کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4642981 | 1341362 | 2007 | 10 صفحه PDF | دانلود رایگان |
We consider a system of the form x˙=Pn(x,y)+xRm(x,y), y˙=Qn(x,y)+yRm(x,y), where Pn(x,y)Pn(x,y), Qn(x,y)Qn(x,y) and Rm(x,y)Rm(x,y) are homogeneous polynomials of degrees n, n and m , respectively, with n⩽mn⩽m. We prove that this system has at most one limit cycle and that when it exists it can be explicitly found and given by quadratures. Then we study a particular case, with n=3n=3 and m=4m=4. We prove that this quintic polynomial system has an explicit limit cycle which is not algebraic. To our knowledge, there are no such type of examples in the literature.The method that we introduce to prove that this limit cycle is not algebraic can be also used to detect algebraic solutions for other families of polynomial vector fields or for probing the absence of such type of solutions.
Journal: Journal of Computational and Applied Mathematics - Volume 200, Issue 1, 1 March 2007, Pages 448–457