Quasiperiodic phenomena in the Van der Pol–Mathieu equation

The Van der Pol–Mathieu equation, combining self-excitation and parametric excitation, is analysed near and at 1:2 resonance, using the averaging method. We analytically prove the existence of stable and unstable periodic solutions near the parametric resonance frequency. Above a certain detuning threshold, quasiperiodic solutions arise with basic periods of order 1 and order 1 / ε where ε is the (small) detuning parameter.