Investigation of Not-in-Phase Synchronization on Coupled System of Four Oscillators Hiroo Sekiya, 1

Recently, synchronization phenomena on four three-dimensional chaotic oscillators full-coupled by capacitors have been investigated. In this system, four-phase synchro-nization can be observed. Four-phase synchronization means that a pair of anti-phase synchronizations are locked at 90°, and are 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 necessary and sufficient conditions for generating an independent pair of anti-phase synchronizations in the coupled system of four oscillators with odd functions as nonlinear characteristics. The systems that do not satisfy the conditions generate a locked pair of anti-phase synchroni