On the Mechanism of Chaos Generation in the Extended Liénard Systems

Tosiro KOGA  Masaharu SHINAGAWA  

IEICE TRANSACTIONS (1976-1990)   Vol.E73   No.6   pp.784-792
Publication Date: 1990/06/25
Online ISSN: 
Print ISSN: 0000-0000
Type of Manuscript: Special Section PAPER (Special Issue on Engineering Chaos)
Category: Chaos, Analysis and Numerical Method

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This paper discusses the behavior of a dynamical system described by the extended Liénard equation with an external force e(t)
where f and g are not necessarily even or odd with respect to x, respectively. First, basic theorems on the existence of limit cycles and properties of singularities are proved in the case where e(t) is equal to a constant bias denoted by e(t)const.A cos ωτ, and effects of f and g on the portrait of trajectories of the systems are clarified. Then, the dynamical behavior of the system, where the external force is periodic, i.e., e(t)A cos ωt, is represented in relation to the singularity which varies periodically in time; an obtained result makes it clear and easy to understand the dynamical behavior. Further, some conditions which are necessary for the system mentioned above to generate a chaotic solution are presented. Finally, the results of the argument above are applied to the periodically forced van der Pol equation, and it is concluded that chaotic solutions hardly exist in this case.