Abstract

We consider planar real analytic differential systems and we are interested in studying the cyclicity of limit cycles and of homoclinic loops, whose critical point is a hyperbolic saddle and whose Poincaré return map is not the identity. We assume that the system has an analytic inverse integrating factor defined in a neighborhood of a limit cycle and we show that the vanishing multiplicity of this inverse integrating factor coincides with the cyclicity of the limit cycle. The key tool to show this coincidence is an ordinary differential equation for the Poincaré return map associated to the limit cycle which is written using the inverse integrating factor. In fact, this ordinary differential equation also applies to the Poincaré map of any regular orbit of the system, provided an analytic inverse integrating factor is given in a neighborhood of this regular orbit. Analogous techniques allow the determination of the cyclicity of a homoclinic loop. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.

This talk is based upon a joint work with Isaac A. Garcia (Universitat de Lleida) and Hector Giacomini (Universite de Tours, France); a preprint of which can be downloaded in: http://xxx.lanl.gov/abs/0710.3238