4.) Logistic Map (12 pts.) Consider the logistic map defined by the iterative equation Pn+1 = p In (1 - In) (1) where n

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4 Logistic Map 12 Pts Consider The Logistic Map Defined By The Iterative Equation Pn 1 P In 1 In 1 Where N 1 (65.57 KiB) Viewed 30 times

4.) Logistic Map (12 pts.) Consider the logistic map defined by the iterative equation Pn+1 = p In (1 - In) (1) where n denotes the time step and u is the "growth” parameter. (a) Find the first-order fixpoint, 2*, defined by r = f(x) and determine the range of u values for which the fixpoint is stable. (hint consider small deviations from the fixpoint, f(x* + Ax) = f(x*) + f'(2*)Ar; what condition should the derivative f'(x*) satisfy to render the fixpoint stable?) (b) The definition for second-order fixpoints is given by i = f(f(x)]. For p=3.3 it has the solutions * = 0.479,0.700,0.824. Sketch both sides of the equation, r = f[f(x)], graphically vs. r and determine whether the fixpoints are stable or unstable. =