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CHAOS Lucy Calvillo Michael Dinse John Donich Elizabeth Gutierrez Maria Uribe

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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.

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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.

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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.

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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

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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.

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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

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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.

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a= 1.5x 0 =

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a= 1.5x 0 =

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a = 1.5x 0 =

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a = 2.1x 0 =

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a = 2.1x 0 =

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a = 2.1x 0 =

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a = 2.8x 0 =

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a = 2.8x 0 =

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a = 2.8x 0 =

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a = 3.1x 0 =

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a = 3.1x 0 =

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a = 3.1x 0 =

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a = 3.6x 0 =

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a = 3.6x 0 =

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a = 3.6x 0 =

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

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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)

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

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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.

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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

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a= 1.5x 0 =

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a= 1.5x 0 =

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a = 1.5x 0 =

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a = 2.1x 0 =

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a = 2.1x 0 =

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a = 2.1x 0 =

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a = 2.8x 0 =

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a = 2.8x 0 =

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a = 2.8x 0 =

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a = 3.1x 0 =

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a = 3.1x 0 =

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a = 3.1x 0 =

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a = 3.6x 0 =

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a = 3.6x 0 =

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a = 3.6x 0 =

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Conclusions

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Work Cited knot.com/blue/chaos.shtml

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