A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals

DOI10.1090/S0002-9939-01-06253-0zbMath0999.34029MaRDI QIDQ2782632

Jaume Llibre, Colin Christopher

Publication date: 8 April 2002

Published in: Proceedings of the American Mathematical Society (Search for Journal in Brave)

zbMATH Keywords

invariant algebraic curvequadratic systemrational first integral

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) Symmetries, invariants of ordinary differential equations (34C14)

Related Items

AN ALGORITHM FOR FINDING INVARIANT ALGEBRAIC CURVES OF A GIVEN DEGREE FOR POLYNOMIAL PLANAR VECTOR FIELDS, Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems, On the Darboux integrability of polynomial differential systems, Number of limit cycles for planar systems with invariant algebraic curves, 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, Integrability of planar polynomial differential systems through linear differential equations, On some open problems in planar differential systems and Hilbert's 16th problem, Perturbation from an asymmetric cubic Hamiltonian, Polynomial inverse integrating factors for quadratic differential systems, Transcendental limit cycles via the structure of arbitrary degree invariant algebraic curves of polynomial planar vector fields

Cites Work

• Holomorphic foliations in \(CP(2)\) having an invariant algebraic curve
• Equations de Pfaff algébriques
• A rational approach to the Prelle-Singer algorithm
• The Poincaré problem in the nondicritical case
• On non-integrability of general systems of differential equations
• Elementary First Integrals of Differential Equations
• Liouvillian First Integrals of Differential Equations
• Invariant algebraic curves and conditions for a centre
• Proximity inequalities and bounds for the degree of invariant curves by foliations of ℙ_{ℂ}²
• Algorithmic Derivation of Centre Conditions
• Planar analytic vector fields with generalized rational first integrals
• About a conjecture on quadratic vector fields
• Unnamed Item
• Unnamed Item