CHAOS

1
CHAOS

• Lucy Calvillo
• Michael Dinse
• John Donich
• Elizabeth Gutierrez
• Maria Uribe

2
Problem Statement

• Consider the function f(x)ax(1-x) on
  the interval 0,1 where a is a real number 1
  lt a lt 5
• This function is also known as the logistic
  function.

3
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.

4
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 x1.
• 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.

5
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 lt a lt 5

6
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.

7
Problem Statement

• Compute the first twenty values of the sequence
  given by xn1 f(xn) Using the
  starting values of x00.3
  x00.6 x00.9 For a 1.5, 2.1, 2.8,
  3.1 3.6

8
Iterations

• Iteration making repititions, iterations are
  functions that are repeated. For instance the
  first iteration yields xn1 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.

9
a 1.5 x00.3
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 0.33333329
4
10
a 1.5 x00.6
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
11
a 1.5 x00.9
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

12
a 2.1 x00.3
0.3 0.441 0.51769 0.524343 0.523756 0.523815
0.523809 0.52381 0.52381 0.52381 0.52381 0.5
2381 0.52381 0.52381 0.52381 0.52381 0.52381
0.52381 0.52381 0.52381 0.52381
13
a 2.1 x00.6
0.6 0.504 0.524966 0.523691 0.523821 0.523808
0.52381 0.52381 0.52381 0.52381 0.52381 0.5
2381 0.52381 0.52381 0.52381 0.52381 0.52381
0.52381 0.52381 0.52381 0.52381
14
a 2.1 x00.9
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 0.523
81 0.52381 0.52381 0.52381 0.52381
15
a 2.8 x00.3
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
16
a 2.8 x00.6
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

17
a 2.8 x00.9
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

18
a 3.1 x00.3
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
19
a 3.1 x00.6
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
20
a 3.1 x00.9
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
21
a 3.6 x00.3
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
22
a 3.6 x00.6
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 0.34063
3 0.808568 0.557228 0.88821 0.357455
23
a 3.6 x00.9
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

24
Problem Statement

• Compute f(x) and explain the behavior

25
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 lt -1, the iterative path is repelled and
  spirals away from fixed point
• -1 lt m, the iterative path is attracted and
  spirals into the fixed point
• 0 lt m lt1, the iterative path is attracted and
  staircases into the fixed point
• m gt1, the iterative path is repelled and
  staircases away

26
Problem Statement

• Consider g(x) f(f(x)) and compute all fixed
  points.

27
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.

28
Problem Statement

• Investigate the sequence xn1 g(xn) for the
  values of Using the starting values of
  x00.3 x00.6 x00.9 For
  a 1.5, 2.1, 2.8, 3.1 3.6

29
a 1.5 x00.3
0.3 0.3236625 0.330808345 0.332694877 0.333173
26 0.333293286 0.33332332 0.33333083 0.3333327
07 0.333333177 0.333333294 0.333333324 0.33333
3331 0.333333333 0.333333333 0.333333333 0.333
333333 0.333333333 0.333333333 0.333333333 0.3
33333333
30
a 1.5 x00.6
0.6 0.3456 0.336235 0.334049 0.333512 0.33337
8 0.333344 0.333336 0.333334 0.333334 0.33333
3 0.333333 0.333333 0.333333 0.333333 0.33333
3 0.333333 0.333333 0.333333 0.333333 0.33333
3
31
a 1.5 x00.9
0.9 0.175163 0.254629 0.305462 0.325441 0.331
289 0.332818 0.333204 0.333301 0.333325 0.333
331 0.333333 0.333333 0.333333 0.333333 0.333
333 0.333333 0.333333 0.333333 0.333333 0.333
333
32
a 2.1 x00.3
0.3 0.51769 0.523756 0.523809 0.52381 0.52381
0.52381 0.52381 0.52381 0.52381 0.52381 0.5
2381 0.52381 0.52381 0.52381 0.52381 0.52381
0.52381 0.52381 0.52381 0.52381
33
a 2.1 x00.6
0.6 0.524966 0.523821 0.52381 0.52381 0.52381
0.52381 0.52381 0.52381 0.52381 0.52381 0.5
2381 0.52381 0.52381 0.52381 0.52381 0.52381
0.52381 0.52381 0.52381 0.52381
34
a 2.1 x00.9
0.9 0.321886 0.521362 0.523786 0.523809 0.523
81 0.52381 0.52381 0.52381 0.52381 0.52381 0
.52381 0.52381 0.52381 0.52381 0.52381 0.5238
1 0.52381 0.52381 0.52381 0.52381
35
a 2.8 x00.3
0.3 0.678317 0.665521 0.65744 0.652246 0.6488
95 0.646735 0.645345 0.644452 0.643879 0.6435
11 0.643276 0.643125 0.643029 0.642967 0.6429
27 0.642902 0.642886 0.642876 0.642869 0.6428
65
36
a 2.8 x00.6
0.6 0.617165 0.626913 0.632816 0.636489 0.638
803 0.64027 0.641205 0.641801 0.642182 0.6424
25 0.642581 0.64268 0.642744 0.642785 0.64281
1 0.642827 0.642838 0.642845 0.642849 0.64285
2
37
a 2.8 x00.9
0.9 0.527789 0.590409 0.612166 0.62398 0.6310
17 0.635363 0.638091 0.639818 0.640917 0.6416
17 0.642064 0.64235 0.642533 0.64265 0.642724
0.642772 0.642803 0.642822 0.642835 0.642843

38
a 3.1 x00.3
0.3 0.704317 0.709292 0.714923 0.721145 0.727
799 0.734618 0.741239 0.747262 0.752354 0.756
34 0.759241 0.761225 0.762515 0.763326 0.7638
24 0.764124 0.764304 0.764411 0.764475 0.7645
12
39
a 3.1 x00.6
0.6 0.590438 0.5818 0.574595 0.569051 0.56508
7 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

40
a 3.1 x00.9
0.9 0.623593 0.614348 0.60458 0.594806 0.5856
63 0.577744 0.571421 0.566748 0.56352 0.56140
3 0.560067 0.559245 0.558748 0.558449 0.55827
2 0.558166 0.558104 0.558067 0.558045 0.55803
3
41
a 3.6 x00.3
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

42
a 3.6 x00.6
0.6 0.423014 0.38381 0.455466 0.344379 0.5477
27 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

43
a 3.6 x00.9
0.9 0.788486 0.863717 0.879072 0.850462 0.893
599 0.810455 0.889878 0.821977 0.897416 0.797
689 0.876396 0.856419 0.88817 0.826966 0.8991
75 0.791473 0.868084 0.87228 0.864764 0.87753
8
44
Conclusions
45
Work Cited

• http//www.ukmail.org/oswin/logistic.html
• http//www.cut-the- knot.com/blue/chaos.shtml