Integrability of planar polynomial differential systems through linear differential equations

DOI10.1216/rmjm/1181069462zbMath1140.34001arXivmath/0506035OpenAlexW2047849888MaRDI QIDQ2477833

Maite Grau, Jaume Giné, Hector J. Giacomini

Publication date: 14 March 2008

Published in: Rocky Mountain Journal of Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0506035

zbMATH Keywords

Darboux integrabilityinvariant algebraic curveintegrating factorpolynomial vector fieldrational first integralLiouvillian function

Mathematics Subject Classification ID

Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Transformation and reduction of ordinary differential equations and systems, normal forms (34C20) Explicit solutions, first integrals of ordinary differential equations (34A05)

Related Items (11)

Explicit non-algebraic limit cycles for polynomial systems ⋮ Nonlocal deformations of autonomous invariant curves for Liénard equations with quadratic damping ⋮ A survey on the inverse integrating factor ⋮ Exactly solvable quadratic differential equation systems through generalized inversion ⋮ Linearizabiliy and Lax representations for cubic autonomous and non-autonomous nonlinear oscillators ⋮ On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory ⋮ Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields ⋮ On some open problems in planar differential systems and Hilbert's 16th problem ⋮ A simple method for constructing non-liouvillian first integrals of autonomous planar systems ⋮ Invariant curves and integrability of planar \(\mathcal C^r\) differential systems ⋮ Polynomial first integrals via the Poincaré series

Cites Work

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• Holomorphic foliations in \(CP(2)\) having an invariant algebraic curve
• Equations de Pfaff algébriques
• On the integrability of two-dimensional flows
• Remarks on ``The classification of reversible cubic systems with center
• Invariant algebraic curves linear in one variable for planar real quadratic systems.
• A family of non-Darboux-integrable quadratic polynomial differential systems with algebraic solutions of arbitrarily high degree
• Generalized cofactors and nonlinear superposition principles
• On real analytic foliations and the center problem
• The Poincaré problem in the nondicritical case
• A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals
• Elementary First Integrals of Differential Equations
• Algebraic Particular Integrals, Integrability and the Problem of the Center
• Some examples for the Poincaré and Painlevé problems
• Liouvillian First Integrals of Differential Equations
• Invariant algebraic curves and conditions for a centre
• About a conjecture on quadratic vector fields

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