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 < a < 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 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.

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 < a < 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: x n+1 = f(x n ) Using the starting values of x 0 =0.3 x 0 =0.6 x 0 =0.9 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: x n+1 = f(x n )f(x) = ax (1-x) x 1 = f(0.3) x 1 = (1.5)(0.3)(1-0.3) x 1 = 0.315 Iterations allowed us to compare the convergence behavior.

9. a= 1.5x 0 =0.3 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 0.333333294

10. a= 1.5x 0 =0.6 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 0.333333

11. a = 1.5x 0 =0.9 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 = 2.1x 0 =0.3 0.3 0.441 0.51769 0.524343 0.523756 0.523815 0.523809 0.52381

13. a = 2.1x 0 =0.6 0.6 0.504 0.524966 0.523691 0.523821 0.523808 0.52381

14. a = 2.1x 0 =0.9 0.9 0.189 0.321886 0.458378 0.521362 0.524042 0.523786 0.523812 0.523809 0.52381

15. a = 2.8x 0 =0.3 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 = 2.8x 0 =0.6 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 = 2.8x 0 =0.9 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 = 3.1x 0 =0.3 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 = 3.1x 0 =0.6 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 = 3.1x 0 =0.9 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 = 3.6x 0 =0.3 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 = 3.6x 0 =0.6 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 0.340633 0.808568 0.557228 0.88821 0.357455

23. a = 3.6x 0 =0.9 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. 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 f(x) = ax(1-x) f(x) = ax - ax 2 f ’(x) = a - 2ax f ’(x) = a (1 - 2x)

26. Problem Statement Consider g(x) = f(f(x)) and compute all fixed points.

27. g(x) = f(f(x)) f(x)=ax - ax 2 f(f(x))=a(ax - ax 2 ) - a(ax - ax 2 ) 2 g(x) = f(f(x)) g(x) = a(ax - ax 2 ) - a(ax - ax 2 ) 2 The fixed points of the function are: 0 (a - 1) / a 1/2 + 1/2a + 1/2a (a 2 - 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 x n+1 = g(x n ) for the values of: Using the starting values of x 0 =0.3 x 0 =0.6 x 0 =0.9 For a= 1.5, 2.1, 2.8, 3.1 & 3.6

29. a= 1.5x 0 =0.3 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

30. a= 1.5x 0 =0.6 0.6 0.3456 0.336235 0.334049 0.333512 0.333378 0.333344 0.333336 0.333334 0.333333

31. a = 1.5x 0 =0.9 0.9 0.175163 0.254629 0.305462 0.325441 0.331289 0.332818 0.333204 0.333301 0.333325 0.333331 0.333333

32. a = 2.1x 0 =0.3 0.3 0.51769 0.523756 0.523809 0.52381

33. a = 2.1x 0 =0.6 0.6 0.524966 0.523821 0.52381

34. a = 2.1x 0 =0.9 0.9 0.321886 0.521362 0.523786 0.523809 0.52381

35. a = 2.8x 0 =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

36. a = 2.8x 0 =0.6 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

37. a = 2.8x 0 =0.9 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

38. a = 3.1x 0 =0.3 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

39. a = 3.1x 0 =0.6 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

40. a = 3.1x 0 =0.9 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

41. a = 3.6x 0 =0.3 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 = 3.6x 0 =0.6 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

43. a = 3.6x 0 =0.9 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

44. Conclusions

45. Work Cited http://www.ukmail.org/~oswin/logistic.html http://www.cut-the- knot.com/blue/chaos.shtml