Mathematics 118A, Winter 1998

Homework Assignment 6, due Monday, March 2

1. The Lienard form of the Van der Pol equation,

ẋ_{1} = x_{2} − (\frac{1}{3}x_{1}^{3} − x_{1})
ẋ_{2} = −x_{1},

is a special case of the system characterized by Theorem 4.6 in Verhulst’s book. Re-do the proof of this Theorem for this special case only. You should strive for a very clear exposition of the argument.

2. What happens if you add a fifth order term to the Van der Pol equation? Consider the system

ẋ_{1} = x_{2} − (\frac{1}{5}x_{1}^{5} − x_{1}^{3} + x_{1})
ẋ_{2} = −x_{1}.

Is this case still covered by Theorem 4.6? How many periodic solutions to you expect to find? Use Mathematica to plot the periodic orbits in the phase space. (How do you find these orbits if you can’t solve the equation explicitly?) Try to conjecture how the addition of higher order terms to the system will change the phase portrait.