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Local analytic integrability for nilpotent centers

Published online by Cambridge University Press:  22 September 2003

JAVIER CHAVARRIGA
Affiliation:
Departament de Matemàtica, Universitat de Lleida, Av. Jaume II, 69, 25001 Lleida, Spain (e-mail: chava@eup.udl.es, gine@eup.udl.es)
HECTOR GIACOMIN
Affiliation:
Laboratoire de Mathématique et Physique Théorique, CNRS(UMR 6083), Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont, 37200 Tours, France (e-mail: giacomin@celfi.phys.univ–tours.fr)
JAUME GINÉ
Affiliation:
Departament de Matemàtica, Universitat de Lleida, Av. Jaume II, 69, 25001 Lleida, Spain (e-mail: chava@eup.udl.es, gine@eup.udl.es)
JAUME LLIBRE
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain (e-mail: jllibre@mat.uab.es)

Abstract

Let X(x,y) and Y(x,y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system \dot{x}=y+ X(x,y), \dot{y}=Y(x,y) has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following.

  1. If X= y f(x,y^2) and Y= g(x,y^2), then the systemhas a local analytic first integral of the form H=y^2+F(x,y),where F starts with terms of order higher than two.

  2. If the system has a formal first integral, then it hasa formal first integral of the form H=y^2+F(x,y), where Fstarts with terms of order higher than two. In particular, if thesystem has a local analytic first integral defined at the origin,then it has a local analytic first integral of the formH=y^2+F(x,y), where F starts with terms of order higher than two.

  3. As an application we characterize the nilpotent centersfor the differential systems \dot{x}=y+P_3(x,y),\dot{y}=Q_3(x,y), which have a local analytic first integral,where P_3 and Q_3 are homogeneous polynomials of degree three.

Type
Research Article
Copyright
2003 Cambridge University Press

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