Phase synchronization in the forced Lorenz system

We demonstrate that the dynamics of phase synchronization in a chaotic system under weak periodic forcing depends crucially on the distribution of intrinsic characteristic times of this system. Under the external periodic action, the frequency of every unstable periodic orbit is locked to the frequency of the force. In systems which in the autonomous case displays nearly isochronous chaotic rotations, the locking ratio is the same for all periodic orbits; since a typical chaotic orbit wanders between the periodic ones, its;phase follows the phase of the force. For the Lorenz attractor with its unbounded times of return onto a Poincare surface, such state of perfect phase synchronization is inaccessible. Analysis with the help of unstable periodic orbits shows that this state is replaced by another one, which we call ''imperfect phase synchronization," and in which we observe alternation of temporal segments, corresponding to different rational values of frequency lockings. [S1063-651X(99)12212-8].