, 2.2383307638, 0.460449869125, 4.74659218985, 1.00632848364, 1.48529818444, 0.763445542138, 2.31227613724, 0.454560963612, 4.82805529784, 1.03639584272, 1.41384480852, 0.829796022866, 2.0312443343, 0.493549363618, 4.32619402777, 0.857547187581, 1.92812335628, 0.520978725785, 4.01988188742, 0.75723152737, 2.34141594179, 0.452983948587, 4.85024319839, 1.04464728488, 1.39510634214, 0.8489268805, 1.95933669708, 0.511798758374, 4.11857831239, 0.788695847159, 2.19909560503, 0.464744972087, 4.68852569949, 0.98512283821, 1.53886692177, 0.7199197729, 2.52809088206, 0.451711640924, 4.86826092985, 1.05136694921, 1.38011157813, 0.8647930778, 1.90243448221, 0.529144083029, 3.93510534297, 0.730921080594, 2.47087082432, 0.450582250785, 4.88434300109, 1.05737897929, 1.36689361302, 0.879204300019, 1.85277836427, 0.546579074382, 3.76297386597, 0.679728717504, 2.75438981787, 0.467365359878, 4.65364364184, 0.972477972292, 1.57214176593, 0.695295742208, 2.66335860192, 0.459076624189, 4.76539511362, 1.0132361215, 1.46842841517, 0.778214716845, 2.24508457109, 0.459794843079, 4.75554650179, 1.00961557341, 1.4772358025, 0.770437892837, 2.28010841875, 0.456781117504, 4.79708761329, 1.02492325389, 1.44051005579, 0.803850694573, 2.13496168221, 0.473658285271, 4.57150912521, 0.942992998937, 1.65385627985, 0.641870522071, 2.99577507272, 0.500297831584, 4.2475770644, 0.831073539985, 2.0263256222, 0.494675107413, 4.31292409756, 0.853045817969, 1.94433283503, 0.516112124111, 4.07174648309, 0.773657816043, 2.26551221683, 0.457959958526, 4.78077103457, 1.01889935936, 1.45480328284, 0.790536113746, 2.19116110203, 0.4657176707, 4.67552980628, 0.980403417769, 1.55116656175, 0.710611562543, 2.57802149508, 0.453693829983, 4.84023571497, 1.04092238286, 1.40352045868, 0.840242430931, 1.99151934053, 0.503147250312, 4.21504285884, 0.820257964604, 2.06851498702, 0.485573308878, 4.42205046167, 0.890443540622, 1.81532925792, 0.561257395211, 3.62676984959, 0.641567272839, 2.99783262926, 0.500639293865, 4.24365791675, 0.82976624193, 2.03135920134, 0.493523281502, 4.32650222391, 0.857651888087, 1.92774863467, 0.521093808841, 4.01866769361, 0.756849913504, 2.34322272758, 0.452899389631, 4.85143742516, 1.04509214237, 1.39410635805, 0.849969338563, 1.9555237525, 0.512877734553, 4.10678618319, 0.78489131239, 2.21563090993, 0.462831589401, 4.71425512611, 0.99449532843, 1.51485308632, 0.738816606784, 2.43095590404, 0.450560394657, 4.88465504691, 1.05749576485, 1.36663867642, 0.879486247915, 1.85182545425, 0.546935820075, 3.75957217618, 0.678749223099, 2.76026917335, 0.467985111604, 4.64545290083, 0.969519242647, 1.58007675093, 0.689680323231, 2.69568152862, 0.461731871123, 4.72914322488, 0.999935990372, 1.50116003636, 0.75003449827, 2.37583326682, 0.45163150566, 4.86939928034, 1.05179206293, 1.37917083534, 0.86580542895, 1.89888445251, 0.530317187071, 3.92315043378, 0.727266770436, 2.48966717193, 0.450813927845, 4.88103721091, 1.05614206976, 1.36959797334, 0.876222788693, 1.86289817405, 0.542843064427, 3.79888064461, 0.690146628096, 2.69297566475, 0.461497156436, 4.73233034381, 1.00110231958, 1.49824784236, 0.752462863493, 2.36413915245, 0.452030414208, 4.86373677651, 1.04967808373, 1.38385813028, 0.860781343025, 1.91659632137, 0.524573000541, 3.98222361084, 0.745459757011, 2.39809198764, 0.451040540928, 4.87780707903, 1.05493401354, 1.37224677072, 0.873319093021, 1.87283006319, 0.539268856436, 3.83372223534, 0.700390423613, 2.63452048134, 0.456987390923, 4.79422624517, 1.02386581897, 1.4430043548, 0.801497982246, 2.14474032292, 0.472140897451, 4.59110589916, 0.949990283828, 1.63392217661, 0.654033848039, 2.91493548947, 0.487650040822, 4.39677775057, 0.881706738547, 1.84434522835, 0.549766135759, 3.73274882382, 0.671071671381, 2.80699723041, 0.473256349448, 4.57668734107, 0.944839616932, 1.64856186034, 0.6450488577, 2.97433492822, 0.496796308003, 4.28809037388, 0.844657307587, 1.97506492962, 0.507469870365, 4.16641595349, 0.804251302588, 2.13330301047, 0.473921493735, 4.56812310426, 0.941786410792, 1.6573288473, 0.639806089189, 3.00982372024, 0.502647974539, 4.22071568874, 0.822137773945, 2.0610923172, 0.487085037167, 4.40363132462, 0.884071602885, 1.83642609948, 0.552820849612, 3.70412325972, 0.662969883353, 2.85757737096, 0.479629025933, 4.49564864058, 0.916136256754, 1.73369019238, 0.598258069156, 3.31468739103, 0.563406419715, 3.60745616797, 0.636339019868, 3.03363759429, 0.506730592558, 4.17467096572, 0.80695499195, 2.12215695756, 0.475734518363, 4.54490536279, 0.933532432352, 1.68136640406, 0.625946282795, 3.1067166762, 0.520008029616, 4.03014569944, 0.760462810711, 2.32619749409, 0.453757087586, 4.83934552045, 1.04059129824, 1.40427190029, 0.839474361093, 1.99440201064, 0.502411281882, 4.22340913136, 0.823031201506, 2.05757789549, 0.487813942364, 4.39479269495, 0.881022402498, 1.84664595321, 0.548889935305, 3.74102159247, 0.673430786764, 2.79251618207, 0.471558042907, 4.59866817832, 0.952696878486, 1.62630386016, 0.658825655262, 2.88396956307, 0.483217673871, 4.45099102621, 0.900502713374, 1.78272382806, 0.575180056011, 3.50434806524, 0.609253378664, 3.22968637461, 0.544725949292, 3.78071947914, 0.684859499749, 2.7238902693, 0.4643110868, 4.6943409109, 0.987237804189, 1.53340042694, 0.724136345411, 2.50593438707, 0.451125433383, 4.87659789095, 1.05448192079, 1.37323995074, 0.872234542094, 1.87655913584, 0.537950210747, 3.8466993822, 0.70423907208, 2.61303654283, 0.455607841089, 4.81341403794, 1.03096526693, 1.42637697833, 0.817419293907, 2.07979673979, 0.483347020095, 4.44939430334, 0.899946228166, 1.78450561426, 0.574390695422, 3.51112117835, 0.610988519062, 3.21657015523, 0.541955548093, 3.80748711712, 0.692664806281, 2.67843161996, 0.460274271672, 4.74899004363, 1.00720828557, 1.48313404864, 0.765310688486, 2.30363195653, 0.455107341039, 4.82040515143, 1.03355690191, 1.42037626097, 0.823304534032, 2.05650442134) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: |
#PART F - INTERVAL OF STABILITY OF THE EIGHT CYCLE [SLIGHTLY LARGER THAN BEFORE];
c=1.911;
print "c value chosen from interval of stability of the eight cycle:";
print c;
print "";
N=2000;
print "Iterations computed:";
print N;
print "";
eightcycleA=CompIterate(c,N);
print "Values for all iterations:";
print eightcycleA;
print "";
eightcycleA_lyapunov=LyapunovEst(c,N);
print "Estimate of Lyapunov exponent:";
print eightcycleA_lyapunov;
print "";
num_intervals=100;
max_value=VectorMax(eightcycleA,N);
min_value=VectorMin(eightcycleA,N);
eightcycleA_bars=BinIt(eightcycleA,N,max_value,min_value,num_intervals);
print "Graphical approximation of frequency distribution across 100 equal intervals of value range:";
bar_chart(eightcycleA_bars);
c value chosen from interval of stability of the eight cycle: 1.91100000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.589, 0.771479861548, 2.36337027555, 0.540058807672, 3.91398000807, 0.812473224138, 2.18766598804, 0.554157509005, 3.77970204977, 0.772564381256, 2.35845370719, 0.540246732914, 3.91214167684, 0.811914232095, 2.18992868005, 0.553871966392, 3.78235021254, 0.77333275467, 2.35497962494, 0.540386184323, 3.91077839732, 0.811499910444, 2.19160803418, 0.55366193019, 3.78429995568, 0.773898976369, 2.35242444371, 0.540492287925, 3.90974162192, 0.811184943223, 2.19288596813, 0.553503176706, 3.78577468173, 0.774327528728, 2.35049328088, 0.540574475284, 3.90893883454, 0.810941134528, 2.19387595228, 0.5533808331, 3.7869117899, 0.774658134251, 2.34900511267, 0.540638984782, 3.90830890019, 0.810749867093, 2.19465306095, 0.553285187121, 3.78780112922, 0.774916802436, 2.34784174272, 0.540690128723, 3.90780959263, 0.8105982903, 2.19526920134, 0.553209596652, 3.78850421684, 0.775121360318, 2.34692234684, 0.540730990928, 3.90741073484, 0.810477225457, 2.19576150044, 0.553149354254, 3.78906469285, 0.775284465531, 2.34618964843, 0.540763836333, 3.90709017533, 0.810379937994, 2.19615723071, 0.553101028528, 3.78951439295, 0.775415358688, 2.34560190021, 0.540790364179, 3.90683130284, 0.810301379637, 2.1964768556, 0.553062061384, 3.78987706609, 0.775520937352, 2.34512798206, 0.540811871295, 3.90662144484, 0.810237700335, 2.1967359942, 0.553030510903, 3.79017075068, 0.775606443214, 2.34474427116, 0.540829361214, 3.906450798, 0.810185922535, 2.19694673416, 0.553004881046, 3.79040934971, 0.775675917988, 2.34443256962, 0.54084361932, 3.90631169213, 0.810143717089, 2.19711853621, 0.552984005349, 3.7906037075, 0.775732515325, 2.34417868912, 0.540855265957, 3.90619807015, 0.810109245036, 2.19725887353, 0.552966965329, 3.79076236567, 0.775778719956, 2.34397145804, 0.540864794829, 3.90610511244, 0.810081043331, 2.19737369394, 0.552953031904, 3.79089210626, 0.775816505232, 2.34380200877, 0.540872601292, 3.90602896005, 0.810057940706, 2.19746776065, 0.552941622491, 3.79099834973, 0.775847448656, 2.34366325566, 0.540879003559, 3.90596650738, 0.810038994651, 2.1975449076, 0.552932269022, 3.79108545188, 0.775872818093, 2.34354950599, 0.540884258825, 3.9059152446, 0.810023443531, 2.19760823364, 0.552924593744, 3.79115692855, 0.775893637067, 2.34345616554, 0.540888575684, 3.90587313634, 0.810010669731, 2.19766025215, 0.552918290665, 3.79121562807, 0.775910734867, 2.34337951288, 0.540892123795, 3.90583852728, 0.81000017099, 2.19770300735, 0.552913111165, 3.79126386494, 0.775924785426, 2.34331652439, 0.540895041467, 3.90581006801, 0.809991537895, 2.19773816581, 0.552908852734, 3.79130352458, 0.77593633778, 2.34326473722, 0.540897441672, 3.90578665641, 0.809984436068, 2.19776708877, 0.552905350075, 3.79133614604, 0.775945840129, 2.34322214115, 0.54089941683, 3.90576739088, 0.809978591979, 2.19779088989, 0.552902468042, 3.79136298772, 0.775953658967, 2.3431870926, 0.540901042647, 3.9057515329, 0.809973781579, 2.19781048142, 0.552900095982, 3.79138508002, 0.775960094395, 2.34315824588, 0.540902381207, 3.90573847687, 0.809969821147, 2.19782661143, 0.552898143189, 3.7914032676, 0.775965392428, 2.34313449792, 0.540903483462, 3.90572772575, 0.809966559902, 2.19783989394, 0.552896535243, 3.79141824351, 0.775969754932, 2.34311494366, 0.540904391266, 3.9057188713, 0.809963874001, 2.19785083325, 0.552895211033, 3.79143057686, 0.775973347673, 2.34309883993, 0.540905139014, 3.90571157801, 0.809961661669, 2.19785984385, 0.552894120345, 3.79144073528, 0.775976306863, 2.34308557608, 0.540905754987, 3.90570557002, 0.809959839221, 2.19786726653, 0.552893221902, 3.79144910322, 0.775978744488, 2.34307465011, 0.540906262451, 3.90570062041, 0.80995833782, 2.19787338164, 0.55289248175, 3.79145599688, 0.775980752651, 2.34306564913, 0.540906680549, 3.90569654244, 0.809957100824, 2.19787841986, 0.552891871957, 3.79146167642, 0.775982407138, 2.34305823345, 0.540907025038, 3.90569318243, 0.809956081611, 2.19788257108, 0.552891369532, 3.79146635595, 0.775983770319, 2.34305212347, 0.540907308891, 3.90569041385, 0.8099552418, 2.1978859916, 0.552890955551, 3.79147021174, 0.77598489354, 2.34304708904, 0.54090754279, 3.90568813249, 0.809954549785, 2.19788881017, 0.552890614429, 3.79147338892, 0.775985819079, 2.34304294066, 0.540907735532, 3.90568625257, 0.809953979539, 2.19789113278, 0.552890333334, 3.79147600702, 0.775986581753, 2.34303952227, 0.540907894363, 3.9056847034, 0.809953509621, 2.19789304675, 0.552890101697, 3.79147816447, 0.775987210238, 2.34303670534, 0.540908025253, 3.90568342676, 0.809953122371, 2.19789462402, 0.552889910811, 3.79147994237, 0.775987728156, 2.34303438398, 0.540908133118, 3.90568237469, 0.809952803242, 2.19789592383, 0.552889753504, 3.79148140752, 0.775988154967, 2.34303247098, 0.540908222011, 3.90568150768, 0.809952540246, 2.19789699502, 0.552889623867, 3.79148261495, 0.775988506703, 2.34303089447, 0.540908295268, 3.90568079316, 0.809952323507, 2.1978978778, 0.552889517032, 3.79148361001, 0.775988796573, 2.34302959525, 0.540908355641, 3.9056802043, 0.809952144887, 2.19789860532, 0.552889428987, 3.79148443006, 0.775989035461, 2.34302852454, 0.540908405397, 3.90567971901, 0.809951997681, 2.19789920489, 0.552889356427, 3.79148510588, 0.775989232335, 2.34302764213, 0.540908446402, 3.90567931906, 0.809951876363, 2.19789969902, 0.552889296627, 3.79148566286, 0.775989394586, 2.34302691491, 0.540908480196, 3.90567898945, 0.809951776381, 2.19790010625, 0.552889247344, 3.79148612188, 0.775989528304, 2.34302631558, 0.540908508048, 3.9056787178, 0.80995169398, 2.19790044187, 0.552889206728, 3.79148650018, 0.775989638506, 2.34302582165, 0.540908531001, 3.90567849392, 0.80995162607, 2.19790071847, 0.552889173254, 3.79148681195, 0.775989729329, 2.34302541457, 0.540908549918, 3.90567830941, 0.809951570102, 2.19790094643, 0.552889145666, 3.7914870689, 0.775989804181, 2.34302507908, 0.540908565509, 3.90567815735, 0.809951523976, 2.1979011343, 0.55288912293, 3.79148728066, 0.77598986587, 2.34302480259, 0.540908578358, 3.90567803203, 0.809951485961, 2.19790128914, 0.552889104192, 3.79148745519, 0.775989916711, 2.34302457471, 0.540908588948, 3.90567792874, 0.809951454631, 2.19790141675, 0.552889088749, 3.79148759903, 0.775989958612, 2.34302438691, 0.540908597675, 3.90567784362, 0.80995142881, 2.19790152192, 0.552889076022, 3.79148771757, 0.775989993145, 2.34302423213, 0.540908604868, 3.90567777346, 0.809951407529, 2.19790160859, 0.552889065532, 3.79148781527, 0.775990021606, 2.34302410457, 0.540908610796, 3.90567771564, 0.80995138999, 2.19790168003, 0.552889056887, 3.79148789579, 0.775990045062, 2.34302399944, 0.540908615682, 3.90567766799, 0.809951375536, 2.1979017389, 0.552889049762, 3.79148796215, 0.775990064393, 2.34302391279, 0.540908619708, 3.90567762872, 0.809951363623, 2.19790178742, 0.55288904389, 3.79148801684, 0.775990080326, 2.34302384138, 0.540908623027, 3.90567759635, 0.809951353805, 2.19790182741, 0.552889039051, 3.79148806192, 0.775990093456, 2.34302378253, 0.540908625762, 3.90567756967, 0.809951345713, 2.19790186037, 0.552889035062, 3.79148809907, 0.775990104278, 2.34302373403, 0.540908628016, 3.90567754769, 0.809951339044, 2.19790188753, 0.552889031775, 3.79148812968, 0.775990113197, 2.34302369405, 0.540908629874, 3.90567752957, 0.809951333548, 2.19790190992, 0.552889029066, 3.79148815492, 0.775990120548, 2.3430236611, 0.540908631405, 3.90567751463, 0.809951329018, 2.19790192837, 0.552889026833, 3.79148817571, 0.775990126606, 2.34302363395, 0.540908632667, 3.90567750233, 0.809951325285, 2.19790194358, 0.552889024993, 3.79148819285, 0.775990131599, 2.34302361157, 0.540908633707, 3.90567749218, 0.809951322208, 2.19790195611, 0.552889023476, 3.79148820698, 0.775990135714, 2.34302359313, 0.540908634564, 3.90567748382, 0.809951319672, 2.19790196644, 0.552889022226, 3.79148821862, 0.775990139105, 2.34302357793, 0.54090863527, 3.90567747693, 0.809951317582, 2.19790197495, 0.552889021196, 3.79148822821, 0.7759901419, 2.3430235654, 0.540908635852, 3.90567747126, 0.80995131586, 2.19790198196, 0.552889020347, 3.79148823612, 0.775990144204, 2.34302355508, 0.540908636332, 3.90567746658, 0.80995131444, 2.19790198775, 0.552889019648, 3.79148824264, 0.775990146102, 2.34302354657, 0.540908636727, 3.90567746272, 0.809951313271, 2.19790199251, 0.552889019071, 3.79148824801, 0.775990147667, 2.34302353956, 0.540908637053, 3.90567745954, 0.809951312306, 2.19790199644, 0.552889018596, 3.79148825244, 0.775990148956, 2.34302353378, 0.540908637322, 3.90567745692, 0.809951311512, 2.19790199968, 0.552889018204, 3.79148825608, 0.775990150019, 2.34302352901, 0.540908637543, 3.90567745476, 0.809951310857, 2.19790200234, 0.552889017881, 3.79148825909, 0.775990150895, 2.34302352509, 0.540908637726, 3.90567745298, 0.809951310317, 2.19790200454, 0.552889017615, 3.79148826157, 0.775990151617, 2.34302352185, 0.540908637876, 3.90567745152, 0.809951309872, 2.19790200635, 0.552889017396, 3.79148826361, 0.775990152212, 2.34302351918, 0.540908638, 3.90567745031, 0.809951309505, 2.19790200785, 0.552889017215, 3.79148826529, 0.775990152702, 2.34302351699, 0.540908638102, 3.90567744931, 0.809951309203, 2.19790200908, 0.552889017066, 3.79148826668, 0.775990153106, 2.34302351518, 0.540908638186, 3.90567744849, 0.809951308954, 2.19790201009, 0.552889016943, 3.79148826782, 0.775990153439, 2.34302351368, 0.540908638256, 3.90567744781, 0.809951308749, 2.19790201093, 0.552889016842, 3.79148826877, 0.775990153714, 2.34302351245, 0.540908638313, 3.90567744726, 0.80995130858, 2.19790201162, 0.552889016759, 3.79148826954, 0.77599015394, 2.34302351144, 0.54090863836, 3.9056774468, 0.80995130844, 2.19790201218, 0.55288901669, 3.79148827018, 0.775990154127, 2.3430235106, 0.540908638399, 3.90567744642, 0.809951308326, 2.19790201265, 0.552889016633, 3.79148827071, 0.77599015428, 2.34302350991, 0.540908638431, 3.90567744611, 0.809951308231, 2.19790201304, 0.552889016587, 3.79148827115, 0.775990154407, 2.34302350935, 0.540908638457, 3.90567744585, 0.809951308153, 2.19790201336, 0.552889016548, 3.7914882715, 0.775990154511, 2.34302350888, 0.540908638479, 3.90567744564, 0.809951308088, 2.19790201362, 0.552889016517, 3.7914882718, 0.775990154597, 2.34302350849, 0.540908638497, 3.90567744546, 0.809951308035, 2.19790201383, 0.55288901649, 3.79148827204, 0.775990154668, 2.34302350818, 0.540908638512, 3.90567744532, 0.809951307992, 2.19790201401, 0.552889016469, 3.79148827224, 0.775990154727, 2.34302350791, 0.540908638524, 3.9056774452, 0.809951307956, 2.19790201416, 0.552889016451, 3.79148827241, 0.775990154775, 2.3430235077, 0.540908638534, 3.9056774451, 0.809951307926, 2.19790201428, 0.552889016437, 3.79148827254, 0.775990154814, 2.34302350752, 0.540908638542, 3.90567744502, 0.809951307902, 2.19790201438, 0.552889016424, 3.79148827266, 0.775990154847, 2.34302350737, 0.540908638549, 3.90567744495, 0.809951307881, 2.19790201446, 0.552889016415, 3.79148827275, 0.775990154874, 2.34302350725, 0.540908638555, 3.9056774449, 0.809951307865, 2.19790201453, 0.552889016406, 3.79148827283, 0.775990154896, 2.34302350715, 0.540908638559, 3.90567744485, 0.809951307851, 2.19790201458, 0.5528890164, 3.79148827289, 0.775990154915, 2.34302350707, 0.540908638563, 3.90567744482, 0.80995130784, 2.19790201463, 0.552889016394, 3.79148827294, 0.77599015493, 2.343023507, 0.540908638566, 3.90567744479, 0.809951307831, 2.19790201467, 0.552889016389, 3.79148827298, 0.775990154942, 2.34302350695, 0.540908638569, 3.90567744476, 0.809951307823, 2.1979020147, 0.552889016386, 3.79148827302, 0.775990154952, 2.3430235069, 0.540908638571, 3.90567744474, 0.809951307817, 2.19790201473, 0.552889016383, 3.79148827305, 0.775990154961, 2.34302350686, 0.540908638573, 3.90567744472, 0.809951307811, 2.19790201475, 0.55288901638, 3.79148827307, 0.775990154968, 2.34302350683, 0.540908638574, 3.90567744471, 0.809951307807, 2.19790201476, 0.552889016378, 3.79148827309, 0.775990154974, 2.34302350681, 0.540908638575, 3.9056774447, 0.809951307804, 2.19790201478, 0.552889016376, 3.79148827311, 0.775990154978, 2.34302350679, 0.540908638576, 3.90567744469, 0.809951307801, 2.19790201479, 0.552889016375, 3.79148827312, 0.775990154982, 2.34302350677, 0.540908638577, 3.90567744468, 0.809951307798, 2.1979020148, 0.552889016374, 3.79148827313, 0.775990154985, 2.34302350675, 0.540908638578, 3.90567744467, 0.809951307796, 2.19790201481, 0.552889016373, 3.79148827314, 0.775990154988, 2.34302350674, 0.540908638578, 3.90567744467, 0.809951307795, 2.19790201481, 0.552889016372, 3.79148827315, 0.77599015499, 2.34302350673, 0.540908638579, 3.90567744466, 0.809951307793, 2.19790201482, 0.552889016371, 3.79148827315, 0.775990154992, 2.34302350672, 0.540908638579, 3.90567744466, 0.809951307792, 2.19790201482, 0.552889016371, 3.79148827316, 0.775990154993, 2.34302350672, 0.540908638579, 3.90567744466, 0.809951307791, 2.19790201483, 0.55288901637, 3.79148827316, 0.775990154995, 2.34302350671, 0.54090863858, 3.90567744465, 0.809951307791, 2.19790201483, 0.55288901637, 3.79148827317, 0.775990154996, 2.34302350671, 0.54090863858, 3.90567744465, 0.80995130779, 2.19790201483, 0.552889016369, 3.79148827317, 0.775990154997, 2.3430235067, 0.54090863858, 3.90567744465, 0.809951307789, 2.19790201484, 0.552889016369, 3.79148827317, 0.775990154997, 2.3430235067, 0.54090863858, 3.90567744465, 0.809951307789, 2.19790201484, 0.552889016369, 3.79148827317, 0.775990154998, 2.3430235067, 0.54090863858, 3.90567744465, 0.809951307789, 2.19790201484, 0.552889016369, 3.79148827318, 0.775990154998, 2.3430235067, 0.54090863858, 3.90567744465, 0.809951307788, 2.19790201484, 0.552889016369, 3.79148827318, 0.775990154999, 2.34302350669, 0.540908638581, 3.90567744465, 0.809951307788, 2.19790201484, 0.552889016369, 3.79148827318, 0.775990154999, 2.34302350669, 0.540908638581, 3.90567744465, 0.809951307788, 2.19790201484, 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3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: c value chosen from interval of stability of the eight cycle: 1.91100000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.589, 0.771479861548, 2.36337027555, 0.540058807672, 3.91398000807, 0.812473224138, 2.18766598804, 0.554157509005, 3.77970204977, 0.772564381256, 2.35845370719, 0.540246732914, 3.91214167684, 0.811914232095, 2.18992868005, 0.553871966392, 3.78235021254, 0.77333275467, 2.35497962494, 0.540386184323, 3.91077839732, 0.811499910444, 2.19160803418, 0.55366193019, 3.78429995568, 0.773898976369, 2.35242444371, 0.540492287925, 3.90974162192, 0.811184943223, 2.19288596813, 0.553503176706, 3.78577468173, 0.774327528728, 2.35049328088, 0.540574475284, 3.90893883454, 0.810941134528, 2.19387595228, 0.5533808331, 3.7869117899, 0.774658134251, 2.34900511267, 0.540638984782, 3.90830890019, 0.810749867093, 2.19465306095, 0.553285187121, 3.78780112922, 0.774916802436, 2.34784174272, 0.540690128723, 3.90780959263, 0.8105982903, 2.19526920134, 0.553209596652, 3.78850421684, 0.775121360318, 2.34692234684, 0.540730990928, 3.90741073484, 0.810477225457, 2.19576150044, 0.553149354254, 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3.90567803203, 0.809951485961, 2.19790128914, 0.552889104192, 3.79148745519, 0.775989916711, 2.34302457471, 0.540908588948, 3.90567792874, 0.809951454631, 2.19790141675, 0.552889088749, 3.79148759903, 0.775989958612, 2.34302438691, 0.540908597675, 3.90567784362, 0.80995142881, 2.19790152192, 0.552889076022, 3.79148771757, 0.775989993145, 2.34302423213, 0.540908604868, 3.90567777346, 0.809951407529, 2.19790160859, 0.552889065532, 3.79148781527, 0.775990021606, 2.34302410457, 0.540908610796, 3.90567771564, 0.80995138999, 2.19790168003, 0.552889056887, 3.79148789579, 0.775990045062, 2.34302399944, 0.540908615682, 3.90567766799, 0.809951375536, 2.1979017389, 0.552889049762, 3.79148796215, 0.775990064393, 2.34302391279, 0.540908619708, 3.90567762872, 0.809951363623, 2.19790178742, 0.55288904389, 3.79148801684, 0.775990080326, 2.34302384138, 0.540908623027, 3.90567759635, 0.809951353805, 2.19790182741, 0.552889039051, 3.79148806192, 0.775990093456, 2.34302378253, 0.540908625762, 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2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485, 0.552889016368, 3.79148827318, 0.775990155, 2.34302350669, 0.540908638581, 3.90567744464, 0.809951307787, 2.19790201485) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: |
#PART F - INTERVAL OF STABILITY OF THE EIGHT CYCLE [SLIGHTLY SMALLER THAN BEFORE];
c=1.909;
print "c value chosen from interval of stability of the eight cycle:";
print c;
print "";
N=2000;
print "Iterations computed:";
print N;
print "";
eightcycleB=CompIterate(c,N);
print "Values for all iterations:";
print eightcycleB;
print "";
eightcycleB_lyapunov=LyapunovEst(c,N);
print "Estimate of Lyapunov exponent:";
print eightcycleB_lyapunov;
print "";
num_intervals=100;
max_value=VectorMax(eightcycleB,N);
min_value=VectorMin(eightcycleB,N);
eightcycleB_bars=BinIt(eightcycleB,N,max_value,min_value,num_intervals);
print "Graphical approximation of frequency distribution across 100 equal intervals of value range:";
bar_chart(eightcycleB_bars);
c value chosen from interval of stability of the eight cycle: 1.90900000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.591, 0.772106536769, 2.36252743177, 0.54209024135, 3.89617905478, 0.809074673816, 2.20347690922, 0.554223109814, 3.7810940648, 0.774968182094, 2.34961076349, 0.542612607272, 3.89111260079, 0.807544012082, 2.20973980361, 0.553495806493, 3.7878431683, 0.776929031847, 2.34082105819, 0.543012123332, 3.88724459161, 0.806377180094, 2.2145319654, 0.552954037274, 3.79288274456, 0.778396485328, 2.33427525425, 0.543333104624, 3.88414126914, 0.805442129871, 2.21838344444, 0.552527815206, 3.79685481224, 0.779555063971, 2.329126595, 0.543599768528, 3.88156603114, 0.804666943328, 2.22158405394, 0.552179825568, 3.80010260374, 0.780503670089, 2.32492370357, 0.543826776618, 3.87937585206, 0.80400820084, 2.2243093245, 0.551887937319, 3.80283012804, 0.781301209072, 2.3213989182, 0.544023659167, 3.87747788165, 0.803437744937, 2.22667339341, 0.55163801574, 3.80516791495, 0.781985433323, 2.3183813144, 0.544196945711, 3.87580857467, 0.802936322769, 2.22875448314, 0.551420523903, 3.80720416806, 0.782581889571, 2.31575558277, 0.544351295722, 3.87432262882, 0.802490219722, 2.23060842728, 0.551228747821, 3.80900106339, 0.783108607954, 2.31344055593, 0.544490144089, 3.87298667631, 0.802089341376, 2.23227639457, 0.551057797608, 3.81060393598, 0.78357874889, 2.31137712552, 0.544616091279, 3.87177547442, 0.801726056174, 2.23378955885, 0.550904011479, 3.81204676906, 0.784002186666, 2.30952103298, 0.544731150297, 3.87066949629, 0.80139446401, 2.23517205586, 0.55076458288, 3.81335563223, 0.78438650246, 2.30783835998, 0.544836909075, 3.86965334801, 0.801089916006, 2.23644292771, 0.550637317947, 3.81455092043, 0.784737631669, 2.30630259514, 0.544934640711, 3.86871469477, 0.800808688901, 2.23761744326, 0.550520472442, 3.81564886058, 0.785060299893, 2.30489266237, 0.545025380387, 3.86784351213, 0.800547758148, 2.2387080196, 0.550412639034, 3.81666255407, 0.78535832488, 2.30359155779, 0.545109980323, 3.86703155139, 0.800304635995, 2.23972488033, 0.550312667444, 3.81760271589, 0.785634830747, 2.30238538435, 0.545189149846, 3.86627195001, 0.800077253527, 2.24067653582, 0.550219606667, 3.81847820978, 0.785892403191, 2.30126265334, 0.545263485149, 3.86555894238, 0.799863873139, 2.24157013983, 0.550132662388, 3.81929644296, 0.786133204068, 2.30021376956, 0.54533349176, 3.86488764159, 0.799663022544, 2.24241175873, 0.550051165058, 3.82006366221, 0.786359057371, 2.29923064545, 0.545399601734, 3.86425387252, 0.799473444272, 2.24320657776, 0.549974545609, 3.82078517928, 0.786571514712, 2.29830640759, 0.545462187006, 3.86365404227, 0.799294056531, 2.24395906091, 0.549902316731, 3.8214655447, 0.786771905872, 2.29743517039, 0.545521569858, 3.86308503856, 0.7991239225, 2.2446730766, 0.549834058261, 3.82210868355, 0.786961378278, 2.29661185949, 0.545578031233, 3.86254414917, 0.798962225984, 2.24535199729, 0.549769405657, 3.82271800263, 0.787140928189, 2.29583207249, 0.545631817398, 3.86202899734, 0.79880825192, 2.24599877936, 0.549708040802, 3.8232964759, 0.78731142557, 2.29509196783, 0.545683145323, 3.86153748962, 0.798661370628, 2.24661602768, 0.549649684619, 3.82384671324, 0.787473634123, 2.29438817554, 0.545732207075, 3.86106777335, 0.798521024975, 2.24720604826, 0.54959409107, 3.8243710163, 0.787628227545, 2.29371772475, 0.545779173418, 3.86061820171, 0.798386719841, 2.24777089152, 0.549541042253, 3.82487142413, 0.787775802839, 2.29307798442, 0.545824196805, 3.86018730489, 0.798258013407, 2.24831238804, 0.549490344363, 3.82534975082, 0.787916891281, 2.29246661458, 0.545867413855, 3.85977376599, 0.798134509899, 2.24883217839, 0.549441824342, 3.8258076167, 0.788051967526, 2.29188152581, 0.545908947436, 3.85937640093, 0.798015853522, 2.24933173813, 0.549395327083, 3.82624647441, 0.788181457203, 2.29132084552, 0.545948908424, 3.85899414142, 0.797901723338, 2.24981239881, 0.549350713084, 3.82666763079, 0.788305743309, 2.29078288953, 0.545987397184, 3.85862602064, 0.797791828934, 2.25027536584, 0.549307856464, 3.82707226532, 0.788425171596, 2.29026613822, 0.546024504841, 3.85827116095, 0.797685906736, 2.25072173368, 0.549266643288, 3.82746144585, 0.788540055153, 2.28976921616, 0.546060314359, 3.85792876345, 0.797583716852, 2.25115249884, 0.549226970127, 3.82783614195, 0.788650678314, 2.28929087484, 0.546094901475, 3.85759809898, 0.797485040365, 2.25156857102, 0.549188742835, 3.82819723635, 0.7887573, 2.2888299778, 0.5461283355, 3.85727850034, 0.797389676989, 2.25197078283, 0.549151875489, 3.82854553487, 0.788860156608, 2.28838548792, 0.546160680017, 3.85696935558, 0.797297443053, 2.25235989807, 0.54911628948, 3.82888177486, 0.788959464497, 2.28795645634, 0.546191993488, 3.85667010213, 0.797208169728, 2.25273661902, 0.549081912721, 3.82920663264, 0.789055422151, 2.28754201296, 0.546222329779, 3.85638022173, 0.797121701503, 2.25310159283, 0.549048678955, 3.82952072994, 0.789148212058, 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0.79338843779, 2.26894418059, 0.547672973712, 3.84255794473, 0.793008903611, 2.27056412029, 0.547539616006, 3.84382540928, 0.793385197683, 2.26895800278, 0.547671830135, 3.84256881081, 0.793012128889, 2.27055034676, 0.547540744206, 3.84381468387, 0.793382012732, 2.26897158981, 0.54767070611, 3.84257949115, 0.79301529905, 2.27053680873, 0.54754185321, 3.84380414098, 0.793378881995, 2.26898494568, 0.547669601301, 3.84258998895, 0.79301841504, 2.27052350215, 0.547542943346, 3.84379377751, 0.793375804549, 2.26899807433, 0.547668515377, 3.84260030735, 0.79302147779, 2.27051042304, 0.547544014935, 3.84378359041, 0.793372779484, 2.26901097962, 0.547667448014, 3.84261044943, 0.793024488215, 2.27049756749, 0.547545068293, 3.84377357666, 0.793369805908, 2.26902366536, 0.547666398893, 3.84262041821, 0.793027447211, 2.27048493166, 0.54754610373, 3.84376373331, 0.793366882944, 2.26903613528, 0.5476653677, 3.84263021667, 0.793030355664, 2.27047251178, 0.547547121551, 3.84375405747, 0.793364009729, 2.26904839307, 0.547664354128, 3.84263984774, 0.793033214439, 2.27046030413, 0.547548122057, 3.84374454627, 0.793361185415, 2.26906044233, 0.547663357874, 3.84264931429, 0.79303602439, 2.27044830506, 0.547549105542, 3.84373519691, 0.793358409168, 2.26907228662, 0.54766237864, 3.84265861915, 0.793038786354, 2.270436511, 0.547550072296, 3.84372600664, 0.793355680171, 2.26908392941, 0.547661416135, 3.84266776508, 0.793041501155, 2.27042491843, 0.547551022603, 3.84371697275, 0.793352997617, 2.26909537415, 0.54766047007, 3.84267675482, 0.7930441696, 2.27041352389, 0.547551956745, 3.84370809257, 0.793350360716, 2.26910662421, 0.547659540165, 3.84268559105, 0.793046792485, 2.27040232398, 0.547552874995, 3.84369936349, 0.793347768691, 2.26911768288, 0.547658626142, 3.84269427639, 0.793049370591, 2.27039131536, 0.547553777624, 3.84369078293, 0.793345220776, 2.26912855345, 0.547657727728, 3.84270281343, 0.793051904686, 2.27038049474, 0.547554664898, 3.84368234837, 0.79334271622, 2.26913923909, 0.547656844656, 3.84271120472, 0.793054395523, 2.27036985892, 0.547555537079, 3.84367405731, 0.793340254286, 2.26914974297, 0.547655976663, 3.84271945275, 0.793056843844, 2.2703594047, 0.547556394423, 3.84366590733, 0.793337834247, 2.26916006817, 0.54765512349, 3.84272755999, 0.793059250377, 2.270349129, 0.547557237183, 3.84365789601, 0.79333545539, 2.26917021774, 0.547654284883, 3.84273552883, 0.793061615837, 2.27033902873, 0.547558065606, 3.84365002099, 0.793333117014, 2.26918019466, 0.547653460592, 3.84274336166, 0.793063940929, 2.2703291009, 0.547558879937, 3.84364227997, 0.79333081843, 2.26919000187, 0.547652650373, 3.84275106079, 0.793066226344, 2.27031934255, 0.547559680416, 3.84363467064, 0.793328558959, 2.26919964225, 0.547651853984, 3.84275862853, 0.79306847276, 2.27030975078, 0.547560467278, 3.84362719078, 0.793326337936, 2.26920911865, 0.547651071188, 3.84276606713, 0.793070680845, 2.27030032274, 0.547561240755, 3.84361983818, 0.793324154708, 2.26921843385, 0.547650301753, 3.84277337878, 0.793072851256, 2.27029105561, 0.547562001075, 3.84361261068, 0.793322008629, 2.2692275906, 0.54764954545, 3.84278056567, 0.793074984637, 2.27028194664, 0.547562748461, 3.84360550614, 0.793319899068, 2.2692365916, 0.547648802053, 3.84278762993, 0.793077081622, 2.27027299314, 0.547563483133, 3.84359852247, 0.793317825404, 2.26924543948, 0.547648071342, 3.84279457366, 0.793079142833, 2.27026419242, 0.547564205308, 3.84359165762, 0.793315787025, 2.26925413686, 0.5476473531, 3.84280139893, 0.793081168883, 2.27025554189, 0.547564915198, 3.84358490957, 0.793313783331, 2.26926268629, 0.547646647113, 3.84280810775, 0.793083160372, 2.27024703896, 0.547565613013, 3.84357827632, 0.793311813731, 2.2692710903, 0.547645953172, 3.84281470212, 0.793085117892, 2.27023868111, 0.547566298957, 3.84357175593, 0.793309877645, 2.26927935135, 0.547645271071, 3.842821184, 0.793087042024, 2.27023046587, 0.547566973233, 3.84356534647, 0.793307974503, 2.26928747188, 0.547644600606, 3.84282755532, 0.793088933339, 2.27022239078, 0.547567636038, 3.84355904606, 0.793306103744, 2.26929545427, 0.547643941579, 3.84283381796, 0.793090792398, 2.27021445346, 0.547568287569, 3.84355285283, 0.793304264818, 2.26930330088, 0.547643293795, 3.84283997378, 0.793092619752, 2.27020665154, 0.547568928016, 3.84354676498, 0.793302457181, 2.26931101401, 0.54764265706, 3.84284602461, 0.793094415943, 2.27019898271, 0.547569557569, 3.84354078071, 0.793300680303, 2.26931859594, 0.547642031186, 3.84285197224, 0.793096181504, 2.27019144469, 0.547570176412, 3.84353489824, 0.793298933659, 2.2693260489, 0.547641415988, 3.84285781844, 0.793097916958, 2.27018403525, 0.547570784728, 3.84352911586, 0.793297216736, 2.26933337507, 0.547640811282, 3.84286356494, 0.79309962282, 2.27017675219, 0.547571382696, 3.84352343185, 0.793295529026, 2.26934057663, 0.54764021689, 3.84286921345, 0.793101299596, 2.27016959334, 0.547571970492, 3.84351784456, 0.793293870034, 2.26934765567, 0.547639632635, 3.84287476563, 0.793102947781, 2.27016255659, 0.547572548289, 3.84351235232, 0.793292239271, 2.2693546143) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: c value chosen from interval of stability of the eight cycle: 1.90900000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.591, 0.772106536769, 2.36252743177, 0.54209024135, 3.89617905478, 0.809074673816, 2.20347690922, 0.554223109814, 3.7810940648, 0.774968182094, 2.34961076349, 0.542612607272, 3.89111260079, 0.807544012082, 2.20973980361, 0.553495806493, 3.7878431683, 0.776929031847, 2.34082105819, 0.543012123332, 3.88724459161, 0.806377180094, 2.2145319654, 0.552954037274, 3.79288274456, 0.778396485328, 2.33427525425, 0.543333104624, 3.88414126914, 0.805442129871, 2.21838344444, 0.552527815206, 3.79685481224, 0.779555063971, 2.329126595, 0.543599768528, 3.88156603114, 0.804666943328, 2.22158405394, 0.552179825568, 3.80010260374, 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3.84362719078, 0.793326337936, 2.26920911865, 0.547651071188, 3.84276606713, 0.793070680845, 2.27030032274, 0.547561240755, 3.84361983818, 0.793324154708, 2.26921843385, 0.547650301753, 3.84277337878, 0.793072851256, 2.27029105561, 0.547562001075, 3.84361261068, 0.793322008629, 2.2692275906, 0.54764954545, 3.84278056567, 0.793074984637, 2.27028194664, 0.547562748461, 3.84360550614, 0.793319899068, 2.2692365916, 0.547648802053, 3.84278762993, 0.793077081622, 2.27027299314, 0.547563483133, 3.84359852247, 0.793317825404, 2.26924543948, 0.547648071342, 3.84279457366, 0.793079142833, 2.27026419242, 0.547564205308, 3.84359165762, 0.793315787025, 2.26925413686, 0.5476473531, 3.84280139893, 0.793081168883, 2.27025554189, 0.547564915198, 3.84358490957, 0.793313783331, 2.26926268629, 0.547646647113, 3.84280810775, 0.793083160372, 2.27024703896, 0.547565613013, 3.84357827632, 0.793311813731, 2.2692710903, 0.547645953172, 3.84281470212, 0.793085117892, 2.27023868111, 0.547566298957, 3.84357175593, 0.793309877645, 2.26927935135, 0.547645271071, 3.842821184, 0.793087042024, 2.27023046587, 0.547566973233, 3.84356534647, 0.793307974503, 2.26928747188, 0.547644600606, 3.84282755532, 0.793088933339, 2.27022239078, 0.547567636038, 3.84355904606, 0.793306103744, 2.26929545427, 0.547643941579, 3.84283381796, 0.793090792398, 2.27021445346, 0.547568287569, 3.84355285283, 0.793304264818, 2.26930330088, 0.547643293795, 3.84283997378, 0.793092619752, 2.27020665154, 0.547568928016, 3.84354676498, 0.793302457181, 2.26931101401, 0.54764265706, 3.84284602461, 0.793094415943, 2.27019898271, 0.547569557569, 3.84354078071, 0.793300680303, 2.26931859594, 0.547642031186, 3.84285197224, 0.793096181504, 2.27019144469, 0.547570176412, 3.84353489824, 0.793298933659, 2.2693260489, 0.547641415988, 3.84285781844, 0.793097916958, 2.27018403525, 0.547570784728, 3.84352911586, 0.793297216736, 2.26933337507, 0.547640811282, 3.84286356494, 0.79309962282, 2.27017675219, 0.547571382696, 3.84352343185, 0.793295529026, 2.26934057663, 0.54764021689, 3.84286921345, 0.793101299596, 2.27016959334, 0.547571970492, 3.84351784456, 0.793293870034, 2.26934765567, 0.547639632635, 3.84287476563, 0.793102947781, 2.27016255659, 0.547572548289, 3.84351235232, 0.793292239271, 2.2693546143) Estimate of Lyapunov exponent: 0.897033441501 Graphical approximation of frequency distribution across 100 equal intervals of value range: |
#PART F CONCLUSIONS;
print "Our estimates of the Lyapunov exponent in the interval of stability of the eight-cycle was almost identical when a c value was chosen slight larger or slightly smaller than our initial choice. The graph of the frequency distribution however varied, most likely as an artifact of bin grouping definitions."
Our estimates of the Lyapunov exponent in the interval of stability of the eight-cycle was almost identical when a c value was chosen slight larger or slightly smaller than our initial choice. The graph of the frequency distribution however varied, most likely as an artifact of bin grouping definitions. Our estimates of the Lyapunov exponent in the interval of stability of the eight-cycle was almost identical when a c value was chosen slight larger or slightly smaller than our initial choice. The graph of the frequency distribution however varied, most likely as an artifact of bin grouping definitions. |
#PART G DISCUSSION;
print "As c increases, the Lyapunov exponent appears to reamin constant and the number number of modes in the graph of the frequency distribution appears to increase."
As c increases, the Lyapunov exponent appears to reamin constant and the number number of modes in the graph of the frequency distribution appears to increase. As c increases, the Lyapunov exponent appears to reamin constant and the number number of modes in the graph of the frequency distribution appears to increase. |
#Problem 3
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# Solves for p(c), the fixed point of fc(x)
var('x c')
solve([3/x + x/2 - c == x],x)
[x == -c - sqrt(c^2 + 6), x == -c + sqrt(c^2 + 6)] [x == -c - sqrt(c^2 + 6), x == -c + sqrt(c^2 + 6)] |
# f = -c + sqrt(c^2 + 6) (one of the roots solved for above) exists for all 0 < c < 2 and is positive over the whole interval, making it our positive p(c).
f = -c + sqrt(c^2 + 6)
plot(f,0,2);
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# g = -c - sqrt(c^2 + 6), the other root, exists over 0 < c < 2, but is negative over the interval, so we discard this root.
g = -c - sqrt(c^2 + 6)
plot(g,0,2);
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# In order to determine the interval of stability, we first differentiate fc(x) to get fc'(x), shown here.
F(x, c) = diff(3/x + x/2 - c, x)
print F(x, c);
print F(-c + sqrt(c^2 + 6));
-3/x^2 + 1/2 -3/(c - sqrt(c^2 + 6))^2 + 1/2 -3/x^2 + 1/2 -3/(c - sqrt(c^2 + 6))^2 + 1/2 |
# We use a graph to show that the graph of the derivative of p(c) is equal to -1 at √2.
F = -3/(c - sqrt(c^2 + 6))^2 + 1/2
G = sqrt(2)*x - 3
P = plot(F,0,2) + plot(G, 0, 2)
P.show;
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<bound method Graphics.show of ># Here, g(x) is fc(x), and we use it to create the 2-cycle, fc(fc(x)).
var('x c')
g(x) = 3/x + x/2 - c
g(g)
-3/2*c + 1/4*x - 6/(2*c - x - 6/x) + 3/2/x -3/2*c + 1/4*x - 6/(2*c - x - 6/x) + 3/2/x |
# We solve the 2-cycle. We discard the last two solutions, x == -c - sqrt(c^2 +6) and x == -c + sqrt(c^2 + 6), because they were solutions of fc(x), and are left with the first two.
solve([-3/2*c + 1/4*x - 6/(2*c - x - 6/x) + 3/2/x
== x],x)
[x == c - sqrt(c^2 - 2), x == c + sqrt(c^2 - 2), x == -c - sqrt(c^2 + 6), x == -c + sqrt(c^2 + 6)] [x == c - sqrt(c^2 - 2), x == c + sqrt(c^2 - 2), x == -c - sqrt(c^2 + 6), x == -c + sqrt(c^2 + 6)] |
# Plotting p1(c), the first root given, for 0 < c < 2, we determine that is exists after √2, and is positive the whole time.
p = c - sqrt(c^2 - 2)
plot(p,0,2);
verbose 0 (3989: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 141 points. verbose 0 (3989: plot.py, generate_plot_points) Last error message: '' verbose 0 (3989: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 141 points. verbose 0 (3989: plot.py, generate_plot_points) Last error message: '' |
# p2(c) also has its interval of existence beginning at √2, and is positive for √2 < c < 2.
q = c + sqrt(c^2 - 2)
plot(q,0,2);
verbose 0 (3989: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 141 points. verbose 0 (3989: plot.py, generate_plot_points) Last error message: '' verbose 0 (3989: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 141 points. verbose 0 (3989: plot.py, generate_plot_points) Last error message: '' |
# Here, we differentiate fc(fc(x)).
diff(-3/2*c + 1/4*x - 6/(2*c - x - 6/x) + 3/2/x, x)
6*(6/x^2 - 1)/(2*c - x - 6/x)^2 - 3/2/x^2 + 1/4 6*(6/x^2 - 1)/(2*c - x - 6/x)^2 - 3/2/x^2 + 1/4 |
# To get the interval of stability, we first multiply the derivatives of p1(c) and p2(c), giving the expression below
var('x c')
r(x) = 6*(6/x^2 - 1)/(2*c - x - 6/x)^2 - 3/2/x^2 + 1/4
r(c - sqrt(c^2 - 2)) * r(c + sqrt(c^2 - 2))
1/16*(24*(6/(c - sqrt(c^2 - 2))^2 - 1)/(c + sqrt(c^2 - 2) - 6/(c - sqrt(c^2 - 2)))^2 - 6/(c - sqrt(c^2 - 2))^2 + 1)*(24*(6/(c + sqrt(c^2 - 2))^2 - 1)/(c - sqrt(c^2 - 2) - 6/(c + sqrt(c^2 - 2)))^2 - 6/(c + sqrt(c^2 - 2))^2 + 1) 1/16*(24*(6/(c - sqrt(c^2 - 2))^2 - 1)/(c + sqrt(c^2 - 2) - 6/(c - sqrt(c^2 - 2)))^2 - 6/(c - sqrt(c^2 - 2))^2 + 1)*(24*(6/(c + sqrt(c^2 - 2))^2 - 1)/(c - sqrt(c^2 - 2) - 6/(c + sqrt(c^2 - 2)))^2 - 6/(c + sqrt(c^2 - 2))^2 + 1) |
# We solve that expression so its absolute value is less than 1, giving an interval of stability in between √2 and √10/√3
solve_ineq (-1<-1/16*(24*(6/(c - sqrt(c^2 - 2))^2 - 1)/(c + sqrt(c^2 - 2) - 6/(c - sqrt(c^2 - 2)))^2 - 6/(c - sqrt(c^2 - 2))^2 + 1)*(24*(6/(c + sqrt(c^2 - 2))^2 - 1)/(c - sqrt(c^2 - 2) - 6/(c + sqrt(c^2 - 2)))^2 - 6/(c + sqrt(c^2 - 2))^2 + 1)<1)
[[c > -1/3*sqrt(3)*sqrt(10), c < -sqrt(2)], [c > sqrt(2), c < 1/3*sqrt(3)*sqrt(10)]] [[c > -1/3*sqrt(3)*sqrt(10), c < -sqrt(2)], [c > sqrt(2), c < 1/3*sqrt(3)*sqrt(10)]] |
# For part e, we note that the interval of stability for fc(x) at p(c) ends at √2, where the interval of stability for the 2-cycle begins. We also create three functions corresponding to fc(p(c), p1(c), and p2(c) and determine that at √2, they are all equal.
var('x c')
s(x,c) = 3/x + x/2 - c
t(x) = x - sqrt(x^2 - 2)
u(x) = x + sqrt(x^2 - 2)
s(sqrt(2),sqrt(2)), t(sqrt(2)), u(sqrt(2));
(sqrt(2), sqrt(2), sqrt(2)) (sqrt(2), sqrt(2), sqrt(2)) |
# We finally determine that the derivative of fc(x) is equal to -1 at √2.
var('x c')
v(x) = diff(3/x + x/2 - c, x)
v(sqrt(2))
-1 -1 |
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