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2 Simple Maps     continued...

This famous two-dimensional map, widely used to model accelerator dynamics, is given by

Notice in Eq. (20) that the restrictions (to and ) keep the values of q and p between 0 and 1. Write scripts to iterate this map and to plot it. For each k-value, you should generate a plot using several starting points, each of which lies in a different part of the unit square. For small values of k there is no chaos and for values of k above chaos reigns. Study the behavior of this map for different k values. The k region between .9 and 1.1 is thought to be especially important. In particular, the onset of widespread chaotic behavior is believed to occur at [7,8].