Chapter 12.3, Problem 21E

van der Pol’s Equation.

a. Show that van der Pol’s equation

$\frac{d^{2}x}{d{t}^2}+\epsilon \left(x^2-1\right)\frac{dx}{dt}+x=0$

Is equivalent to the system

$\frac{dx}{dt}=y$,

$\frac{dy}{dt}=-x+\epsilon y-\epsilon x^{2}y$

Where $y:=dx/dt$.

b. Describe how the stability of the zero solution of van der Pol’s equation (equivalently, the origin of the system) depends on the parameter $\epsilon$.