Synchronized States Observed in Coupled Four Oscillators

Hiroyuki KITAJIMA  Hiroshi KAWAKAMI  Tetsuo HATTORI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E88-A   No.3   pp.712-717
Publication Date: 2005/03/01
Online ISSN: 
DOI: 10.1093/ietfec/e88-a.3.712
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Nonlinear Problems
coupled oscillator,  bifurcation,  phase synchronization,  symmetry,  

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In this paper, we examine oscillatory modes generated by the Hopf bifurcations of equilibrium points except for the origin in a system of coupled four oscillators. (The bifurcation analyses of the origin for many coupled oscillators were already done.) The Hopf bifurcations of the equilibrium points with strong symmetrical property and the generated oscillatory modes are classified. We observe four-phase, in-phase and a pair of anti-phase synchronized states. Even in a system of four coupled oscillators, we discover the existence of a stable three-phase oscillation. By the numerical bifurcation analysis of generated periodic oscillations we find out successive period-doubling bifurcations as the route to chaos and show some of them are symmetry-breaking bifurcations. As a result of the symmetry-breaking period-doubling bifurcations, a periodic solution with complete synchronization becomes a chaotic solution with phase synchronization.