Phase Synchronization of Chaotic Oscillators

TLDR

The paper introduces phase synchronization as a new effect in weakly coupled self‑sustained chaotic oscillators. The authors characterize the phenomenon using analytic signal methods (Hilbert transform, partial Poincaré maps) and analyze its relation to the Lyapunov spectrum. In coupled Rössler attractors, phases lock while amplitudes remain chaotic and uncorrelated, and when a chaotic oscillator is coupled to a hyperchaotic one, frequencies entrain but phase differences diverge.

Abstract

We present the new effect of phase synchronization of weakly coupled self-sustained chaotic oscillators. To characterize this phenomenon, we use the analytic signal approach based on the Hilbert transform and partial Poincar\'e maps. For coupled R\"ossler attractors, in the synchronous regime the phases are locked, while the amplitudes vary chaotically and are practically uncorrelated. Coupling a chaotic oscillator with a hyperchaotic one, we observe another new type of synchronization, where the frequencies are entrained, while the phase difference is unbounded. A relation between the phase synchronization and the properties of the Lyapunov spectrum is studied.