Symmetry Breaking and Recovering in a System of n Hybridly Coupled Oscillators

Olivier PAPY
Hiroshi KAWAKAMI

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E79-A    No.10    pp.1568-1574
Publication Date: 1996/10/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and its Applications (NOLTA))
Category: Nonlinear Circuits and Bifurcation
Keyword: 
bifurcation,  symmetry breaking,  oriented coupling,  

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Summary: 
We consider a ring of n Rayleigh oscillators coupled hybridly. Using the symmetrical property of the system we demonstrate the degeneracy of the Hopf bifurcation of the equilibrium at the origin. The degeneracy implies the exstence and stability of the n-phase oscillation. We discuss some consequences of the perturbation of the symmetry. Then we study the case n = 3. We show the bifurcation diagram of the equilibria and of hte periodic solutions. Especially, we analyze the mechanism for the symmetry breaking bifurcation of the fully symmetric solution. We report and explain the occurrence of both chaotic attractors and repellors and show two types of symmetry recovering crisis they undergo.