Differential Equations with Symbolic Computation

Estimating Limit Cycle Bifurcations from Centers (p. 23)

Colin Christopher

Abstract.

We consider a simple computational approach to estimating the cyclicity of centers in various classes of planar polynomial systems. Among the results we establish are confirmation of , Zoladek’s result that at least 11 limit cycles can bifurcate from a cubic center, a quartic system with 17 limit cycles bifurcating from a non-degenerate center, and another quartic system with at least 22 limit cycles globally.

Mathematics Subject Classification (2000). 34C07.

Keywords. Limit cycle, center, multiple bifurcation.

1. Introduction
The use of multiple Hopf bifurcations of limit cycles from critical points is now a well-established technique in the analysis of planar dynamical systems. For many small classes of systems, the maximum number, or cyclicity, of bifurcating limit cycles is known and has been used to obtain important estimates on the general behavior of these systems.

In particular, quadratic systems can have at most three such limit cycles [1], symmetric cubic systems (those without quadratic terms) and projective quadratic systems at most five [11, 15, 8]. Results are also known explicitly for several large classes of Lienard systems [3].

The idea behind the method is to calculate the successive coeficients ?i in the return map for the vector field about a non-degenerate monodromic critical point.
The cyclicity can then be found from examining these coeficients and their common zeros. The terms ?2k are merely analytic functions of the previous ?i, so the only interesting functions are the ones of the form ?2i+1. If ?2k+1 is the first non-zero one of these, then at most k limit cycles can bifurcate from the origin, and, provided we have suficient choice in the coeficients ?i, we can also obtain that many limit cycles in a simultaneous bifurcation from the critical point.

We call the functions ?2i+1 the Liapunov quantities of the system. If all the ?2i+1 vanish then the critical point is a center. It is possible to analyze this case also, but to do fully requires a more intimate knowledge, not only of the common zeros of the polynomials ?i, but also of the ideal they generate in the ring of coeficients. The papers [1, 15] cover the case of a center also. We call the set of coe.cients for which all the ?i vanish the center variety.

In the cases we consider here, when ?0 = 0, the remaining coeficients are polynomials in the parameters of the system. By the Hilbert Basis Theorem, the center variety is then an algebraic set. Unfortunately, although the calculation of the Liapunov quantities is straight forward, the computational complexity of finding their common zeros grows very quickly. The result is that some very simple systems have remained intractable (to direct calculation at least) at present, for example, the system of Kukles’ [9]: