scholarly journals Multiplicity of invariant algebraic curves in polynomial vector fields

2007 ◽  
Vol 229 (1) ◽  
pp. 63-117 ◽  
Author(s):  
Colin Christopher ◽  
Jaume Llibre ◽  
Jorge Vitório Pereira
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Invariant Algebraic Curves

scholarly journals Inverse problems for invariant algebraic curves: explicit computations

2009 ◽  
Vol 139 (2) ◽  
pp. 287-302 ◽  
Author(s):  
Colin Christopher ◽  
Jaume Llibre ◽  
Chara Pantazi ◽  
Sebastian Walcher
Keyword(s):  
Inverse Problems ◽  
Algebraic Curve ◽  
Affine Plane ◽  
Vector Fields ◽  
Algebraic Curves ◽  
General Setting ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Algorithmic Approach ◽  
Invariant Algebraic Curves

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are non-degenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

scholarly journals Cofactors and equilibria for polynomial vector fields

2014 ◽  
Vol 144 (4) ◽  
pp. 753-760
Author(s):  
Antoni Ferragut ◽  
Jaume Llibre
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Equilibrium Points ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Differential Systems ◽  
Existence Of Equilibrium ◽  
Invariant Algebraic Curves ◽  
Exponential Factors

We present a relationship between the existence of equilibrium points of differential systems and the cofactors of the invariant algebraic curves and the exponential factors of the system.

scholarly journals On a Result of Darboux

2001 ◽  
Vol 4 ◽  
pp. 197-210 ◽  
Author(s):  
Javier Chavarriga ◽  
Jaume Llibre ◽  
Jean Moulin Ollagnier
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Enumerative Geometry ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Multiple Points ◽  
Intersection Index ◽  
Basic Facts ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractThis paper is concerned with a relation of Darboux in enumerative geometry, which has very useful applications in the study of polynomial vector fields. The original statement of Darboux was not correct. The present paper gives two different elementary proofs of this relation. The first one follows the ideas of Darboux, and uses basic facts about the intersection index of two plane algebraic curves; the second proof is rather more sophisticated, and gives a stronger result, which should also be very useful. The power of the relation of Darboux is then illustrated by the provision of new, simple proofs of two known results. First, it is shown that an irreducible invariant algebraic curve of degree n > 1 without multiple points for a polynomial vector field of degree m satisfies n ≤ m + 1. Second, a proof is given that quadratic polynomial vector fields have no algebraic limit cycles of degree 3.

Plane polynomial vector fields with prescribed invariant curves

2000 ◽  
Vol 130 (3) ◽  
pp. 633-649 ◽  
Author(s):  
S. Walcher
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Invariant Sets ◽  
Invariant Curves ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Integrating Factors

The main result of this paper is the determination of all plane polynomial vector fields that admit a prescribed collection of algebraic curves as invariant sets. As an application, the polynomial vector fields admitting certain types of algebraic integrating factors are characterized.

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields
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