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CHAOS

CHAOS. Lucy Calvillo Michael Dinse John Donich Elizabeth Gutierrez Maria Uribe. Problem Statement. Consider the function: f ( x )= a x (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.

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CHAOS

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  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: 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

  8. 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.

  9. a= 1.5 x0=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= 1.5 x0=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 0.333334 0.333334 0.333333 0.333333

  11. a = 1.5 x0=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 = 2.1 x0=0.3 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

  13. a = 2.1 x0=0.6 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

  14. a = 2.1 x0=0.9 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 0.52381 0.52381 0.52381 0.52381 0.52381

  15. a = 2.8 x0=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 = 2.8 x0=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 = 2.8 x0=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 = 3.1 x0=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 = 3.1 x0=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 = 3.1 x0=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 = 3.6 x0=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 = 3.6 x0=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 = 3.6 x0=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. 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

  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 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

  29. a= 1.5 x0=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 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333 0.333333333

  30. a= 1.5 x0=0.6 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

  31. a = 1.5 x0=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 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333

  32. a = 2.1 x0=0.3 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

  33. a = 2.1 x0=0.6 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

  34. a = 2.1 x0=0.9 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

  35. 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

  36. a = 2.8 x0=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 = 2.8 x0=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 = 3.1 x0=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 = 3.1 x0=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 = 3.1 x0=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 = 3.6 x0=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 = 3.6 x0=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 = 3.6 x0=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

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