Math 6410 • ODE

Consider the van der Pol equation \[ \ddot {x}+x+\epsilon (x^2-1)\dot {x}=\Gamma \cos (\omega t), \qquad 0<\epsilon \ll 1 \] with \(\Gamma =O(1)\) and \(\omega \ne 1/3,1,3\). Using the method of multiple scales show that the solution is attracted to \[ x(t)=\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2\ge 2(1-\omega ^2)^2\) and \[ x(t)=2\left [1-\frac {\Gamma ^2}{2(1-\omega ^2)^2}\right ]^{1/2}\cos t +\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2<2(1-\omega ^2)^2\). Explain why this result breaks down when \(\omega =1/3,1,3\).