scholarly journals Planar polynomial vector fields having a polynomial first integral can be obtained from linear systems

2011 ◽  
Vol 24 (7) ◽  
pp. 1115-1119 ◽  
Author(s):  
Belen García ◽  
Héctor Giacomini ◽  
Jesús S. Pérez del Río
Keyword(s):  
Linear Systems ◽  
Vector Fields ◽  
First Integral ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Polynomial First Integral

scholarly journals RATIONAL FIRST INTEGRALS FOR POLYNOMIAL VECTOR FIELDS ON ALGEBRAIC HYPERSURFACES OF ℝn+1

2012 ◽  
Vol 22 (11) ◽  
pp. 1250270 ◽  
Author(s):  
JAUME LLIBRE ◽  
YUDY BOLAÑOS
Keyword(s):  
Algebraic Geometry ◽  
Vector Field ◽  
Linear Algebra ◽  
Vector Fields ◽  
First Integral ◽  
First Integrals ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Algebraic Hypersurfaces ◽  
Rational First Integral

Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in ℝn of degree m is at least [Formula: see text], then the vector field has a rational first integral. Llibre and Zhang used only Linear Algebra to provide a shorter and easier proof of the result given by Jouanolou. We use ideas of Llibre and Zhang to extend the Jouanolou result to polynomial vector fields defined on algebraic regular hypersurfaces of ℝn+1, this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝn+1.

scholarly journals Dynamical and Algebraic Analysis of Planar Polynomial Vector Fields Linked to Orthogonal Polynomials

2020 ◽  
Vol 55 (4) ◽  
Author(s):  
Jorge Rodríguez Contreras ◽  
Alberto Reyes Linero ◽  
Maria Campo Donado ◽  
Primitivo B. Acosta-Humánez
Keyword(s):  
Orthogonal Polynomials ◽  
Vector Fields ◽  
Galois Theory ◽  
First Integral ◽  
Galois Groups ◽  
Phase Portraits ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Points Of View ◽  
Darboux First Integral

In the present work, our goal is to establish a study of some families of quadratic polynomial vector fields connected to orthogonal polynomials that relate, via two different points of view, the qualitative and the algebraic ones. We extend those results that contain some details related to differential Galois theory as well as the inclusion of Darboux theory of integrability and the qualitative theory of dynamical systems. We conclude this study with the construction of differential Galois groups, the calculation of Darboux first integral, and the construction of the global phase portraits.

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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