On the classification of Liénard systems with amplitude-independent periods

Abstract

We consider conditions under which the second-order differential equationx þ f ðxÞ 'x þ gðxÞ ¼ 0 has a family of periodic orbits with constant period. This condition is equivalent to seeking conditions under which the two-dimensional autonomous system 'x ¼ y; ' y ¼ ÀgðxÞ À f ðxÞy has a center with constant period: i.e., an isochronous center. In turn, this is equivalent to the latter system being locally linearizable. We give a simple necessary and sufficient condition for this when f and g are analytic functions. In the case where when f and g are polynomials, we show that this reduces to a finitely determinable system of equations. A complete classification is given of all such systems of degree 34 or less. r 2004 Published by Elsevier Inc.

Local bifurcations of critical periods for cubic Liénard equations with cubic damping

“…In this paper we discuss system (1.1) with (1.2), where λ := (a 1 , a 2 , a 3 , b 2 , b 3 ) ∈ R 5 is regarded as the parameter. We first apply the results in [4,5] on centers of polynomial Liénard equations to give a necessary and sufficient condition for coefficients under which the cubic Liénard equation with cubic damping has a center at O and finding the set of coefficients in which the center is isochronous. Then, we identify the weak centers of various possible order and discuss the local bifurcation of critical periods.…”

The center conditions and local bifurcation of critical periods for a Liénard system

“…A special case of centers is the isochronous centers, a center is called a isochronous center if the period of all periodic solutions near the center is constant. In 2004, Christopher and Devlin obtain a criteria for an isochronous center of system (1.1) in [10]. Other works concerning isochronous center conditions for some systems such as Liénard systems and time-reversible systems can be found in [3,[11][12][13][14] for instance.…”

On the reversible quadratic centers with monotonic period function

“…Apart from the monotonicity problem, several other questions related to the behaviour of the period function have been extensively studied. Let us quote, for instance, the isochronicity problem (see [5,6,10]) and the bifurcation of critical periods (see [11,15,19]). …”