An example of rapid evolution of complex limit cycles

Nikolay Dimitrov

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arXiv · 1106.2786

An example of rapid evolution of complex limit cycles

Nikolay Dimitrov

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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.

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Nikolay Dimitrov. 2011-06-14. An example of rapid evolution of complex limit cycles. https://arxiv.org/abs/1106.2786

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