AN EXAMPLE OF RAPID EVOLUTION OF COMPLEX LIMIT CYCLES

作者：Dimitrov Nikolay^*

来源：Discrete and Continuous Dynamical Systems, 2011, 31(3): 709-735.

DOI：10.3934/dcds.2011.31.709

• Holomorphic foliation
• complex limit cycle
• Poincare map
• covering space
• fiber bundle

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 de fined 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.

• 出版日期2011-11