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Cyclicity of rigid centres on centre manifolds of three-dimensional systems

Part of: Qualitative theory General theory in ordinary differential equations

Published online by Cambridge University Press:  09 February 2023

Claudio Pessoa Open the ORCID record for Claudio Pessoa [Opens in a new window] ,
Lucas Queiroz Open the ORCID record for Lucas Queiroz [Opens in a new window]  and
Jarne D. Ribeiro Open the ORCID record for Jarne D. Ribeiro [Opens in a new window]
Claudio Pessoa
Affiliation:
Universidade Estadual Paulista (UNESP), Instituto de Biociências Letras e Ciências Exatas, R. Cristovão Colombo, 2265, 15.054-000, S. J. Rio Preto, SP, Brasil (c.pessoa@unesp.br, lucas.queiroz@unesp.br)
Lucas Queiroz
Affiliation:
Universidade Estadual Paulista (UNESP), Instituto de Biociências Letras e Ciências Exatas, R. Cristovão Colombo, 2265, 15.054-000, S. J. Rio Preto, SP, Brasil (c.pessoa@unesp.br, lucas.queiroz@unesp.br)
Jarne D. Ribeiro
Affiliation:
Instituto Federal de Educação, Ciência e Tecnologia do Sul de Minas Gerais - IFSULDEMINAS, R. Mario Ribola, 409, Penha II, 37903-358 Passos, MG, Brasil (jarne.ribeiro@ifsuldeminas.edu.br)
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Abstract

We work with polynomial three-dimensional rigid differential systems. Using the Lyapunov constants, we obtain lower bounds for the cyclicity of the known rigid centres on their centre manifolds. Moreover, we obtain an example of a quadratic rigid centre from which is possible to bifurcate 13 limit cycles, which is a new lower bound for three-dimensional quadratic systems.

Keywords

Limit cyclesrigid centresHopf singularitiescentre manifoldsthree-dimensional systemsLyapunov constants

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 34C05: Location of integral curves, singular points, limit cycles 34A34: Nonlinear equations and systems, general
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 1 , February 2024 , pp. 188 - 200
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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