FM 06-15; arXiv:1506.07573

Authors: Leonardo De Carlo, Guido Gentile, Alessandro Giuliani

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

Sommario: 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.

Keywords: partially hyperbolic systems; Anosov systems; synchronization; phase-locking; Lyapunov exponents; fractal attractor; SRB measure; tree expansion; perturbation theory.

Leonardo De Carlo
Gran Sasso Science Institute (GSSI), Viale Francesco Crispi 7, L'Aquila, 67100, Italy
neoleodeo@gmail.com

Guido Gentile
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, 00146 Roma Italy
gentile@mat.uniroma3.it

Alessandro Giuliani
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, 00146 Roma Italy
giuliani@mat.uniroma3.it