Bifurcations of Quasi-Periodic Responses in Coupled van der Pol Oscillators with External Force (Special Section on Nonlinear Theory and Its Applications)

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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 observed. 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 desynchronizationto 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.
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1994-11-25