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

DOI10.1017/prm.2019.63zbMath1465.34038OpenAlexW2999992619MaRDI QIDQ4965394

Maria V. Demina, Claudia Valls

Publication date: 1 March 2021

Published in: Proceedings of the Royal Society of Edinburgh: Section A Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1017/prm.2019.63

zbMATH Keywords

Poincaré problemDarboux polynomialsLiénard systemsinvariant algebraic curvesLiouvillian first integrals

Mathematics Subject Classification ID

Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Nonlinear ordinary differential equations and systems (34A34) Explicit solutions, first integrals of ordinary differential equations (34A05) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Invariant manifolds for ordinary differential equations (34C45) Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) (34C08)

Related Items (8)

The method of Puiseux series and invariant algebraic curves ⋮ Liouvillian integrability and the Poincaré problem for nonlinear oscillators with quadratic damping and polynomial forces ⋮ Integrability and solvability of polynomial Liénard differential systems ⋮ Classification of Invariant Algebraic Curves and Nonexistence of Algebraic Limit Cycles in Quadratic Systems from Family (I) of the Chinese Classification ⋮ A class of new solvable nonlinear isochronous systems and their classical dynamics ⋮ Integrable geodesic flows and metrisable second-order ordinary differential equations ⋮ A characterization of the generalized Liénard polynomial differential systems having invariant algebraic curves ⋮ Puiseux integrability of differential equations

Cites Work

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• Invariant algebraic curves for Liénard dynamical systems revisited
• Resolution of the Poincaré problem and nonexistence of algebraic limit cycles in family (I) of Chinese classification
• Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems
• Liouvillian integrability of some quadratic Liénard polynomial differential systems
• Qualitative theory of planar differential systems
• Liouvillian First Integrals of Differential Equations
• Invariant algebraic curves and conditions for a centre
• Algebraic invariant curves for the Liénard equation
• Asymptotic behaviour and expansions of solutions of an ordinary differential equation

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