Darboux integrability and algebraic limit cycles for a class of polynomial differential systems |
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Authors: | JinLong Cao Jaume Llibre Xiang Zhang |
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Institution: | 1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China 2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, 08193, Bellaterra, Spain 3. Ministry of Education Key Lab of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, 200240, China
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Abstract: | 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. |
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Keywords: | Darboux first integral algebraic limit cycles Abel differential equation |
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