On a Result of Darboux

DOI10.1112/S1461157000000863zbMath1068.13013OpenAlexW2317242021MaRDI QIDQ4827592

Jaume Llibre, Jean Moulin Ollagnier, Javier Chavarriga

Publication date: 18 November 2004

Published in: LMS Journal of Computation and Mathematics (Search for Journal in Brave)

Full work available at URL: http://www.lms.ac.uk/jcm/4/lms2000-025/

Mathematics Subject Classification ID

Polynomial rings and ideals; rings of integer-valued polynomials (13F20) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) (34C08)

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Algebraic limit cycles of degree 4 for quadratic systems ⋮ On the birth and death of algebraic limit cycles in quadratic differential systems ⋮ Relationships between limit cycles and algebraic invariant curves for quadratic systems ⋮ Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems ⋮ Algebraic Limit Cycles Bifurcating from Algebraic Ovals of Quadratic Centers ⋮ Classification of Invariant Algebraic Curves and Nonexistence of Algebraic Limit Cycles in Quadratic Systems from Family (I) of the Chinese Classification ⋮ Some remarks about the integration of polynomial planar vector fields. ⋮ On the limit cycles bifurcating from an ellipse of a quadratic center ⋮ Simple Darboux points of polynomial planar vector fields. ⋮ On some open problems in planar differential systems and Hilbert's 16th problem ⋮ On the stability of limit cycles for planar differential systems ⋮ Coexistence of limit cycles and invariant algebraic curves for a Kukles system ⋮ Invariant algebraic curves and rational first integrals of holomorphic foliations in \(\mathbb C P(2)\) ⋮ Invariant algebraic curves of large degree for quadratic system ⋮ Liouvillian integration of the Lotka-Volterra system ⋮ Classification of quadratic systems admitting the existence of an algebraic limit cycle ⋮ Quadratic planar differential systems with algebraic limit cycles via quadratic plane Cremona maps ⋮ On the classification of foliations of degree three with one singularity

Cites Work

• Algebraic invariant curves of plane polynomial differential systems
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