Synchronization of oscillations.
From simple and evident to complex and improbable
V.S. Anishchenko
Laboratory of Nonlinear Dynamics,
Saratov State University, Saratov, Russia
The study of interaction of oscillatory systems is a subject of an enhanced attention in modern science and especially for research in physics and biology. This interest is caused, in particular, by the effect of synchronization. Today a much class of phenomena are attributed to the problem of synchronization comparing to the first description of clock synchronization made 300 years ago by Huygens.
The classic phenomenon of synchroniziny the periodic oscillations which exhibits itself in establishing the rational retios of frequencies and phases of interacting oscillators has an explicit mathematical description. In this case the effect of synchronization corresponds to the rough resonance on a smooth two-dimensional torus possessing a retional Poincare winding number.
But synchronization also takes place when the chaotic oscillations with continuous spectrum interact. In the regime of chaotic synchronization the interacting subsystems evolve in the same way ("synchronously"). With this, the effects of locking and supressing the basic frequencies of subsystems, in case when such frequencies may be revealed, may take place.
More complicated synchronization phenomena are observed in chaotic systems exhibiting bistable properties. For such systems the motion of mean switching frequency, as the natural frequency in statistical sense, is useful to introduce.
It was found out that the synchronization effects are also observed with respect to the mean switching frequency of deterministic chaotic system. With this, in particular, the effect of locking the mean switching frequency is realized.
One may make the problem more complicated and consider the bistable systems influenced by noise, i.e., begin to study the dynamics of stochastic systems. Here we can await for still more interesting results. It was discovered that in stochastic bistable systems the effects of synchronization such as stochastic resonance and mean switching frequency locking may be observed. With this, and this is very important, the role of control parameter of synchronization noise influence leads to the increase of order in system! The synchronization phenomena under control, the latter being connected with noise influence, are obviously of special interest not only for physics, but also for biology, chemistry and medicine.
Note also, that a certain class of tasks which are considered in the frames of controlling chaos may be ascribed to the problem of synchronization. Studying the process of controlling chaos in the presence of external noise is of a special interest because the latter allows one to approach the understanding of a surprising property of alive organisms to extract weak signals on the background of environmental noise.
In the present paper, the phenomena of synchronizing the periodic, chaotic and stochastic oscillations by means of numerical simulation and full-scale experiments are subsequently studied and analyzed in detail. A special attention is payed to the role of external noise in the process of synchronization. The effects of synchronization of osciilations are discussed also from the viewpoint of general knowledge about the phenomenon of selforganization of complex systems when nonlinear dissipative oscillators interact.