Chapter 3.6, Problem 24P

An asymptotically stable limit cycle is a closed trajectory $C$ in the phase plane such that all nonclosed trajectories, that start sufficiently near $C$, from either the inside or the outside, spiral toward $C$ as $t\to \infty$. Demonstrate numerically that the Van der Pol equation $u"-3\left(u^2-1\right)u\text{'}+u=0$ has such a limit cycle by drawing a direction field and phase portrait for the equivalent dynamical system. Describe the speed of the motion around $C$ by viewing the component plots of $u\left(t\right)$ and $u\text{'}\left(t\right)$.