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Highest weak focus order for trigonometric Liénard equations

  • Armengol Gasull
  • Jaume GinéEmail author
  • Claudia Valls
Article
  • 19 Downloads

Abstract

Given a planar analytic differential equation with a critical point which is a weak focus of order k,  it is well known that at most k limit cycles can bifurcate from it. Moreover, in case of analytic Liénard differential equations this order can be computed as one half of the multiplicity of an associated planar analytic map. By using this approach, we can give an upper bound of the maximum order of the weak focus of pure trigonometric Liénard equations only in terms of the degrees of the involved trigonometric polynomials. Our result extends to this trigonometric Liénard case a similar result known for polynomial Liénard equations.

Keywords

Trigonometric Liénard equation Weak focus Cyclicity 

Mathematics Subject Classification

Primary 34C07 Secondary 13H15 34C25 37C27 

Notes

Acknowledgements

The first author is partially supported by “Agencia Estatal de Investigación” and “Ministerio de Ciencia, Innovación y Universidades”, Grant number MTM2016-77278-P and AGAUR, Generalitat de Catalunya, grant 2017-SGR-1617. The second author is partially supported by a MINECO/FEDER grant number MTM2017-84383-P and an AGAUR grant number 2017SGR-1276. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Departament de MatemàticaUniversitat de LleidaLleidaSpain
  3. 3.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

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