Fixed and moving limit cycles for Liénard equations

Abstract

We consider a family of planar vector fields that writes as a Liénard system in suitable coordinates. It has an explicit solution that often contains periodic orbits of the system. We prove a general result that gives the hyperbolicity of these periodic orbits and we also study the coexistence of them with other non explicit periodic orbits. Our family contains the celebrated Wilson polynomial Liénard equation, as well as all polynomial Liénard systems having hyperelliptic limit cycles. As an illustrative example we study in more detail a natural 1-parametric extension of Wilson example that has at least two limit cycles, one of them algebraic, presents a transcritical bifurcation of limit cycles and for a given parameter has a non-hyperbolic double algebraic limit cycle. In order to prove that for some values of the parameter the system has exactly two hyperbolic limit cycles we use several suitable Dulac functions.

Effectiveness of the Bendixson-Dulac theorem

“…The following result allows to extend, and to prove in an easier way, the recent results about the maximum number of limit cycles of the above system when B(x) = x 3 − bx given in [3,21].…”

“…Liénard systems with an explicit solution. We study a family of Liénard type equations introduced recently in [21] that includes the Wilson family of Liénard equations ( [28]), which gave the first example of such equations having an algebraic limit cycle. More concretely, we consider systems…”

“…obtained in [3,21]. It also solves in the best possible way the some times called Coppel's problem for polynomial systems, which in his own words (when restricted to quadratic systems) says:"Ideally one might hope to characterize the phase portraits of quadratic systems by means of algebraic inequalities on the coefficients," see [9].…”

“…It also solves in the best possible way the some times called Coppel's problem for polynomial systems, which in his own words (when restricted to quadratic systems) says:"Ideally one might hope to characterize the phase portraits of quadratic systems by means of algebraic inequalities on the coefficients," see [9]. The relevant values describing the bifurcations of the limit cycles of this system are b and b * , see again [21]. We will show below that the value b ≈ −1.44 is algebraic.…”

“…The function Z was obtained in [21] from the integral of the divergence of the system on the algebraic limit cycle after some algebraic manipulations. In fact, the discriminant with respect to x of the denominator of the integrand gives the polynomial of the left-hand side of ( 9) that determines b.…”