Construction of the Lyapunov spectrum in a chaotic system displaying phase synchronization

TL;DR

This paper constructs the Lyapunov spectrum for a chaotic system with phase synchronization, revealing how weak coupling leads to a lower-dimensional attractor and providing explicit calculations of Lyapunov exponents.

Contribution

It introduces a convergent perturbative method to construct the invariant manifold and compute the Lyapunov spectrum in a coupled chaotic system with phase synchronization.

Findings

Invariant manifold of dimension less than three constructed

Lyapunov exponents computed explicitly via convergent series

Synchronization causes the system to settle on a lower-dimensional attractor

Abstract

We consider a three-dimensional chaotic system consisting of the suspension of Arnold's cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.