#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, 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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) 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);
c value chosen from interval of stability of the two cycle: 1.55000000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.95, 0.963461538462, 2.04550322432, 0.939383375108, 2.11327598923, 0.926234874194, 2.15203672591, 0.920046617157, 2.17072765122, 0.917388879888, 2.17884526232, 0.916298746175, 2.18219074063, 0.915860620732, 2.18353789965, 0.915686022883, 2.18407517512, 0.915616681437, 2.18428861988, 0.915589180135, 2.18437328397, 0.915578278852, 2.18440684574, 0.915573958607, 2.18442014673, 0.915572246613, 2.18442541758, 0.915571568221, 2.18442750621, 0.915571299406, 2.18442833384, 0.915571192887, 2.18442866178, 0.915571150679, 2.18442879173, 0.915571133954, 2.18442884323, 0.915571127327, 2.18442886363, 0.915571124701, 2.18442887172, 0.91557112366, 2.18442887492, 0.915571123248, 2.18442887619, 0.915571123085, 2.18442887669, 0.91557112302, 2.18442887689, 0.915571122994, 2.18442887697, 0.915571122984, 2.184428877, 0.91557112298, 2.18442887701, 0.915571122979, 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2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702) 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 two cycle: 1.55000000000000 Iterations computed: 2000 Values for all iterations: (1.0, 1.95, 0.963461538462, 2.04550322432, 0.939383375108, 2.11327598923, 0.926234874194, 2.15203672591, 0.920046617157, 2.17072765122, 0.917388879888, 2.17884526232, 0.916298746175, 2.18219074063, 0.915860620732, 2.18353789965, 0.915686022883, 2.18407517512, 0.915616681437, 2.18428861988, 0.915589180135, 2.18437328397, 0.915578278852, 2.18440684574, 0.915573958607, 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2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702, 0.915571122978, 2.18442887702) 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 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, 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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, 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, 3.88340591411, 0.804220707285, 2.2224296215, 0.551088827446, 3.80931291453, 0.782200055629, 2.31643598619, 0.543310979858, 3.88335505283, 0.804205394505, 2.22249299368, 0.55108202322, 3.8093767268, 0.782218769329, 2.31635358696, 0.543315850353, 3.88330798941, 0.804191225381, 2.22255163524, 0.551075728872, 3.80943575872, 0.782236081533, 2.31627736264, 0.543320358777, 3.88326442546, 0.804178110025, 2.22260591763, 0.551069904094, 3.80949038803, 0.782252102925, 2.31620682502, 0.54332453335, 3.88322408812, 0.804165966244, 2.22265618061, 0.551064512057, 3.80954095984, 0.782266934645, 2.31614152804, 0.543328399912, 3.88318672756, 0.80415471879, 2.22270273521, 0.551059519072, 3.80958778988, 0.782280669223, 2.31608106372, 0.543331982143, 3.88315211483, 0.804144298702, 2.22274586655, 0.551054894289, 3.80963116729, 0.782293391406, 2.31602505842, 0.543335301774, 3.88312003987, 0.804134642713, 2.2227858362, 0.551050609427, 3.80967135715, 0.782305178894, 2.31597316962, 0.543338378757, 3.88309030979, 0.804125692731, 2.2228228844, 0.551046638533, 3.80970860273, 0.782316102997, 2.31592508302, 0.543341231432, 3.88306274731, 0.804117395367, 2.22285723193, 0.551042957771, 3.80974312748, 0.782326229216, 2.31588050994, 0.54334387667, 3.88303718936, 0.804109701512, 2.22288908191, 0.551039545229, 3.80977513681, 0.782335617771, 2.31583918502, 0.543346329999, 3.88301348583, 0.804102565966, 2.22291862128, 0.55103638075, 3.80980481972, 0.782344324067, 2.31580086414, 0.543348605725, 3.88299149847, 0.804095947093, 2.22294602228, 0.551033445776, 3.80983235018, 0.782352399117, 2.31576532255, 0.543350717035, 3.88297109982, 0.804089806517, 2.22297144366, 0.551030723209, 3.80985788851, 0.78235988992, 2.3157323532, 0.543352676089, 3.88295217236, 0.804084108847, 2.22299503185, 0.551028197289, 3.80988158249, 0.782366839809, 2.3157017652, 0.543354494111, 3.88293460761, 0.804078821423, 2.22301692196, 0.551025853477, 3.80990356843, 0.782373288754, 2.31567338251, 0.543356181464, 3.88291830543, 0.804073914091, 2.22303723877, 0.551023678356, 3.80992397216, 0.782379273648, 2.31564704265, 0.543357747719, 3.88290317331, 0.804069359002, 2.22305609751, 0.551021659537, 3.80994290984, 0.782384828558, 2.31562259556, 0.543359201722, 3.88288912578, 0.804065130414, 2.2230736047, 0.551019785576, 3.80996048882, 0.782389984957, 2.31559990267, 0.543360551653, 3.88287608379, 0.804061204535, 2.2230898588, 0.551018045893, 3.80997680827, 0.782394771936, 2.31557883586, 0.543361805076, 3.88286397425, 0.804057559355, 2.2231049509, 0.55101643071, 3.80999195994, 0.782399216392, 2.3155592767, 0.54336296899, 3.88285272953, 0.804054174514, 2.22311896525, 0.55101493098, 3.81000602865, 0.782403343205, 2.31554111564, 0.543364049871, 3.88284228703, 0.804051031168, 2.22313197986, 0.551013538334, 3.8100190929, 0.782407175398, 2.31552425133, 0.543365053718, 3.88283258882, 0.804048111871, 2.22314406693, 0.551012245023, 3.81003122536, 0.782410734281, 2.31550858993, 0.543365986084, 3.88282358121, 0.804045400466, 2.22315529332, 0.551011043875, 3.81004249329, 0.782414039583, 2.3154940446, 0.543366852114, 3.88281521451, 0.804042881989, 2.22316572098, 0.551009928247, 3.81005295901, 0.782417109578, 2.31548053488, 0.543367656575, 3.88280744265, 0.804040542573, 2.22317540732, 0.551008891985, 3.81006268024, 0.782419961195, 2.31546798626, 0.543368403885, 3.88280022295, 0.804038369367, 2.22318440552, 0.551007929387, 3.81007171046, 0.782422610121, 2.31545632967, 0.54336909814, 3.88279351583, 0.804036350459, 2.2231927649, 0.551007035168, 3.81008009924, 0.782425070894, 2.31544550113, 0.543369743135, 3.88278728461, 0.804034474805, 2.22320053116, 0.551006204429, 3.81008789252, 0.782427356993, 2.31543544131, 0.543370342394, 3.88278149527, 0.804032732164, 2.22320774672, 0.551005432628, 3.81009513293, 0.782429480915, 2.31542609519, 0.543370899181, 3.88277611625, 0.804031113036, 2.22321445089, 0.551004715552, 3.81010185997, 0.782431454249, 2.31541741177, 0.543371416526, 3.88277111827, 0.80402960861, 2.22322068015, 0.551004049294, 3.81010811029, 0.782433287745, 2.31540934374, 0.543371897241, 3.8827664742, 0.804028210711, 2.22322646835, 0.551003430229, 3.81011391789, 0.782434991375, 2.31540184719, 0.543372343932, 3.88276215881, 0.804026911753, 2.22323184689, 0.551002854996, 3.81011931432, 0.782436574393, 2.31539488142, 0.543372759021, 3.88275814874, 0.804025704696, 2.22323684491, 0.551002320472, 3.81012432885, 0.782438045385, 2.31538840862, 0.543373144755, 3.88275442226, 0.804024583008, 2.22324148947, 0.551001823763, 3.81012898863, 0.782439412318, 2.31538239374, 0.543373503219, 3.88275095925, 0.804023540621, 2.22324580567, 0.551001362179, 3.8101333189, 0.782440682589, 2.31537680421, 0.543373836348, 3.88274774098, 0.804022571907, 2.22324981684, 0.551000933227, 3.81013734305, 0.782441863064, 2.31537160983, 0.543374145941, 3.8827447501, 0.804021671638, 2.2232535446, 0.55100053459, 3.81014108281, 0.782442960117, 2.31536678255, 0.543374433666, 3.88274197048, 0.804020834961, 2.22325700905, 0.551000164116, 3.81014455837, 0.782443979665, 2.31536229632, 0.543374701073, 3.88273938714, 0.804020057367, 2.22326022886, 0.55099981981, 3.81014778844, 0.782444927203, 2.31535812696, 0.543374949602, 3.88273698619, 0.804019334672, 2.22326322136, 0.550999499816, 3.81015079043, 0.782445807834, 2.31535425201, 0.543375180588, 3.88273475471, 0.804018662988, 2.22326600264, 0.550999202412, 3.8101535805, 0.782446626298, 2.31535065062, 0.543375395274, 3.8827326807, 0.804018038705, 2.22326858765, 0.550998926, 3.81015617364, 0.782447386995, 2.31534730342, 0.543375594813, 3.88273075303, 0.80401745847, 2.22327099027, 0.550998669093, 3.81015858379, 0.782448094013, 2.31534419242, 0.543375780276, 3.88272896134, 0.804016919166, 2.2232732234, 0.550998430312, 3.8101608239, 0.78244875115, 2.31534130092, 0.543375952658, 3.88272729602, 0.804016417901, 2.22327529903, 0.550998208375, 3.81016290599, 0.782449361933, 2.31533861339, 0.543376112883, 3.88272574814, 0.804015951986, 2.22327722829, 0.550998002091, 3.81016484124, 0.782449929638, 2.3153361154, 0.543376261812, 3.88272430941, 0.804015518923, 2.22327902151, 0.550997810354, 3.81016664001, 0.78245045731, 2.31533379358, 0.54337640024, 3.8827229721, 0.804015116391, 2.22328068831, 0.550997632136, 3.81016831196, 0.782450947778, 2.31533163546, 0.54337652891, 3.88272172907, 0.804014742236, 2.22328223762, 0.550997466483, 3.81016986603, 0.782451403668, 2.31532962949, 0.543376648512, 3.88272057365, 0.804014394452, 2.22328367772, 0.550997312506, 3.81017131056, 0.782451827423, 2.31532776492, 0.543376759684, 3.88271949966, 0.80401407118, 2.22328501633, 0.550997169382, 3.81017265328, 0.78245222131, 2.31532603177, 0.543376863022, 3.88271850136, 0.804013770688, 2.22328626061, 0.550997036345, 3.81017390136, 0.782452587439, 2.31532442076, 0.543376959078, 3.88271757339, 0.80401349137, 2.22328741722, 0.550996912683, 3.8101750615, 0.782452927767, 2.31532292329, 0.543377048367, 3.88271671081, 0.804013231732, 2.22328849233, 0.550996797734, 3.81017613989, 0.782453244115, 2.31532153132, 0.543377131365, 3.882715909, 0.804012990386, 2.2232894917, 0.550996690884, 3.81017714231, 0.782453538175, 2.31532023743, 0.543377208516, 3.88271516368, 0.804012766043, 2.22329042067, 0.550996591562, 3.8101780741, 0.782453811518, 2.3153190347, 0.543377280232, 3.88271447086, 0.804012557502, 2.2232912842, 0.550996499237, 3.81017894025, 0.782454065606, 2.31531791669, 0.543377346897, 3.88271382683, 0.80401236365, 2.22329208691, 0.550996413415, 3.81017974539, 0.782454301796, 2.31531687743, 0.543377408867, 3.88271322817, 0.804012183452, 2.22329283308, 0.550996333638, 3.81018049382, 0.782454521349, 2.31531591138, 0.543377466472, 3.88271267167, 0.804012015945, 2.2232935267, 0.55099625948, 3.81018118954, 0.78245472544, 2.31531501337, 0.543377520021, 3.88271215436, 0.804011860235, 2.22329417147, 0.550996190544, 3.81018183625, 0.782454915156, 2.3153141786, 0.543377569798, 3.88271167349, 0.80401171549, 2.22329477083, 0.550996126464, 3.81018243743, 0.782455091512, 2.31531340262, 0.54337761607, 3.88271122647, 0.804011580938, 2.22329532799, 0.550996066896, 3.81018299627, 0.782455255449, 2.31531268129, 0.543377659084, 3.88271081093, 0.804011455861, 2.22329584592, 0.550996011523, 3.81018351575, 0.782455407842, 2.31531201075, 0.543377699069, 3.88271042465, 0.80401133959, 2.22329632737, 0.550995960048, 3.81018399866, 0.782455549504, 2.31531138743, 0.543377736239, 3.88271006557, 0.804011231507, 2.22329677493, 0.550995912199, 3.81018444756, 0.782455681191, 2.315310808, 0.543377770792, 3.88270973177, 0.804011131033, 2.22329719098, 0.550995867718, 3.81018486486, 0.782455803606, 2.31531026936, 0.543377802912, 3.88270942148, 0.804011037634, 2.22329757773, 0.550995826369, 3.81018525278, 0.782455917402, 2.31530976865, 0.543377832771, 3.88270913303, 0.80401095081, 2.22329793725, 0.550995787932, 3.81018561338, 0.782456023187, 2.31530930319, 0.543377860527, 3.88270886488, 0.804010870098, 2.22329827147, 0.5509957522, 3.8101859486, 0.782456121524, 2.31530887051, 0.54337788633, 3.88270861562, 0.804010795069, 2.22329858215, 0.550995718984, 3.81018626021, 0.782456212937, 2.31530846828, 0.543377910316, 3.8827083839, 0.804010725322, 2.22329887097, 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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: 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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0.804125692731, 2.2228228844, 0.551046638533, 3.80970860273, 0.782316102997, 2.31592508302, 0.543341231432, 3.88306274731, 0.804117395367, 2.22285723193, 0.551042957771, 3.80974312748, 0.782326229216, 2.31588050994, 0.54334387667, 3.88303718936, 0.804109701512, 2.22288908191, 0.551039545229, 3.80977513681, 0.782335617771, 2.31583918502, 0.543346329999, 3.88301348583, 0.804102565966, 2.22291862128, 0.55103638075, 3.80980481972, 0.782344324067, 2.31580086414, 0.543348605725, 3.88299149847, 0.804095947093, 2.22294602228, 0.551033445776, 3.80983235018, 0.782352399117, 2.31576532255, 0.543350717035, 3.88297109982, 0.804089806517, 2.22297144366, 0.551030723209, 3.80985788851, 0.78235988992, 2.3157323532, 0.543352676089, 3.88295217236, 0.804084108847, 2.22299503185, 0.551028197289, 3.80988158249, 0.782366839809, 2.3157017652, 0.543354494111, 3.88293460761, 0.804078821423, 2.22301692196, 0.551025853477, 3.80990356843, 0.782373288754, 2.31567338251, 0.543356181464, 3.88291830543, 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0.804009805001, 2.22330268188, 0.550995280679, 3.8101903722, 0.782457419201, 2.31530316066, 0.543378226833, 3.88270532616, 0.804009804943, 2.22330268212, 0.550995280653, 3.81019037244, 0.782457419273, 2.31530316034, 0.543378226852, 3.88270532598, 0.804009804888, 2.22330268235, 0.550995280629, 3.81019037267, 0.782457419339, 2.31530316005, 0.543378226869, 3.88270532581, 0.804009804837, 2.22330268256, 0.550995280606, 3.81019037288, 0.782457419401, 2.31530315978, 0.543378226886, 3.88270532566, 0.80400980479, 2.22330268275, 0.550995280585, 3.81019037307, 0.782457419459, 2.31530315953, 0.543378226901, 3.88270532551, 0.804009804746, 2.22330268294, 0.550995280566, 3.81019037326, 0.782457419512, 2.31530315929, 0.543378226915, 3.88270532538, 0.804009804705, 2.22330268311, 0.550995280548, 3.81019037343, 0.782457419562, 2.31530315907, 0.543378226928, 3.88270532525, 0.804009804667, 2.22330268326, 0.550995280531, 3.81019037358, 0.782457419608, 2.31530315887, 0.54337822694, 3.88270532513, 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, 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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: 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, 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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, 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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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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, 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, 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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, 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, 2.28714135811, 0.546251738628, 3.85609923591, 0.797037894827, 2.25345541702, 0.549016527158, 3.8298246395, 0.789238002357, 2.28675375526, 0.546280266047, 3.8558267021, 0.796956616938, 2.25379864441, 0.548985401008, 3.83011889, 0.78932494828, 2.28637852466, 0.546307954683, 3.85556221016, 0.796877744811, 2.25413178739, 0.548955248427, 3.83040397039, 0.789409193414, 2.28601503777, 0.546334844136, 3.85530537931, 0.796801164248, 2.25445532176, 0.548926021164, 3.83068033374, 0.789490870827, 2.2856627123, 0.54636097124, 3.85505585542, 0.796726769053, 2.25476969009, 0.548897674442, 3.83094840059, 0.789570104051, 2.28532100787, 0.54638637031, 3.8548133086, 0.796654460312, 2.25507530471, 0.548870166633, 3.83120856196, 0.789647007963, 2.2849894221, 0.546411073368, 3.85457743106, 0.796584145745, 2.2553725504, 0.548843458973, 3.831461182, 0.789721689563, 2.28466748723, 0.546435110342, 3.85434793519, 0.796515739133, 2.25566178679, 0.548817515312, 3.83170660038, 0.78979424867, 2.28435476701, 0.546458509242, 3.85412455182, 0.796449159796, 2.25594335046, 0.548792301882, 3.83194513443, 0.789864778547, 2.28405085395, 0.546481296319, 3.85390702871, 0.796384332134, 2.25621755689, 0.548767787098, 3.832177081, 0.789933366454, 2.28375536693, 0.546503496212, 3.85369512915, 0.796321185207, 2.25648470218, 0.548743941374, 3.8324027182, 0.790000094147, 2.28346794895, 0.546525132078, 3.85348863071, 0.796259652364, 2.25674506462, 0.548720736958, 3.83262230692, 0.790065038334, 2.28318826514, 0.546546225704, 3.85328732411, 0.796199670902, 2.25699890605, 0.548698147787, 3.83283609221, 0.790128271073, 2.28291600102, 0.546566797618, 3.85309101221, 0.796141181758, 2.25724647317, 0.548676149352, 3.83304430456, 0.790189860146, 2.28265086088, 0.546586867182, 3.85289950909, 0.796084129233, 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0.792973398548, 2.27071575272, 0.547527202114, 3.84394342704, 0.793420244292, 2.2688085017, 0.547684204293, 3.84245123618, 0.792977230953, 2.27069938484, 0.547528541558, 3.8439306928, 0.793416462654, 2.26882463262, 0.547682868589, 3.84246392727, 0.792980997791, 2.27068329715, 0.547529858205, 3.84391817534, 0.793412745412, 2.26884048903, 0.547681555747, 3.8424764012, 0.792984700193, 2.27066748484, 0.547531152446, 3.84390587098, 0.793409091466, 2.26885607559, 0.547680265372, 3.84248866173, 0.792988339267, 2.27065194314, 0.547532424661, 3.84389377607, 0.793405499736, 2.26887139692, 0.547678997077, 3.84250071252, 0.792991916106, 2.2706366674, 0.547533675226, 3.84388188704, 0.793401969161, 2.26888645753, 0.547677750481, 3.84251255719, 0.79299543178, 2.27062165302, 0.547534904511, 3.84387020038, 0.793398498695, 2.26890126187, 0.547676525211, 3.84252419929, 0.792998887342, 2.27060689551, 0.547536112878, 3.84385871264, 0.793395087314, 2.26891581431, 0.547675320899, 3.84253564231, 0.793002283827, 2.27059239043, 0.547537300683, 3.84384742042, 0.793391734009, 2.26893011914, 0.547674137184, 3.84254688966, 0.79300562225, 2.27057813345, 0.547538468278, 3.84383632039, 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: |
#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. |
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