The van der Pol equation is a second-order differential equation describing oscillatory dynamics in a variable x:

dx2 dt 2 2 s1 2 x2d dx dt 1 x − 0 where  is a positive constant. This equation was first obtained by an electrical engineer named Balthasar van der Pol, but has since been used as a model for a variety of phenomena, including sustained oscillatory dynamics of neural impulses.

(a) Convert the Van der Pol equation into a system of two first-order differential equations by defining the new variable y as y − x 2 x3 3

2 dxydt



(b) Construct the phase plane for the equations obtained in part (a), including nullclines, the equilibrium, and the direction of movement.

(c) What does the phase plane analysis from part

(b) tell you about the dynamics of x?