Synchronized States Observed in Coupled Four Oscillators

Systems of coupled oscillators are widely used as models for biological rhythmic oscillations such as human circadian rhythms[1, 2], finger movements, animal locomotion[3], swarms of fireflies that flash in synchrony, synchronous firing of cardiac pacemaker cells[5, 6], and so on. Using these coupled oscillator models, many investigators have studied the mechanism of generation of synchronous oscillation and phase transitions between distinct oscillatory modes. From the standpoint of bifurcation, the former and the latter correspond to Hopf bifurcation of an equilibrium point (or tangent bifurcation of a periodic solution) and pitchfork bifurcation (or period-doubling bifurcation) of a periodic solution, respectively. Using group theoretic discussion applied to the coupled oscillators, we can derive some general theorems concerning with the bifurcations of equilibrium points and periodic solutions[7]. In the study of coupled oscillator system, the four-coupled oscillator system is one of the most interesting system, because there exists an irregular degenerate oscillatory mode (or an independent pair of anti-phase mode) [8, 9] when the equation of the single oscillator is invariant under inversion of state variables. Mishima and Kawakami studied the oscillatory modes generated by the Hopf bifurcations of the origin (equilibrium point) in several systems of coupled four BVP (Bonhöffervan der Pol) oscillators [10]. However, they considered the Hopf bifurcation of the origin, because only the Hopf bifurcation of the origin is supercritical. Tsumoto et al. investigated bifurcations of the Modified BVP (MBVP) equation[11]. In the MVBP system, the supercritical Hopf bifurcation of nonorigin equilibrium points occurs. In this paper, we examine the oscillatory modes generated by the Hopf bifurcations of non-origin equilibrium points in the four-coupled oscillator system . The Hopf bifurcations of the equilibrium points with strong symmetrical property and the generated oscillatory modes are classified. We observe three-phase, four-phase, in-phase and a pair of antiphase synchronized states. The three-phase and four-phase solutions meet the Neimark-Sacker bifurcations, and threephase and four-phase quasi-periodic solutions are generated, respectively.