Bifurcations of the Quasi–Periodic Solutions of a Coupled Forced van der Pol Oscillator

Olivier PAPY; Hiroshi KAWAKAMI

Bifurcations of the Quasi–Periodic Solutions of a Coupled Forced van der Pol Oscillator

Olivier PAPY, Hiroshi KAWAKAMI

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In this paper we study the bifurcation phenomena of quasi–periodic states of a model of the human circadian rhythm, which is described by a system of coupled van der Pol equations with a periodic external forcing term. In the system a periodic or quasi–periodic solution corresponds to a synchronized or desynchronized state of the circadian rhythm, respectively. By using a stroboscopic mapping, called a Poincar mapping, the periodic or quasi–periodic solution is reduced to a fixed point or an invariant closed curve (ab. ICC). Hence we can discuss the bifurcations for the periodic and quasi–periodic solutions by considering that of the fixed point and ICC of the mapping. At first, the geometrical behavior of the 3 generic bifurcations, i.e., tangent, Hopf and period doublig bifurcations, of the periodic solutions is given, Then, we use a qualitative approach to bring out the similar behavior for the bifurcations of the periodic and quasi–periodic solutions in the phase space and in the Poincarsection respectively. At last, we show bifurcation diagrams concerning both periodic and quasi–periodic solutions, in different parameter planes. For the ICC, we concentrate our attention on the period doubling cascade route to chaos, the folding of the parameter plane, the windows in the chaos and the occurrence of the type I intermittency.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E77-A No.11 pp.1788-1793

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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Olivier PAPY, Hiroshi KAWAKAMI, "Bifurcations of the Quasi–Periodic Solutions of a Coupled Forced van der Pol Oscillator" in IEICE TRANSACTIONS on Fundamentals, vol. E77-A, no. 11, pp. 1788-1793, November 1994, doi: .
Abstract: In this paper we study the bifurcation phenomena of quasi–periodic states of a model of the human circadian rhythm, which is described by a system of coupled van der Pol equations with a periodic external forcing term. In the system a periodic or quasi–periodic solution corresponds to a synchronized or desynchronized state of the circadian rhythm, respectively. By using a stroboscopic mapping, called a Poincar mapping, the periodic or quasi–periodic solution is reduced to a fixed point or an invariant closed curve (ab. ICC). Hence we can discuss the bifurcations for the periodic and quasi–periodic solutions by considering that of the fixed point and ICC of the mapping. At first, the geometrical behavior of the 3 generic bifurcations, i.e., tangent, Hopf and period doublig bifurcations, of the periodic solutions is given, Then, we use a qualitative approach to bring out the similar behavior for the bifurcations of the periodic and quasi–periodic solutions in the phase space and in the Poincarsection respectively. At last, we show bifurcation diagrams concerning both periodic and quasi–periodic solutions, in different parameter planes. For the ICC, we concentrate our attention on the period doubling cascade route to chaos, the folding of the parameter plane, the windows in the chaos and the occurrence of the type I intermittency.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e77-a_11_1788/_p

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@ARTICLE{e77-a_11_1788,
author={Olivier PAPY, Hiroshi KAWAKAMI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Bifurcations of the Quasi–Periodic Solutions of a Coupled Forced van der Pol Oscillator},
year={1994},
volume={E77-A},
number={11},
pages={1788-1793},
abstract={In this paper we study the bifurcation phenomena of quasi–periodic states of a model of the human circadian rhythm, which is described by a system of coupled van der Pol equations with a periodic external forcing term. In the system a periodic or quasi–periodic solution corresponds to a synchronized or desynchronized state of the circadian rhythm, respectively. By using a stroboscopic mapping, called a Poincar mapping, the periodic or quasi–periodic solution is reduced to a fixed point or an invariant closed curve (ab. ICC). Hence we can discuss the bifurcations for the periodic and quasi–periodic solutions by considering that of the fixed point and ICC of the mapping. At first, the geometrical behavior of the 3 generic bifurcations, i.e., tangent, Hopf and period doublig bifurcations, of the periodic solutions is given, Then, we use a qualitative approach to bring out the similar behavior for the bifurcations of the periodic and quasi–periodic solutions in the phase space and in the Poincarsection respectively. At last, we show bifurcation diagrams concerning both periodic and quasi–periodic solutions, in different parameter planes. For the ICC, we concentrate our attention on the period doubling cascade route to chaos, the folding of the parameter plane, the windows in the chaos and the occurrence of the type I intermittency.},
keywords={},
doi={},
ISSN={},
month={November},}

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TY - JOUR
TI - Bifurcations of the Quasi–Periodic Solutions of a Coupled Forced van der Pol Oscillator
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1788
EP - 1793
AU - Olivier PAPY
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 - In this paper we study the bifurcation phenomena of quasi–periodic states of a model of the human circadian rhythm, which is described by a system of coupled van der Pol equations with a periodic external forcing term. In the system a periodic or quasi–periodic solution corresponds to a synchronized or desynchronized state of the circadian rhythm, respectively. By using a stroboscopic mapping, called a Poincar mapping, the periodic or quasi–periodic solution is reduced to a fixed point or an invariant closed curve (ab. ICC). Hence we can discuss the bifurcations for the periodic and quasi–periodic solutions by considering that of the fixed point and ICC of the mapping. At first, the geometrical behavior of the 3 generic bifurcations, i.e., tangent, Hopf and period doublig bifurcations, of the periodic solutions is given, Then, we use a qualitative approach to bring out the similar behavior for the bifurcations of the periodic and quasi–periodic solutions in the phase space and in the Poincarsection respectively. At last, we show bifurcation diagrams concerning both periodic and quasi–periodic solutions, in different parameter planes. For the ICC, we concentrate our attention on the period doubling cascade route to chaos, the folding of the parameter plane, the windows in the chaos and the occurrence of the type I intermittency.
ER -