A note on first integrals of vector fields and endomorphisms

1972 ◽  
Vol 3 (2) ◽  
pp. 127-132 ◽  
Author(s):  
Ricardo Mane
Keyword(s):  
Vector Fields ◽  
First Integrals

scholarly journals First integrals and normal forms for germs of analytic vector fields

2008 ◽  
Vol 245 (5) ◽  
pp. 1167-1184 ◽  
Author(s):  
Jian Chen ◽  
Yingfei Yi ◽  
Xiang Zhang
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
First Integrals ◽  
Analytic Vector

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 Polynomial first integrals of quadratic vector fields

2006 ◽  
Vol 230 (2) ◽  
pp. 393-421 ◽  
Author(s):  
Javier Chavarriga ◽  
Belén García ◽  
Jaume Llibre ◽  
Jesús S. Pérez del Río ◽  
José Angel Rodríguez
Keyword(s):  
Vector Fields ◽  
First Integrals ◽  
Quadratic Vector Fields
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