The local cyclicity problem: Melnikov method using Lyapunov constants

Luiz F.S. Gouveia, Joan Torregrosa

Research output: Contribution to journalArticleResearchpeer-review


In 1991, Chicone and Jacobs showed the equivalence between the computation of the firstorder Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, soM(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp(4) ≥ 43 and Mcp(5) ≥ 65.

Original languageEnglish
Pages (from-to)356-375
Number of pages20
JournalProceedings of the Edinburgh Mathematical Society
Issue number2
Early online date19 Apr 2022
Publication statusPublished - 19 Apr 2022


  • Lyapunov constants
  • Melnikov theory
  • local cyclicity


Dive into the research topics of 'The local cyclicity problem: Melnikov method using Lyapunov constants'. Together they form a unique fingerprint.

Cite this