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

Olivier PAPY
Hiroshi KAWAKAMI

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A    No.11    pp.1788-1793
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,  quasi–periodic solution,  

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Summary: 
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.