Phase synchronization of chaotic oscillators by external driving

We extend the notion of phase locking to the case of chaotic oscillators. Different definitions of the phase are discussed, and the phase dynamics of a single self-sustained chaotic oscillator subjected to external force is investigated. We describe regimes where the amplitude of the oscillator remains chaotic and the phase is synchronized by the external force. This effect is demonstrated for periodic and noisy driving. This phase synchronization is characterized via direct calculation of the phase, as well as by implicit indications, such as the resonant growth of the discrete component in the power spectrum and the appearance of a macroscopic average field in an ensemble of driven oscillators. The R6ssler and the Lorenz systems are shown to provide examples of different phase coherence properties, with different response to the external force. A relation between the phase synchronization and the properties of the Lyapunov spectrum is discussed.