**Exercise 4.9.** **Secondary islands**

In figure 4.3 (section 4.1)
we see a chain of six secondary
islands in the oscillation region. Carry out the Poincaré-Birkhoff
construction to obtain the alternating sequence of stable and unstable
fixed points for this island chain.

**Exercise 4.10.** **Invariant curves of the standard map**

**a**. Make programs that reproduce
figures 4.22
and 4.23. You will need to
develop an effective method of estimating the probability of
visitation. There is one suggestion of how to do that in the text,
but you may find a better way.

**b**. As the parameter *K* is increased beyond the critical value,
the golden invariant curve ceases to exist. Investigate how the
method for finding invariant curves fails beyond the critical value of
*K*.