Oscillators with Parametrically Excited Nonlinearity: Resonance, Anti-resonance and Switch

We discover presence of a hitherto unexplored type of resonance in a parametrically excited Van der Pol oscillator. The oscillator also possesses a state of anti-resonance. In the weak non-linear limit, we explain how to practically get a complete picture of different states of limiting oscillations present in the oscillator when the non-linear term therein is excited by an arbitrary $2\pi$ periodic function of time. We also illustrate how two such oscillators can be coupled to behave like a two-state switch allowing an sharp change of value of amplitude for stable oscillations from one constant to another.