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Algebraic Limit Cycles on Quadratic Polynomial Differential Systems

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  27 February 2018

Jaume Llibre  and
Claudia Valls
Jaume Llibre*
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain ([email protected])
Claudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal ([email protected])
*
*Corresponding author.
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Abstract

Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

Keywords

algebraic limit cyclequadratic polynomial differential systemquadratic polynomial vector field

MSC classification

Primary: 37D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 499 - 512
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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