Investigation of not-in-phase synchronization on coupled system of four oscillators

Recently, synchronization phenomena on four three-dimensional chaotic oscil lators full-coupled by capacitors have been investigated. In this system, f our-phase synchronization can be observed. Four-phase synchronization means that a pair of anti-phase synchronizations are locked at 90 degrees, and a re never observed in the coupled system of van der Pol oscillators. However , there has been no theoretical discussion of the conditions for generating a locked pair of anti-phase synchronizations, such as four-phase synchroni zation on coupled identical oscillators. In this paper, we give the necessa ry and sufficient conditions for generating an independent pair of anti-pha se synchronizations in the coupled system of four oscillators with odd func tions as nonlinear characteristics. The systems that do not satisfy the con ditions generate a locked pair of anti-phase synchronizations. Therefore, i t is the first theoretical clarification of generating a locked pair of ant i-phase synchronizations. We investigate the synchronization phenomena on t he coupled system of three-dimensional oscillators and indicate the validit y of the necessary and sufficient conditions. When each oscillator oscillat es with direct components, when there are variables having direct component s in the coupling terms, and when direct components of oscillations synchro nized at anti-phase are equal in amplitude, the system loses symmetry, whic h is the condition for generating an independent pair of anti-phase synchro nizations. Carrying out numerical calculations, we confirm the generation o f four-phase synchronization and in- and antiphase synchronization. In part icular, generation of stable in- and anti-phase synchronization is the firs t observation. (C) 2001 Scripta Technica, Electron Comm Jpn Pt 3, 85(2): 1- 12,2002.