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A well-known example from the study of discrete dynamical systems is the logistic map. The map is defined by , where . The points x=0 and are both fixed points of the map. Unlike the linear map, calculating the Lyapunov exponent is not trivial since the derivative is not constant for the map. We can, however, do a few computational experiments to better understand this map.

The following links call up the Maple Form Interface. All the examples considered involve the case b=4, so . Please read the explanations in this document before calling up the Maple Form Interface so that you can be clear on what each section of code is actually doing. (Of course, if you are familiar with Maple, you can decipher the code easily.)

1. This bit of code calculates the first N points in the orbits of x0 and y0. Initially, N=10 and which is a fixed point of this map. What happens to the orbit of y0 when y0 is close to ? Try changing x0 and y0 to see how orbits of other neighbouring points behave. (If you increase N, bear in mind that the output may not be easy to read.)
2. This code calculates the finite time analog to the Lyapunov exponent of an orbit orb(x0) over N iterations. Initially, and N=50. Increase N to make a reasonable guess as to what the Lyapunov exponent is. Change x0 to 0, 0.75 and other values in [0,1] to see how the approximation varies.
3. This code works out how errors in the initial condition are propagated. Given x0 and , Maple returns the number of iterations required for the separation between orb(x0) and orb() to be multiplied by 10. Initially, the values and are used; try changing them. Is there any pattern emerging?

Contents Next: Example III: The Up: Lyapunov Exponents Previous: Example I: The