11 Synchronization of Chaotic Systems

The first observation of a synchronization phenomenon in physics is attributed to C. Huygens (1673) during his experiments for developing improved pendulum clocks (Huygens, 1673). Two clocks hanging on the same beam of his room were found to oscillate with exactly the same frequency and opposite phase due to the (weak) coupling in terms of the almost imperceptible oscillations of the beam generated by the clocks. Huygens also observed that both clocks were able to synchronize only in those cases where their individual frequencies almost coincided a prerequisite that turned out to be typical for synchronization of periodic oscillations. Since Huygens' early observation synchronization phenomena were discovered and practically used by many physicists and engineers. Rayleigh investigated synchronous oscillations of vibrating organ tubes and electrically or mechanically connected tuning forks in acoustics (Rayleigh, 1945) and Van der Pol and Van der Mark constructed radio tube oscillators where they observed entrainment when driving such oscillators sinusoidally (Van der Pol and Van der Mark, 1926).* These phenomena were investigated theoretically by Van der Pol (Van der Pol, 1927). Synchronization plays also an important role in celestial mechanics where it explains the locking of revolution periods of planets and satellites. Furthermore, synchronization was not only observed in physics but also in (neuro-) biology where rhythms and cycles may entrain (Glass and Mackey, 1988) or synchronizing clusters of firing neurons are considered for being crucial for information processing in the brain (Schechter, 1996; Schiff et al., 1996). In this and other cases synchronization can play a functional role because it establishes some special relation between coupled systems. For periodic oscillations, for example, the frequencies of interacting systems may become the same or lock with a rational ratio due to synchronization. More complex relations can be expected and have been found for coupled chaotic systems. At first glance, synchronization of chaotic systems seems to be rather surprising because one may naively expect that the sensitive dependence on initial conditions would lead to