Article

Journal of Dynamical and Control Systems

, Volume 4, Issue 4, pp 539-581

First online:

Countable Set of Limit Cycles for the Equation \(\frac{{dw}}{{dz}} = \frac{{P_n \left( {z,w} \right)}}{{Q_n \left( {z,w} \right)}}\)

  • A.A. ShcherbakovAffiliated withRussian Academy of Sciences, Frumkin Inst. of Electrochemistry
  • , E. Rosales-GonzálezAffiliated withInstituto de Matemáticas, Universidad Nacional Autónoma de México, C.P
  • , L. Ortiz-BobadillaAffiliated withInstituto de Matemáticas, Universidad Nacional Autónoma de México, C.P

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Abstract

Differential equations on the complex plane with a rational right-hand side are considered. In a generic case such equation has a countable set of homologically independent limit cycles. It is proved that the exceptional set – the set of equations such that they do not have this property – has the real codimension at least two in the space of equations with right-hand side of degree no greater than a fixed number n.

Polynomial differential equations limit cycles monodromy group conformal maps fixed points