# New sufficient conditions for a center and global phase portraits for polynomial systems.

• Volume: 40, Issue: 2, page 351-372
• ISSN: 0214-1493

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## Abstract

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In this paper we consider cubic polynomial systems of the form: x' = y + P(x, y), y' = −x + Q(x, y), where P and Q are polynomials of degree 3 without linear part. If M(x, y) is an integrating factor of the system, we propose its reciprocal V (x, y) = 1 / M(x,y) as a linear function of certain coefficients of the system. We find in this way several new sets of sufficient conditions for a center. The resulting integrating factors are of Darboux type and the first integrals are in the Liouville form.By induction, we have generalized these results for polynomial systems of arbitrary degree.Moreover, for the cubic case, we have constructed all the phase portraits for each new family with a center.

## How to cite

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Giacomini, Hector, and Ndiaye, Malick. "New sufficient conditions for a center and global phase portraits for polynomial systems.." Publicacions Matemàtiques 40.2 (1996): 351-372. <http://eudml.org/doc/41271>.

@article{Giacomini1996,
abstract = {In this paper we consider cubic polynomial systems of the form: x' = y + P(x, y), y' = −x + Q(x, y), where P and Q are polynomials of degree 3 without linear part. If M(x, y) is an integrating factor of the system, we propose its reciprocal V (x, y) = 1 / M(x,y) as a linear function of certain coefficients of the system. We find in this way several new sets of sufficient conditions for a center. The resulting integrating factors are of Darboux type and the first integrals are in the Liouville form.By induction, we have generalized these results for polynomial systems of arbitrary degree.Moreover, for the cubic case, we have constructed all the phase portraits for each new family with a center.},
author = {Giacomini, Hector, Ndiaye, Malick},
journal = {Publicacions Matemàtiques},
keywords = {Modelo dinámico; Ecuaciones polinómicas; Ecuaciones diferenciales; Resolución de ecuaciones; Integrabilidad; Ecuación de Liouville; integrating factor; center; cubic systems; global phase portraits},
language = {eng},
number = {2},
pages = {351-372},
title = {New sufficient conditions for a center and global phase portraits for polynomial systems.},
url = {http://eudml.org/doc/41271},
volume = {40},
year = {1996},
}

TY - JOUR
AU - Giacomini, Hector
AU - Ndiaye, Malick
TI - New sufficient conditions for a center and global phase portraits for polynomial systems.
JO - Publicacions Matemàtiques
PY - 1996
VL - 40
IS - 2
SP - 351
EP - 372
AB - In this paper we consider cubic polynomial systems of the form: x' = y + P(x, y), y' = −x + Q(x, y), where P and Q are polynomials of degree 3 without linear part. If M(x, y) is an integrating factor of the system, we propose its reciprocal V (x, y) = 1 / M(x,y) as a linear function of certain coefficients of the system. We find in this way several new sets of sufficient conditions for a center. The resulting integrating factors are of Darboux type and the first integrals are in the Liouville form.By induction, we have generalized these results for polynomial systems of arbitrary degree.Moreover, for the cubic case, we have constructed all the phase portraits for each new family with a center.
LA - eng
KW - Modelo dinámico; Ecuaciones polinómicas; Ecuaciones diferenciales; Resolución de ecuaciones; Integrabilidad; Ecuación de Liouville; integrating factor; center; cubic systems; global phase portraits
UR - http://eudml.org/doc/41271
ER -

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