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

Tetsuya YOSHINAGA; Hiroshi KAWAKAMI

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

Tetsuya YOSHINAGA, Hiroshi KAWAKAMI

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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 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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E77-A No.11 pp.1783-1787

Publication Date: 1994/11/25

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Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and Its Applications)

Category: Bifurcation of van der Pol Oscillators

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Tetsuya YOSHINAGA, Hiroshi KAWAKAMI, "Bifurcations of Quasi–Periodic Responses in Coupled van der Pol Oscillators with External Force" in IEICE TRANSACTIONS on Fundamentals, vol. E77-A, no. 11, pp. 1783-1787, November 1994, doi: .
Abstract: 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.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/e77-a_11_1783/_p

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@ARTICLE{e77-a_11_1783,
author={Tetsuya YOSHINAGA, Hiroshi KAWAKAMI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Bifurcations of Quasi–Periodic Responses in Coupled van der Pol Oscillators with External Force},
year={1994},
volume={E77-A},
number={11},
pages={1783-1787},
abstract={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.},
keywords={},
doi={},
ISSN={},
month={November},}

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TY - JOUR
TI - Bifurcations of Quasi–Periodic Responses in Coupled van der Pol Oscillators with External Force
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1783
EP - 1787
AU - Tetsuya YOSHINAGA
AU - Hiroshi KAWAKAMI
PY - 1994
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E77-A
IS - 11
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - November 1994
AB - 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.
ER -