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On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

Part of: Qualitative theory

Published online by Cambridge University Press:  09 January 2020

Maria V. Demina  and
Claudia Valls
Maria V. Demina
Affiliation:
National Research University Higher School of Economics, 34 Tallinskaya Street, 123458, Moscow, Russian Federation (maria_dem@mail.ru)
Claudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001Lisboa, Portugal (cvalls@math.tecnico.ulisboa.pt)
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Abstract

We present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

Keywords

invariant algebraic curvesDarboux polynomialsLiouvillian first integralsLiénard systemsPoincaré problem

MSC classification

Primary: 34C05: Location of integral curves, singular points, limit cycles
Secondary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 34C08: Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3231 - 3251
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Copyright
Copyright © 2020 The Royal Society of Edinburgh

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