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

Recently, synchronization phenomena on four three-dimensional chaotic oscillators full-coupled by capacitors have been investigated. In this system, four-phase synchronization 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 Poi oscillators. However, there has been no theoretical discussion of the conditions for generating a locked pair of anti-phase synchronizations, such as four-phase synchronization 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 synchronizations. Therefore, it is the first theoretical clarification of generating a locked pair of anti-phase synchronizations. We investigate the synchronization phenomena on the coupled system of three-dimensional oscillators and indicate the validity of the necessary and sufficient conditions. When each oscillator oscillates with direct components, when there are variables having direct components in the coupling terms, and when direct components of oscillations synchronized at anti-phase are equal in amplitude, the system loses symmetry, which is the condition for generating an independent pair of anti-phase synchronizations. Carrying out numerical calculations, we confirm the generation of four-phase synchronization and in- and anti-phase synchronization. In particular, generation of stable in- and anti-phase synchronization is the first observation.