摘要
This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.
This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y＋P n＋1（x,y）＋xF2n（x,y）,y=x＋y＋Q n＋1（x,y）＋yF2n（x,y）,where P i（x,y）,Q i（x,y）and F i（x,y）are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.
基金
supported by National Natural Science Foundation of China (Grant No. 11271252)
Ministerio de Economiay Competitidad of Spain (Grant No. MTM2008-03437)
Agència de Gestió d’Ajuts Universitaris i de Recerca of Catalonia (Grant No. 2009SGR410)
ICREA Academia,Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110073110054)
a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme (Grant Nos. FP7-PEOPLE-2012-IRSES-316338 and 318999)
