**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