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].