Phase Synchronization of Regular and Chaotic Oscillators

Synchronization, a basic nonlinear phenomenon, discovered at the beginning of the modern age of science by Huygens [1], is widely encountered in various fields of science, often observed in living nature [2] and finds a lot of engineering applications [3, 4]. In the classical sense, synchronization means adjustment of frequencies of self-sustained oscillators due to a weak interaction. The phase of oscillations may be locked by periodic external force; another situation is the locking of the phases of two interacting oscillators. One can also speak on “frequency entrainment”. Synchronization of periodic systems is pretty well understood [3, 5, 6], effects of noise have been also studied [7]. In the context of interacting chaotic oscillators, several effects are usually referred to as “synchronization”. Due to a strong interaction of two (or a large number) of identical chaotic systems, their states can coincide, while the dynamics in time remains chaotic [8, 9]. This effect is called “complete synchronization” of chaotic oscillators. It can be generalized to the case of non-identical systems [9, 10, 11], or that of the interacting subsystems [12, 13, 14]. Another well-studied effect is the “chaos–destroying” synchronization, when a periodic external force acting on a chaotic system destroys chaos and a periodic regime appears [15], or, in the case of an irregular forcing, the driven system follows the behavior of the force [16]. This effect occurs for a relatively strong forcing as well. A characteristic feature of these phenomena is the existence of a threshold coupling value depending on the Lyapunov exponents of individual systems [8, 9, 17, 18]. Fr om : H an db oo k of C ha os C on tr ol ,e d. H .G .S ch us te r, W ile yV C H ,W ei nh ei m ,1 99 9