h06-5 - of x . Next increment a by . 01 , choose a new...

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CIS6930/4930 Intro to Computational Neuroscience Spring 2006 Home Work Assignment 5: Due Monday 04/17/06 before class The purpose of this assignment is to appreciate the complexity that can result from very simple dynam- ical systems. 1. (50 pts) 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 0 , f ( x 0 ) , f ( f ( x 0 )) , ... 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 [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 initializations
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Unformatted text preview: of x . Next increment a by . 01 , choose a new random starting point x , and do the same, and so on and so forth until you reach a = 4 . . ( a is plotted along the x-axis). One would assume that the non-wandering points for successive values of a should look similar. Do they? In case you nd something interesting happening in a certain range for a , zoom into that range and change a by smaller increments. 2. (50 pts) Consider the 2-D phase space, R 2 and the particular instantiation of the Henon-map given by x n +1 = 1 . 245-x 2 n + 0 . 3 y n y n +1 = x n Find and report a 2-D region in R 2 that gets mapped back into itself. Now plot the attracting set of the map for points beginning in this region. 1...
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