An example of rapid evolution of complex limit cycles

TL;DR

This paper investigates the rapid evolution of complex limit cycles in polynomial holomorphic foliations, revealing the existence of cycles with arbitrary multiplicity and their continuous dependence on perturbation parameters.

Contribution

It introduces the concept of multi-fold cycles and demonstrates their existence and behavior in a specific polynomial family of foliations.

Findings

Existence of limit cycles of any multiplicity in the family.

Continuous dependence of cycles on the perturbation parameter.

Cycles escape from large subdomains as the parameter approaches zero.

Abstract

In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving 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. In addition to that, we introduce the notion of a multi-fold cycle and show that in our example there exists a limit cycle of any multiplicity. Furthermore, such a cycle gives rise to a one-parameter family of cycles continuously depending on the perturbation parameter. As the parameter decreases in absolute value, the cycles from the continuous family escape from a very large subdomain of the complex plane.