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: 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

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 = Iterations allowed us to compare the convergence behavior.

a= 1.5x 0 =

a= 1.5x 0 =

a = 1.5x 0 =

a = 2.1x 0 =

a = 2.1x 0 =

a = 2.1x 0 =

a = 2.8x 0 =

a = 2.8x 0 =

a = 2.8x 0 =

a = 3.1x 0 =

a = 3.1x 0 =

a = 3.1x 0 =

a = 3.6x 0 =

a = 3.6x 0 =

a = 3.6x 0 =

Problem Statement Compute f’(x) and explain the behavior

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)

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

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.

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

a= 1.5x 0 =

a= 1.5x 0 =

a = 1.5x 0 =

a = 2.1x 0 =

a = 2.1x 0 =

a = 2.1x 0 =

a = 2.8x 0 =

a = 2.8x 0 =

a = 2.8x 0 =

a = 3.1x 0 =

a = 3.1x 0 =

a = 3.1x 0 =

a = 3.6x 0 =

a = 3.6x 0 =

a = 3.6x 0 =

Conclusions

Work Cited knot.com/blue/chaos.shtml