Driven oscillations of a limit-cycle oscillator

Abstract

When a chemical oscillator exhibiting a limit cycle is driven by harmonically modulating one of the rate constants, the oscillations may eventually synchronize with the driving frequency, or the two may beat. Near the transition from beating to phase locking, the oscillation frequency is pulled by the drive. The phase slips slowly during most of the beat cycle—almost phase-locked—then slips quickly for a brief time. General properties of a class of nonlinear oscillators predict this behavior. For a three-variable chemical-reaction scheme, stroboscopic equations have analytic solutions if these coupled differential equations are made separable by an additional approximation. This smooths the transition between the beating and phase-locked regimes and makes it.second order, with critical slowing-down of the transients. Computer results for the Tomita-Kitahara model give a weakly first-order transition. Driven biological clocks, including membranes in a.c. fields, should show the predicted behavior. Near the transition, the slow beat frequencies and the long damping times of transients complicate interpretation of experiments in which only a few beat cycles can be monitored.