The Behavior of Synchronized Frequency in Weakly Coupled Nonidentical Periodic Oscillators

Abstract

Interaction between dynamical systems has been a subject of intensive study for the past couple of decades. These studies have mainly focused on synchronization of chaotic systems, conditions of different kinds of synchronized behavior, amplitude death, etc. Synchronization of periodic oscillators and the frequency of the resulting synchronized behavior have remained relatively unexplored. In this paper we consider synchronization of nonidentical periodic oscillators for different coupling schemes, and study the nature of the synchronized frequency. Based on numerical and experimental observations we show that for directly coupled oscillators, the synchronized frequency lies between the individual frequencies and its value does not depend on the coupling constant, while for indirectly coupled oscillators the synchronized frequency lies out of the range and depends on the strength of coupling. We explain the different frequency behaviors of directly and indirectly coupled systems by analytically deriving the expressions of synchronized frequency under certain simplifying assumptions.