Duffing oscillators: control and memory effects

In the first part of this article we study the hysteretic bistable response of Duffing oscillators and show ways to control the switching between stable branches of this nonlinear response. The control mechanism is either applied through a pulse that can be in phase or out of phase with the periodic driving force or through a frequency-modulated driving force. In the second part we show how memory effects in dissipation qualitatively and quantitatively alter the dynamics of Duffing oscillators. We show how memory functions corresponding to different dissipative regimes (diffusion, subdiffusion, and superdiffusion) affect the oscillator. In particular, we obtain universal power laws for the absoption when the driving frequency omega-->0 . For subdiffusive memories the power law exponents nu<2 , for diffusive memories nu=2 , and for superdiffusive memories nu>2 .