Absolute cyclicity, Lyapunov quantities and center conditions

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Abstract

In this paper we consider analytic vector fields X0 having a non-degenerate center point e. We estimate the maximum number of small amplitude limit cycles, i.e., limit cycles that arise after small perturbations of X0 from e. When the perturbation (Xλ) is fixed, this number is referred to as the cyclicity of Xλ at e for λ near 0. In this paper, we study the so-called absolute cyclicity; i.e., an upper bound for the cyclicity of any perturbation Xλ for which the set defined by the center conditions is a fixed linear variety. It is known that the zero-set of the Lyapunov quantities correspond to the center conditions (Caubergh and Dumortier (2004) [6]). If the ideal generated by the Lyapunov quantities is regular, then the absolute cyclicity is the dimension of this so-called Lyapunov ideal minus 1. Here we study the absolute cyclicity in case that the Lyapunov ideal is not regular. © 2010 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)297-309
JournalJournal of Mathematical Analysis and Applications
Volume366
Issue number1
DOIs
Publication statusPublished - 1 Jun 2010

Keywords

  • Absolute cyclicity
  • Bifurcation analysis
  • Center conditions
  • Cyclicity
  • Hilbert's sixteenth problem
  • Lyapunov quantities

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