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Algebraic Limit Cycles on Quadratic Polynomial Differential Systems

Published online by Cambridge University Press:  27 February 2018

Jaume Llibre*
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain (jllibre@mat.uab.cat)
Claudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal (cvalls@math.ist.utl.pt)
*
*Corresponding author.

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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