  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.
