Chapter 7.5, Problem 17P

Consider the van der Pol system

$x\text{'}=y,y\text{'}=-x+\mu \left(1-x^2\right)y$,

where we now allow the parameter $\mu$ to be any real number.

(a) Show that the origin is the only critical point. Determine its type and stability property, and how these depend on $\mu$.

(b) Let $\mu =-1$; draw a phase portrait and conclude that there is a periodic solution that surrounds the origin. Observe that this periodic solution is unstable. Compare your plot with Figure 7.5.4.

(c) Draw a phase portrait for a few other negative values of $\mu$. Describe how the shape of the periodic solution changes with $\mu$.

(d) Consider small positive or negative values of $\mu$. By drawing phase portraits, determine how the periodic solution changes as $\mu \to 0$. Compare the behaviour of the van der Pol system as $\mu$ increases through zero with the behaviour of the system in Problem 16.