Phase Synchronization of Chaotic Self-Oscillatory Systems
J. Kurths, A. Pikovsky and M. Rosenblum
Max-Plank-Arbeitsgruppe ``Nichtlineare Dynamik''
an der Universität Potsdam, Potsdam, Germany
Synchronization of chaotic oscillators is mostly understood as the complete coincidence of states of individual subsystems. Other approaches define synchronization phenomena as a rapprochement of power spectra of respective signals or as a convergence of the partial dimensions. In this work we study phase synchronization of chaotic oscillators; this effect is a direct generalization of the classical phase or frequency locking. Our main finding is that in the chaotic case phase synchronization means that the phases of interacting systems become locked, while the amplitudes vary chaotically, and are practically uncorrelated. We also demonstrate a weaker type of synchronization, where the frequencies are entrained, while the phase difference exhibits a random-walk-type motion.
To introduce instantaneous phases of chaotic systems, we use the analytic signal approach based on the Hilbert transform and partial Poincaré maps. Then, using the Rössler system as a basic model, we have calculated the region of phase locking, which is completely analogous to the synchronization region (the Arnold tongue) for coupled periodic oscillators.
It is remarkable, how the phase synchronization manifests itself
in the Lyapunov spectrum.
In the absence of coupling, each oscillator
has one positive, one negative, and one vanishing Lyapunov
exponents. As the coupling is increased, the positive and the negative
exponents remain, whereas one of the zero exponents becomes negative.
This behavior can be explained as follows: without coupling,
the vanishing exponents correspond to the translation along the trajectory,
i.e. to the shift of the phase of the oscillator.
The coupling produces an ``attraction'' of the phases
such that the phase difference 
decreases. Thus one
of the vanishing exponents becomes negative. For large coupling
the attraction is so strong that the phases remain locked.
It is noteworthy that the phenomenon of phase synchronization is observed even when completely different systems, such as the Rössler oscillator and the Mackey-Glass differential-delay system, or the Rössler and the hyperchaotic Rössler oscillators, interact.
A very interesting manifestation of the described effect is the self-synchronization in a population of non-identical chaotic oscillators, coupled via the mean field. We have studied behavior of large ensembles of Rössler oscillators with Gaussian distributed natural frequencies, and have shown that with the increase of the coupling coefficient, the variance of the mean field sharply increases. This effect takes place because the oscillators synchronize, i.e. oscillate in-phase, and their contributions to the mean field produce a non-zero component. We emphasize, that the amplitudes of the elements of the ensemble remain chaotic and non-correleted.
It is important to know that the phase synchronization is observed already for extremely weak coupling, and in some cases can have no threshold, contrary to other types of synchronization. This phenomenon is a direct generalization of synchronization of periodic self-sustained oscillators. As the latter, it may find practical applications, in particular when a coherent summation of outputs of slightly different generators operating in a chaotic regime is necessary. For this purpose, it is sufficient to synchronize phases, while amplitudes can remain uncorrelated. We expect this to be relevant, e.g. for an important problem of outputs summation in arrays of semiconductor lasers. As another application we are going to investigate the interaction of the cardiac and the respiratory systems.