Antiphase Synchronization in symmetrically Coupled Self-oscillators

Since the beginning of the 90s a new direction appeared in the theory of dynamical systems, which was called “Controlling chaos”. The subject of this field is how to transform chaotic behavior to a regular one or to a more simple chaotic one by means of special small influence on the dynamical system. Pioneer works of this new direction were papers by Hubler and Lucher [1989], Jackson [1990] and the well-known paper by Ott, Grebogi and Yorke [1990]. Now there is a lot of different algorithms of chaos control, which are applied to tasks of hydrodynamics [Singer et al., 1991], mechanics [In & Ditto, 1995; Baretto & Grebogi, 1995], chemistry [Petrov et al., 1993], biology and medicine [Garfinkel et al., 1992; Schiff et al., 1994]. Methods of control of chaos can be applied for synchronization of coupled chaotic oscillators. In scientific literature there is no common view for the problem of chaotic synchronization. Different types of chaotic synchronization are “complete synchronization”, when oscillations of subsystems are equal or nearly equal to each other [Fujisaka & Yamada, 1983; Afraimovich et al., 1986]; “generalized synchronization”, when there is a functional dependence between states of the subsystems [Rulkov et al., 1995]; “frequency synchronization” which means locking of peaks in the spectra of oscillations [Anishchenko et al., 1991, 1992]; “phase synchronization” when phases of oscillations are locked while amplitudes remain uncorrelated [Rosenblum et al., 1996]. The majority of works on controlled synchronization of chaos consider the case of complete in-phase synchronization [Lai & Grebogi, 1993; Malescio, 1996]. Another interesting case of chaotic synchronization, which can be realized by chaos control is antiphase chaotic synchronization, when states of the subsystems satisfy the condition x1 = −x2. The antiphase synchronization of chaos on the example of a two-dimensional map and a six-dimensional flaw was considered in the work [Cao & Lai, 1998]. Following Pecora and Carroll [1990] they used the so-called “master– slave” approach to the task of the antiphase synchronization. In our work we consider antiphase synchronization in symmetrically coupled identical self-oscillators with additional controlling feedback loop. Let us consider the task of antiphase synchronization in symmetrically coupled oscillators. Let the equation of the system be in the form: