Ergodic Theory and Dynamical Systems



Local analytic integrability for nilpotent centers


JAVIER CHAVARRIGA a1, HECTOR GIACOMIN a2, JAUME GINÉ a1 and JAUME LLIBRE a3
a1 Departament de Matemàtica, Universitat de Lleida, Av. Jaume II, 69, 25001 Lleida, Spain (e-mail: chava@eup.udl.es, gine@eup.udl.es)
a2 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)
a3 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain (e-mail: jllibre@mat.uab.es)

Article author query
chavarriga j   [Google Scholar] 
giacomin h   [Google Scholar] 
gine j   [Google Scholar] 
llibre j   [Google Scholar] 
 

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.

  • 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.
  • 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.
  • 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.

(Received June 18 2001)
(Revised May 16 2002)