Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2

2007 ◽  
Vol 67 (2) ◽  
pp. 327-348 ◽  
Author(s):  
Laurent Cairó ◽  
Jaume Llibre
Keyword(s):  
Vector Fields ◽  
First Integral ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Rational 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.

scholarly journals SYMMETRIC PERIODIC ORBITS NEAR HETEROCLINIC LOOPS AT INFINITY FOR A CLASS OF POLYNOMIAL VECTOR FIELDS

2006 ◽  
Vol 16 (11) ◽  
pp. 3401-3410 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Phase Portrait ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Phase Portraits ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Open Set ◽  
Symmetric Periodic Orbits

For polynomial vector fields in ℝ3, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a plane and, second, the existence of two symmetric heteroclinic loops.

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