Phase synchronization and invariant measures in sinusoidally perturbed chaotic systems

We show that, in periodically perturbed chaotic systems, Phase Synchronization appears, associated to a special type of stroboscopic map, in which not only averages quantities are equal to invariants of the perturbation, the angular frequency, but the map is strongly non-equivalent to the attractor. In cases where there is not phase synchronization, basic sets can still be found, but they are almost equivalent to the attractor, meaning that when there is not Phase Synchronization, the observation of the attractor by the stroboscopic map tells one few information about the final slate of a typical initial condition. We base our statements in experimental and numerical results from the sinusoidally perturbed Chua's circuit.