Ageing of an oscillator due to frequency switching

Abstract

If an oscillator is driven by a force that switches between two frequencies, the dynamics it exhibits depends on the precise manner of switching. Here we take a one-dimensional oscillator and consider scenarios in which switching occurs either: (i) between two driving forces which have different frequencies, or (ii) as a single forcing whose frequency switches between two values. The difference is subtle, but its effect on the long term behaviour is severe, and occurs because the expressions of (i) and (ii) are linear and nonlinear, respectively, in terms of a discontinuous quantity (e.g. a sign or Heaviside step function) that represents the switch between frequencies. In scenario (i) the oscillator can be described as a Filippov system, and we will show it has a stable periodic orbit. In scenario (ii) the oscillator exhibits hidden dynamics, which lies outside the theory of Filippov’s systems, and causes the system to be increasingly (as time passes) dominated by sliding along the frequency-switching threshold, and in particular if periodic orbits do exist, they too exhibit sliding. We show that the behaviour persists, at least asymptotically, if the systems are regularized (i.e. if the switch is modelled in the manner of (i) or (ii) but with a smooth rather than discontinuous transition).