Rapid Evolution Of Complex Limit Cycles

Abstract

Limit cycles of planar polynomial vector fields have long been a focus of extensive research. Analogous to the real case, similar problems have been studied in the complex plane where a polynomial differential one-form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the fundamental group of a leaf from the foliation. Whenever the polynomial foliation comes from a perturbation of an exact one-form, one can introduce the notion of a multi-fold cycle. This type of cycle has at least one representative that determines a free homotopy class of loops in an open fibred subdomain of the complex plane. The topology of this subdomain is closely related to the exact one-form mentioned earlier. The current dissertation is an introduction to the notion of multi-fold cycles of a close-to-integrable polynomial foliation. We explore the way these cycles correspond to periodic orbits of certain Poincare maps associated with the foliation. We also discuss the tendency of a continuous family of multi-fold limit cycles to escape from certain large open domains in the complex plane as the foliation converges to its integrable part.