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 also it exists a very large number of non-transient transformations, possibly infinity. In cases where there is not phase synchronization there is either only transitive transformations on the attractor, or a finite number of non-transitive transformations. We base our statements in experimental and numerical results from the sinusoidally perturbed Chua's circuit. 
M.S. Baptista, T. Pereira, J.C. Sartorelli, I.L. Caldas, and J. Kurths