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On the birth and death of algebraic limit cycles in quadratic differential systems

Part of: Qualitative theory Local and nonlocal bifurcation theory Nonlinear dynamics

Published online by Cambridge University Press:  26 May 2020

JAUME LLIBRE ,
REGILENE OLIVEIRA Open the ORCID record for REGILENE OLIVEIRA [Opens in a new window]  and
YULIN ZHAO
JAUME LLIBRE
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. email: jllibre@mat.uab.cat
REGILENE OLIVEIRA
Affiliation:
Departamento de Matemática, Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Avenida Trabalhador São Carlense, 400, 13566-590São Carlos, SP, Brazil. email: regilene@icmc.usp.br
YULIN ZHAO
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai519082, Guangdong, People’s Republic of China. email: mcszyl@mail.sysu.edu.cn
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Abstract

In 1958 started the study of the families of algebraic limit cycles in the class of planar quadratic polynomial differential systems. In the present we known one family of algebraic limit cycles of degree 2 and four families of algebraic limit cycles of degree 4, and that there are no limit cycles of degree 3. All the families of algebraic limit cycles of degree 2 and 4 are known, this is not the case for the families of degree higher than 4. We also know that there exist two families of algebraic limit cycles of degree 5 and one family of degree 6, but we do not know if these families are all the families of degree 5 and 6. Until today it is an open problem to know if there are algebraic limit cycles of degree higher than 6 inside the class of quadratic polynomial differential systems. Here we investigate the birth and death of all the known families of algebraic limit cycles of quadratic polynomial differential systems.

Keywords

Algebraic limit cyclesHopf bifurcationhomoclinic orbitsheteroclinic orbitsquadratic polynomial differential systems

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 34C05: Location of integral curves, singular points, limit cycles 37G15: Bifurcations of limit cycles and periodic orbits 70K05: Phase plane analysis, limit cycles
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 317 - 336
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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