Bifurcations of Quasi–Periodic Responses in Coupled van der Pol Oscillators with External Force

Tetsuya YOSHINAGA  Hiroshi KAWAKAMI  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.11   pp.1783-1787
Publication Date: 1994/11/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and Its Applications)
Category: Bifurcation of van der Pol Oscillators
Keyword: 
bifurcation,  van der Pol equation,  averaging method,  coupled oscillators,  quasi–periodic solution,  

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Summary: 
Bifurcations of quasi–periodic responses in an oscillator described by conductively coupled van der Pol equations with a sinusoidal forcing term are investigated. According to the variation of three base frequencies, i.e., two natural frequencies of oscillators and the forcing frequency, various nonlinear phenomena such as harmonic or subharmonic synchronization, almost synchronization and complete desynchronization are ovserved. The most characteristic phenomenon observed in the four–dimensional nonautonomous system is the occurrence of a double Hopf bifurcation of periodic solutions. A quasi–periodic solution with three base spectra, which is generated by the double Hopf bifurcation, is studied through an investigation of properties of limit cycles observed in an averaged system for the original nonautonomous equations. The oscillatory circuit is particularly motivated by analysis of human circadian rhythms. The transition from an external desynchronization to a complete desynchronization in human rest–activity can be referred to a mechanism of the bifurcation of quasi–periodic solutions with two and three base spectra.