**Exercise 3.14.** **Periodically driven pendulum**

Explore the dynamics of the driven pendulum, using the surface of
section method. We are interested in exploring the regions of
parameter space over which various phenomena occur. Consider a
pendulum of length 9.8 m, mass 1 kg, and acceleration
of gravity *g* = 9.8 m s^{-2}, giving _{0} = 1 rad s^{-1}.
Explore the parameter plane of the amplitude *A* and frequency
of the periodic drive.

Examples of the phenomena to be investigated:

**a**. Inverted equilibrium. Show the
region of parameter space (*A*, ) in which the inverted
equilibrium is stable. If the inverted equilibrium is stable there is
some range of stability, i.e., there is a maximum angle of displacement
from the equilibrium that stable oscillations reach. If you have
enough time, plot contours in the parameter space for different
amplitudes of the stable region.

**b**. Period doubling of the normal equilibrium. For this
case, plot the angular momenta of the stable and unstable equilibria
as functions of the frequency for some given amplitude.

**c**. Transition to large-scale chaos. Show the region of
parameter space (*A*, ) for which the chaotic zones around the
three principal resonance islands are linked.

**Exercise 3.15.** **Spin-orbit surfaces of section**

Write a program to compute surfaces of section for the spin-orbit
problem, with the section points being recorded at pericenter.
Investigate the following:

**a**. Give a Hamiltonian formulation of the spin-orbit problem
introduced in section 2.11.2.

**b**. For out-of-roundness parameter = 0.1 and
eccentricity *e* = 0.1, measure the widths of the regular islands
associated with the 1:1, 3:2, and 1:2 resonances.

**c**. Explore the surfaces of section for a range of
for fixed *e* = 0.1. Estimate the critical value of above
which the main chaotic zones around the 3:2 and the 1:1 resonance
islands are merged.

**d**. For a fixed eccentricity *e* = 0.1 trace the location on the
surface of section of the stable and unstable fixed points associated
with the 1:1 resonance as a function of the out-of-roundness
.