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
.