Frequency stabilization by synchronization of Duffing oscillators

Centro Atómico Bariloche and Instituto Balseiro (Comisión Nacional de Energía Atómica Universidad Nacional de Cuyo), Consejo Nacional de Investigaciones Científicas y Técnicas 8400 San Carlos de Bariloche, Río Negro, Argentina

Received: 15 June 2016
Accepted: 30 July 2016

Abstract

We present analytical and numerical results on the joint dynamics of two coupled Duffing oscillators with nonlinearity of opposite signs (hardening and softening). In particular, we focus on the existence and stability of synchronized oscillations where the frequency is independent of the amplitude. In this regime, the amplitude-frequency interdependence (a-f effect) —a noxious consequence of nonlinearity, which jeopardizes the use of micromechanical oscillators in the design of time-keeping devices— is suppressed. By means of a multiple time scale formulation, we find approximate conditions under which frequency stabilization is achieved, characterize the stability of the resulting oscillations, and compare with numerical solutions to the equations of motion.