1.
Construct a phase diagram for the potential *U*(*x*) =
-(l/3)*x*^{3}.

2. Use numerical calculations to find a solution of the van der Pol oscillator:

Let *a* and w_{0} be equal to 1. Plot the phase
diagram of the solution for the following conditions:

a.
m = 0.007, *x*_{0} = 1.0, *dx*_{0}/*dt* = 0.

b.
m = 0.007, *x*_{0} = 3.0, *dx*_{0}/*dt* = 0.

c.
m = 0.05, *x*_{0} = 1.0, *dx*_{0}/*dt* = 0.

d.
m = 0.05, *x*_{0} = 3.0, *dx*_{0}/*dt* = 0.

In each case, describe whether the motion appears to approach a limit cycle.

3.
Calculate the potential due to a thin circular ring of radius *a* and mass *M* for points lying in the plane of the ring and exterior to it. The result can be expressed as an
elliptical integral (see Appendix *B* for a list of elliptical integrals).

For the region where the distance from the center of the ring is large compared to the radius of the ring, expand the expression for the potential and find the first correction term.

4. The orbital revolution of the Moon about the Earth takes 27.3 days and is in the same direction as the Earth’s rotation (24 hours). Use this information to show that high tides occur everywhere on Earth every 12 hours and 26 minutes.

5.
The so-called *tent* map
is represented by the following iterations:

where 0 < *a* < 1.

a.
Make a map up to 20 iterations for *a* = 0.4 and *a* = 0.7 with *x*_{1} = 0.2.

b.
Plot a bifurcation diagram for the *tent* map.

c.
Show analytically that the Lyapunov exponent for the *tent* map is *l* = ln(2* a*). This indicates that chaotic behavior
occurs for *a* > 1/2.

Practice the material and concepts discussed in Chapter 4.