DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

LIMIT CYCLES OF LIÉNARD SYSTEMS
M. AMAR AND Y. BOUATTIA
We calculate the amplitude and period of the limit cycle of the following classes of Liénard
equations by using the method of Lopez and Lopez-Ruiz: ẍ + ε(x 2 − 1)ẋ + x 2n−1 = 0; ẍ +
ε(x 2m − 1)ẋ + x 2n−1 = 0; ẍ + ε(|x| n − 1)ẋ + sign(x) ·|x| m = 0; ẍ + ε(x 2m − 1)ẋ +
x 1/(2n−1) = 0, where m,n ∈ N.Wegivenumericalresults.
Copyright © 2006 M. Amar and Y. Bouattia. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Limit-cycle behavior is observed in many physical and biological systems. The problem
of determining when a nonlinear dynamical system exhibits limit cycle has been of great
interest for more than a century. Limit cycles cannot occur in linear systems, conservative
systems, and gradient systems. The limit cycles are caused by nonlinearities. It was found
that many oscillatory circuits can be modeled by the Liénard equation
where
ẍ + f (x)ẋ + g(x) = 0, (1.1)
·= d dt . (1.2)
It can be interpreted mechanically as the equation of motion for a unit mass subject
to a nonlinear damping force − f (x)ẋ and a nonlinear restoring force −g(x). Applications
of Liénard’s equation can be found in many important examples including chemical
reactions, growth of a single species, predator-prey systems, and vibration analysis.
In Section 2, we give the theorem of existence and uniqueness of the limit cycle for the
Liénard equation. In Section 3, we give the method of calculation of the amplitude of the
limit cycle of the perturbed centers (see [2]),
ẋ =−y 2l−1 + εP(x, y),
ẏ = x 2k−1 + εQ(x, y),
(1.3)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 743–755