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2 Simple Maps     continued...

This two-dimensional map is given by the equations [1,2]

where and . Note the nonlinear term in the first equation. Write scripts for iterating with these equations and for making an x--y plot of all of the points generated. For sensitivity to initial conditions (the Butterfly effect), consider the following two cases:


and compare the results after 10, 20, 30, 40, and 50 iterations. Next, perform a large number of iterations (400 or more) and plot the resulting object. You can see that the general form of the attractor does not depend on the initial x and y values, even though the Butterfly effect is very obvious. What does this exercise tell us about strange attractors? Examine the behavior of the system for other choices of the constants and . Are there values of and for which the attractor disappears?