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Unformatted text preview: CIS6930/4930 Intro to Computational Neuroscience Fall 2008 Home Work Assignment 5: Due Thursday 12/11/08 in my office The purpose of the first part of the assignment is to appreciate the complexity that can result from very simple dynamical systems. 1. Simulate the time course of the following parameterized class of discrete dynamical systems. The dynamics is that of a single variable x ∈ [0 , 1] that is updated by the equation f ( x n +1 ) = a * x n * (1- x n ) , where a lies in the range [0 , 4] . Note that as long as a ∈ [0 , 4] , f ( x ) ∈ [0 , 1] if x ∈ [0 , 1] . Hence the sequence of points x , f ( x ) , f ( f ( x )) , ... never leaves the unit interval. Your job is to plot the non-wandering (also called recurring) set for the dynamical system for a range of values of the parameter a . Start with a = 0 , choose a random point x ∈ [0 , 1] , and run the dynamical system for 5 , 000 points. Throw away the first 1 , 000 points (which are presumably the transient points until the system settles onto the non-wandering set) and plot the rest on the y-axis. Do this for several random initializationsonto the non-wandering set) and plot the rest on the y-axis....
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