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Limit Cycles of a Perturbation of a Polynomial Hamiltonian Systems of Degree 4 Symmetric with Respect to the Origin

Part of: Qualitative theory

Published online by Cambridge University Press:  18 October 2019

Jaume Llibre ,
Paulina Martínez  and
Claudio Vidal
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Facultat de Ciències Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email: jllibre@mat.uab.cat
Paulina Martínez
Affiliation:
Departamento de Matemática, Facultad de Ciencias, Universidad de Bío-Bío, Casilla 5–C, Concepción, VIII–Región, Chile Email: yohanna.martinez@uab.cat
Claudio Vidal
Affiliation:
Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad de Bío-Bío, Casilla 5–C, Concepción, VIII–región, Chile Email: clvidal@ubiobio.cl
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Abstract

We study the number of limit cycles bifurcating from the origin of a Hamiltonian system of degree 4. We prove, using the averaging theory of order 7, that there are quartic polynomial systems close these Hamiltonian systems having 3 limit cycles.

Keywords

Hamiltonian systemlinear type centerquartic polynomialpolynomial vector fieldphase portrait

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 34C08: Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 3 , September 2020 , pp. 547 - 561
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author J. L. is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911; Y. P. M. was supported by a CONICYT fellowship (Chile); C. V. was partially supported by CONICYT (Chile) through FONDECYT project 1130644.

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