CHAOS
Lucy Calvillo
Michael Dinse
John Donich
Elizabeth Gutierrez
Maria Uribe
Problem Statement
• Consider the function:
f(x)=ax(1-x)
on the interval [0,1]
where a is a real number 1 < a < 5
• This function is also known as the logistic
function.
Logistic Function and the
unrestricted growth function
• The model for unrestricted growth is very
simple: f(x) = ax
• For an example using flies this means that
in each generation there will be a times as
many flies as in the previous generation.
Logistic Function and the
restricted growth function
• In 1845 P.F Verhulst derived a model of restricted growth.
• The model is derived by supposing the factor a decreases
as the number x increases.
• The biggest population that the environment will support is
x=1.
• For our example if there are x insects then 1-x is a measure
of the space nature permits for population growth.
• Consequently replacing a by a(1-x) transforms the model
to:
f(x) = ax(1- x)
which is the initial equation we were given.
Problem Statement
• Compute the fixed points for the function:
f(x)=ax(1-x)
on the interval [0,1]
where a is a real number 1 < a < 5
Fixed Points
• A fixed point is a point which does not
change upon application of a map, system
of differential equations, etc.
• The fixed points can be obtained
graphically as the points of intersection of
the curve f(x) and the line y = x.
• The fixed points of the logistic function are
0 and (a -1) / a.
Problem Statement
• Compute the first twenty values of the
sequence given by:
xn+1= f(xn)
Using the starting values of
x0=0.3
x0=0.6
x0=0.9
For a= 1.5, 2.1, 2.8, 3.1 & 3.6
Iterations
• Iteration: making repititions, iterations are
functions that are repeated. For instance the
first iteration yields:
xn+1 = f(xn)
f(x) = ax (1-x)
x1 = f(0.3)
x1 = (1.5)(0.3)(1-0.3)
x1 = 0.315
• Iterations allowed us to compare the
convergence behavior.
a= 1.5 x0=0.3
y
0.5
0.375
0.25
0.125
0
0
0.25
0.5
0.75
1
x
0.3
0.315
0.3236625
0.328357629
0.330808345
0.332061276
0.332694877
0.333013494
0.33317326
0.333253258
0.333293286
0.333313307
0.33332332
0.333328326
0.33333083
0.333332082
0.333332707
0.33333302
0.333333177
0.333333255
0.333333294
a= 1.5 x0=0.6
y
0.5
0.375
0.25
0.125
0
0
0.25
0.5
0.75
1
x
0.6
0.36
0.3456
0.339241
0.336235
0.334771
0.334049
0.333691
0.333512
0.333422
0.333378
0.333356
0.333344
0.333339
0.333336
0.333335
0.333334
0.333334
0.333334
0.333333
0.333333
a = 1.5 x0=0.9
y
0.5
0.375
0.25
0.125
0
0
0.25
0.5
0.75
1
x
0.9
0.135
0.175163
0.216721
0.254629
0.28469
0.305462
0.318233
0.325441
0.329294
0.331289
0.332305
0.332818
0.333075
0.333204
0.333269
0.333301
0.333317
0.333325
0.333329
0.333331
a = 2.1 x0=0.3
y
0.3
0.441
0.51769
0.524343
0.523756
0.523815
0.523809
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
a = 2.1 x0=0.6
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.6
0.504
0.524966
0.523691
0.523821
0.523808
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
a = 2.1 x0=0.9
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.9
0.189
0.321886
0.458378
0.521362
0.524042
0.523786
0.523812
0.523809
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
a = 2.8 x0=0.3
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.3
0.588
0.678317
0.610969
0.665521
0.623288
0.65744
0.630595
0.652246
0.6351
0.648895
0.637925
0.646735
0.639713
0.645345
0.64085
0.644452
0.641574
0.643879
0.642037
0.643511
a = 2.8 x0=0.6
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.6
0.672
0.617165
0.661563
0.626913
0.654901
0.632816
0.650608
0.636489
0.647838
0.638803
0.646055
0.64027
0.644908
0.641205
0.644171
0.641801
0.643699
0.642182
0.643396
0.642425
a = 2.8 x0=0.9
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.9
0.252
0.527789
0.697838
0.590409
0.677114
0.612166
0.664773
0.62398
0.656961
0.631017
0.651937
0.635363
0.648695
0.638091
0.646606
0.639818
0.645262
0.640917
0.644399
0.641617
a = 3.1 x0=0.3
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.3
0.651
0.704317
0.645589
0.709292
0.639211
0.714923
0.631805
0.721145
0.623394
0.727799
0.614133
0.734618
0.604358
0.741239
0.594592
0.747262
0.58547
0.752354
0.577584
0.75634
a = 3.1 x0=0.6
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.6
0.744
0.590438
0.749645
0.5818
0.754257
0.574595
0.75775
0.569051
0.760219
0.565087
0.761868
0.562419
0.762922
0.560703
0.763577
0.559634
0.763976
0.558982
0.764215
0.55859
a = 3.1 x0=0.9
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.9
0.279
0.623593
0.727647
0.614348
0.734466
0.60458
0.741095
0.594806
0.747136
0.585663
0.752252
0.577744
0.756263
0.571421
0.759187
0.566748
0.761189
0.56352
0.762492
0.561403
a = 3.6 x0=0.3
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.3
0.756
0.66407
0.803091
0.569288
0.882717
0.3727
0.841661
0.479763
0.898526
0.328238
0.793792
0.58927
0.871311
0.403661
0.866588
0.416209
0.874724
0.394494
0.859926
0.433631
a = 3.6 x0=0.6
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.6
0.864
0.423014
0.878664
0.38381
0.8514
0.455466
0.89286
0.344379
0.812816
0.547727
0.8918
0.347375
0.81614
0.5402
0.894182
0.340633
0.808568
0.557228
0.88821
0.357455
a = 3.6 x0=0.9
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.9
0.324
0.788486
0.600392
0.863717
0.423756
0.879072
0.382695
0.850462
0.457835
0.893599
0.342286
0.810455
0.553025
0.889878
0.352782
0.821977
0.526792
0.897416
0.331418
0.797689
Problem Statement
• Compute f’(x) and explain the behavior
f(x) = ax(1-x)
f(x) = ax - ax2
f ’(x) = a - 2ax
f ’(x) = a (1 - 2x)
By evaluating the derivative at the fixed point (x*) it can be determined
•Where f ’(x*) = m, for
• m < -1, the iterative path is repelled and spirals away from fixed point
•-1 < m, the iterative path is attracted and spirals into the fixed point
• 0 < m <1, the iterative path is attracted and staircases into the fixed point
• m >1, the iterative path is repelled and staircases away
Problem Statement
• Consider g(x) = f(f(x)) and compute all fixed
points.
g(x) = f(f(x))
•
f(x)=ax - ax2
f(f(x))=a(ax - ax2) - a(ax - ax2)2
g(x) = f(f(x))
g(x) = a(ax - ax2) - a(ax - ax2)2
• The fixed points of the function are:
0
(a - 1) / a
1/2 + 1/2a + 1/2a (a2 - 2a - 3)0.5
• The first two fixed points are the same as those computed
for the general logistic function.
• The two new fixed points are the numerical values of the
orbit of convergence.
Problem Statement
• Investigate the sequence xn+1 = g(xn) for the
values of:
Using the starting values of
x0=0.3
x0=0.6
x0=0.9
For a= 1.5, 2.1, 2.8, 3.1 & 3.6
a= 1.5 x0=0.3
y
0.3
0.3236625
0.330808345
0.332694877
0.33317326
0.333293286
0.33332332
0.33333083
0.333332707
0.333333177
0.333333294
0.333333324
0.333333331
0.333333333
0.333333333
0.333333333
0.333333333
0.333333333
0.333333333
0.333333333
0.333333333
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
a= 1.5 x0=0.6
y
0.6
0.3456
0.336235
0.334049
0.333512
0.333378
0.333344
0.333336
0.333334
0.333334
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
a = 1.5 x0=0.9
y
0.9
0.175163
0.254629
0.305462
0.325441
0.331289
0.332818
0.333204
0.333301
0.333325
0.333331
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
a = 2.1 x0=0.3
y
0.3
0.51769
0.523756
0.523809
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
a = 2.1 x0=0.6
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.6
0.524966
0.523821
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
a = 2.1 x0=0.9
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.9
0.321886
0.521362
0.523786
0.523809
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
0.52381
a = 2.8 x0=0.3
0.3
0.678317
0.665521
0.65744
0.652246
0.648895
0.646735
0.645345
0.644452
0.643879
0.643511
0.643276
0.643125
0.643029
0.642967
0.642927
0.642902
0.642886
0.642876
0.642869
0.642865
y
0.75
0.5
0.25
0
0.25
0.5
0.75
1
x
a = 2.8 x0=0.6
y
0.6
0.617165
0.626913
0.632816
0.636489
0.638803
0.64027
0.641205
0.641801
0.642182
0.642425
0.642581
0.64268
0.642744
0.642785
0.642811
0.642827
0.642838
0.642845
0.642849
0.642852
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
a = 2.8 x0=0.9
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.9
0.527789
0.590409
0.612166
0.62398
0.631017
0.635363
0.638091
0.639818
0.640917
0.641617
0.642064
0.64235
0.642533
0.64265
0.642724
0.642772
0.642803
0.642822
0.642835
0.642843
a = 3.1 x0=0.3
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.3
0.704317
0.709292
0.714923
0.721145
0.727799
0.734618
0.741239
0.747262
0.752354
0.75634
0.759241
0.761225
0.762515
0.763326
0.763824
0.764124
0.764304
0.764411
0.764475
0.764512
a = 3.1 x0=0.6
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.6
0.590438
0.5818
0.574595
0.569051
0.565087
0.562419
0.560703
0.559634
0.558982
0.55859
0.558355
0.558216
0.558133
0.558085
0.558056
0.558039
0.558029
0.558023
0.558019
0.558017
a = 3.1 x0=0.9
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.9
0.623593
0.614348
0.60458
0.594806
0.585663
0.577744
0.571421
0.566748
0.56352
0.561403
0.560067
0.559245
0.558748
0.558449
0.558272
0.558166
0.558104
0.558067
0.558045
0.558033
a = 3.6 x0=0.3
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.3
0.66407
0.569288
0.3727
0.479763
0.328238
0.58927
0.403661
0.416209
0.394494
0.433631
0.368764
0.488727
0.325317
0.596929
0.417292
0.392741
0.437104
0.364284
0.499137
0.324008
a = 3.6 x0=0.6
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.6
0.423014
0.38381
0.455466
0.344379
0.547727
0.347375
0.5402
0.340633
0.557228
0.357455
0.515405
0.326458
0.593934
0.411851
0.401746
0.419743
0.388846
0.444977
0.354961
0.521457
a = 3.6 x0=0.9
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
0.9
0.788486
0.863717
0.879072
0.850462
0.893599
0.810455
0.889878
0.821977
0.897416
0.797689
0.876396
0.856419
0.88817
0.826966
0.899175
0.791473
0.868084
0.87228
0.864764
0.877538
Conclusions
Work Cited
http://www.ukmail.org/~oswin/logistic.html
http://www.cut-the- knot.com/blue/chaos.shtml