Subharmonic entrainment of a forced relaxation oscillator

Abstract

Using a simplified model of a relaxation oscillator which exhibits the characteristic free response of intervals of slow decay separated by rapid jumps, we determine steady-state response to a harmonic forcing input and obtain regions in the parameter space where subharmonic entrainment occurs. The model relaxation oscillator consists of a one-dimensional flow with jump conditions and is motivated by earlier studies of the flow on the slow manifold of a piecewiselinear relaxation oscillator. The results are obtained by studying the dynamics of the phase mapping which describes how the forcing phase varies between jumps. Details of the phase mapping are obtained both analytically and by numerical integration of the governing flow. The results obtained regarding existence and stability of subharmonics bear strong qualitative resemblance to experimental observations of frequency demultiplication by van der Pol and van der Mark (1927) and to numerical investigations of the forced van der Pol oscillator by Flaherty and Hoppensteadt (1978).