On the Mechanism of Chaos Generation in the Extended Liénard Systems

Tosiro KOGA; Masaharu SHINAGAWA

On the Mechanism of Chaos Generation in the Extended Liénard Systems

Tosiro KOGA, Masaharu SHINAGAWA

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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)
f(x)g(x)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.

Publication: IEICE TRANSACTIONS on transactions Vol.E73-E No.6 pp.784-792

Publication Date: 1990/06/25

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Type of Manuscript: Special Section PAPER (Special Issue on Engineering Chaos)

Category: Chaos, Analysis and Numerical Method

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Tosiro KOGA
Masaharu SHINAGAWA

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Tosiro KOGA, Masaharu SHINAGAWA, "On the Mechanism of Chaos Generation in the Extended Liénard Systems" in IEICE TRANSACTIONS on transactions, vol. E73-E, no. 6, pp. 784-792, June 1990, doi: .
Abstract: This paper discusses the behavior of a dynamical system described by the extended Liénard equation with an external force e(t)
f(x)g(x)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.
URL: https://globals.ieice.org/en_transactions/transactions/10.1587/e73-e_6_784/_p

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@ARTICLE{e73-e_6_784,
author={Tosiro KOGA, Masaharu SHINAGAWA, },
journal={IEICE TRANSACTIONS on transactions},
title={On the Mechanism of Chaos Generation in the Extended Liénard Systems},
year={1990},
volume={E73-E},
number={6},
pages={784-792},
abstract={This paper discusses the behavior of a dynamical system described by the extended Liénard equation with an external force e(t)
f(x)g(x)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.},
keywords={},
doi={},
ISSN={},
month={June},}

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TY - JOUR
TI - On the Mechanism of Chaos Generation in the Extended Liénard Systems
T2 - IEICE TRANSACTIONS on transactions
SP - 784
EP - 792
AU - Tosiro KOGA
AU - Masaharu SHINAGAWA
PY - 1990
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E73-E
IS - 6
JA - IEICE TRANSACTIONS on transactions
Y1 - June 1990
AB - This paper discusses the behavior of a dynamical system described by the extended Liénard equation with an external force e(t)
f(x)g(x)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.
ER -