Counterexample to a Conjecture on the Algebraic Limit Cycles of Polynomial Vector Fields

descriptionPublicationkeyboard_double_arrow_right Article 01 Feb 2005 Spain English Publisher:Springer Science and Business Media LLCJournal:Geometriae Dedicata, volume 110, pages 213-219 (issn: 0046-5755, eissn: 1572-9168,

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Authors: Llibre Saló, Jaume; Pantazi, Chara;

doi: 10.1007/s10711-004-2454-3

handle: 2117/930

Counterexample to a Conjecture on the Algebraic Limit Cycles of Polynomial Vector Fields

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Abstract

In Geometriae Dedicata 79 (2000), 101{108, Rudolf Winkel conjectured: For a given algebraic curve f = 0 of degree m > 4 there is in general no polynomial vector ¯eld of degree less than 2m ¡ 1 leaving invariant f = 0 and having exactly the ovals of f = 0 as limit cycles. Here we show that this conjecture is not true.

Country

Spain

Related Organizations

Universitat Polite`cnica de Catalunya
Spain
Autonomous University of Barcelona
Spain
Universitat Politècnica de Catalunya
Spain

Keywords

Differential equations, :34 Ordinary differential equations::34C Qualitative theory [Classificació AMS], :14 Algebraic geometry::14P Real algebraic and real analytic geometry [Classificació AMS], Classificació AMS::14 Algebraic geometry::14P Real algebraic and real analytic geometry, Classificació AMS::34 Ordinary differential equations::34C Qualitative theory, Geometry, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Topology of real algebraic varieties, Nonlinear ordinary differential equations and systems, algebraic limit cycles, Geometry, Algebraic, Algebraic, Geometria algebraica, polynomial vector fields, Geometria analítica, Equacions diferencials ordinàries

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