Stability Theory of Synchronized Motion in Coupled-Oscillator Systems

TLDR

Weak coupling can lead to various non‑synchronized motions in coupled‑oscillator systems. The study develops a general stability theory for synchronized motions using an extended Lyapunov matrix approach and proposes an experimental method to measure the positive Lyapunov exponent of intrinsic chaos in reaction systems. The method employs an extended Lyapunov matrix approach and examines transition dynamics in a coupled Lorenz model. An explicit formula for the stability parameter of the synchronized state Ψunif is derived, and it is shown that chaotic Ψunif inevitably transitions to a non‑uniform state and then to non‑uniform chaos.

Abstract

The general stability theory of the synchronized motions of the coupled-oscillator systems is developed with the use of the extended Lyapunov matrix approach. We give the explicit formula for a stability parameter of the synchronized state Ψunif. When the coupling strength is weakened, the coupled system may exhibit several types of non-synchronized motion. In particular, if Ψunif is chaotic, we always get a transition from chaotic Ψunif to a certain non-uniform state and finally the non-uniform chaos. Details associated with such transition are investigated for the coupled Lorenz model. As an application of the theory, we propose a new experimental method to directly measure the positive Lyapunov exponent of intrinsic chaos in reaction systems.