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Liouvillian first integrals for a class of generalized Liénard polynomial differential systems

Part of: Stability theory

Published online by Cambridge University Press:  13 July 2016

Jaume Llibre  and
Clàudia Valls
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain (jllibre@mat.uab.cat)
Clàudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal (cvalls@math.ist.utl.pt)
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We study the existence of Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form xʹ = y, yʹ = –g(x) – f(x)y, where f(x) = 3Q(x)Qʹ(x)P(x) + Q(x)2 Pʹ(x) and g(x) = Q(x)Qʹ(x)(Q(x)2 P(x)2 – 1) with P,Q ∈ ℂ[x]. This class of generalized Liénard polynomial differential systems has the invariant algebraic curve (y + Q(x)P(x))2 – Q(x)2 = 0 of hyperelliptic type.

Keywords

Darboux polynomialinvariant algebraic curveexponential factorLiouvillian first integralLiénard polynomial differential system

MSC classification

Secondary: 34D30: Structural stability and analogous concepts
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 6 , December 2016 , pp. 1195 - 1210
Copyright
Copyright © Royal Society of Edinburgh 2016