Investigation of not-in-phase synchronization on coupled system of four oscillators <Abstract> Recently, synchronization phenomena on four three-dimensional chaotic oscillators full-coupled by capacitors are investigated. In this system, four-phase synchronization can be observed. Four-phase synchronization is that a pair of anti-phase synchronization is locked at 90 degrees and it is never observed in the coupled system of van der Pol oscillators. However, there is no theoretical discussion about the conditions for generating a locked pair of anti-phase synchronization on coupled identical oscillators. In this paper, we give the necessary and sufficient conditions for generating an independent pair of anti-phase synchronization in the coupled system of four oscillators with odd functions as nonlinear characteristics. The systems that are not satisfied with the conditions generate a locked pair of anti-phase synchronization. Therefore, it is first theoretical clarification of generating a locked pair of anti-phase synchronization. We investigate the synchronization phenomena on the coupled system of three-dimensional oscillators and indicate the valid 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 has lost symmetry that is the condition for generating an independent pair of anti-phase synchronization. Carrying out numerical calculations, we confirm the generation of four-phase synchronization and in- and anti-phase synchronization. Especially, generation of stable in- and anti-phase synchronization is first observation. |