A frequency entrainment model with relevance to systems displaying adaptive behaviour

Abstract

This paper considers a non-conservative pendulum-type non-linear oscillator which emulates some behavioural features of systems displaying adaptive response, or biological rhythms, under the influence of an environmental excitation (‘ Zeitgeber ’) of electromagnetic origin, for instance. Of special interest are : (a) The entraining or locking-in process of the oscillation under a sudden change Δω of the Zeitgeber frequency ; (b) the time interval within which locking-in is achieved, and (c) the evolution of the oscillation under constant excitation of varying amplitude It is shown that the range of entertainment (pull-in region) increases with the amplitude of excitation and depends on the sign of Δω, and the oscillation damps out for constant excitation beyond a certain critical amplitude The above results are deduced on a quantitative basis by introducing a method of following up the entrainment process in the plane ‘ phase-error—instantaneous frequency offset ’.