Rational limit cycles of Abel equations

Jaume Llibre; Claudia Valls

doi:10.3934/cpaa.2021007

Article Contents

2021, Volume 20, Issue 3: 1077-1089. Doi: 10.3934/cpaa.2021007

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Rational limit cycles of Abel equations

Jaume Llibre^1, , and
Claudia Valls^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2.
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal

^* Corresponding author
^* Corresponding author

Received: May 31, 2020

Revised: October 31, 2020

Early access: January 2021

Published: March 2021

The first author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d'Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2019

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Abstract

We deal with Abel equations $ dy/dx = A(x) y^2 + B(x) y^3 $, where $ A(x) $ and $ B(x) $ are real polynomials. We prove that these Abel equations can have at most two rational limit cycles and we characterize when this happens. Moreover we provide examples of these Abel equations with two nontrivial rational limit cycles.

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Mathematics Subject Classification: Primary: 34C05, 34C07, 34C08.

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