Mixed-mode dynamics of certain bistable oscillators: behavioural mapping, approximations for motion and links with van der Pol oscillators

This study is concerned with a new generalized mathematical model for single degree-of-freedom bistable oscillators with harmonic excitation of low-frequency, linear viscous damping and a restoring force that contains a negative linear term and a positive nonlinear term which is a power-form function of the generalized coordinate. Comprehensive numerical mapping of the range of bifurcatory behaviour shows that such non-autonomous systems can experience mixed-mode oscillations, including bursting oscillations (fast flow oscillations around the outer curves of a slow flow), and relaxation oscillations like a classical (autonomous) van der Pol oscillator. After studying the global system dynamics the focus of the investigations is on cubic oscillators of this type. Approximate techniques are presented to quantify their response, i.e. to determine approximations for both the slow and fast flows. In addition, a clear analogy between the behaviour of two archetypical oscillators—the non-autonomous bistable oscillator operating at low frequency and the strongly damped autonomous van der Pol oscillator—is established for the first time.