Resonances within chaos

In chaotic systems, synchronization phenomena, i.e., resonances, can occur, the simplest being motions asymptotically developing over a low dimension attractor that is visited in synchrony with a weak time-periodic perturbing force. This is analogous to the synchronization (“phase locking”) occurring when an integrable system, perturbed by a periodic forcing and in the presence of friction, develops attracting periodic orbits with period being a simple fraction (1, 3/2, 2,. . .) of the forcing period. Formation of an attractor and existence of dissipation, i.e., of phase volume contraction, are concomitant. A simple illustration of the resonance phenomenon in a chaotic system is exhibited here and exactly solved by deriving the shape of the attractor. This turns out to be a continuous surface with weak smoothness properties, being a strange attractor with an exponent of Hoelder-continuity with a very small lower bound. The peculiarity of the class of systems treated is the possibility of determining analytically an attractor, whose existence can be rather easily seen in simulations.