Nikolai Dimitrov
Nikolai Dimitrov

Ph.D. (2009) Cornell University

First Position
Dissertation
Advisor:
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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 oneform 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 oneform, one can introduce the notion of a multifold 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 oneform mentioned earlier. The current dissertation is an introduction to the notion of multifold cycles of a closetointegrable polynomial foliation. We explore the way these cycles correspond to periodic orbits of certain Poincaré maps associated with the foliation. We also discuss the tendency of a continuous family of multifold limit cycles to escape from certain large open domains in the complex plane as the foliation converges to its integrable part.