A local study of limit cycles in analytic families of planar vector fields
by
Magdalena Caubergh
Departement WNI, Limburgs Universitair Centrum, B-3590 Diepenbeek

The existential part of Hilbert's sixteenth problem asks for the maximum number of isolated periodic orbits (the so-called limit cycles) of a planar polynomial vector field, in function of the degree. Much of the research on this problem has been proceeded by considering particular classes of systems and obtaining upperbounds on the number of limit cycles, which can bifurcate from various systems. Such a local upperbound is often called cyclicity.

In this talk our attention goes to analytic families of planar vector fields ( X_\lambda) _\lambda, depending on a parameter \lambda, which unfold a vector field of center type X_\lambda0. This means that X_\lambda0 contains a continuous band of periodic orbits. There exist several techniques to compute the cyclicity along a given non-isolated periodic orbit of X_\lambda0. For instance, two well-known techniques are the Bautin ideal and the computation of Melnikov functions (abelian integrals). The first one is theoretically very strong, but is in practice very often not easy to compute.

On the other hand, a lot of research is done using the second technique. There is just one obstruction: this technique can only be used in case the parameter is one-dimensional. However, it is possible to reduce the computation of the cyclicity in multi-parameter families to the computation of cyclicity in one-parameter subfamilies (induced by analytic curves through \lambda₀). This reduction will be the main motivation of the talk.

In a natural way, we introduce a family of analytic functions ( f_\lambda) _\lambda, such that periodic orbits of an analytic vector field X_\lambda correspond to zeros of an analytic function f_\lambda. Then the theory of analytic geometry provides an elegant tool (the curve selection lemma) to obtain the desired reduction.

Date received: April 30, 2001