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The local cyclicity problem: Melnikov method using Lyapunov constants

Part of: Qualitative theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  19 April 2022

Luiz F. S. Gouveia Open the ORCID record for Luiz F. S. Gouveia [Opens in a new window]  and
Joan Torregrosa Open the ORCID record for Joan Torregrosa [Opens in a new window]
Luiz F. S. Gouveia
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Departamento de Matemática, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, Brazil (fernandosg@mat.uab.cat; fernando.gouveia@unesp.br)
Joan Torregrosa
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Centre de Recerca Matemàtica, Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain (torre@mat.uab.cat)
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Abstract

In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order 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, so $\mathcal {M}(6) \geq 44$. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that $\mathcal {M}^{c}_{p}(4) \geq 43$ and $\mathcal {M}^{c}_{p}(5) \geq 65.$

Keywords

Melnikov theoryLyapunov constantslocal cyclicity

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 34C23: Bifurcation 37C27: Periodic orbits of vector fields and flows
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 2 , May 2022 , pp. 356 - 375
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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