Explicit non-algebraic limit cycles for polynomial systems
经典分析与常微分方程
2007-05-23 v1 动力系统
摘要
We consider a system of the form x'=P_n(x,y)+xR_m(x,y), y'=Q_n(x,y)+yR_m(x,y), where P_n(x,y), Q_n(x,y) and R_m(x,y) are homogeneous polynomials of degrees n, n and m, respectively, with n<=m. We prove that this system has at most one limit cycle and that when it exists it can be explicitly found. Then we study a particular case, with n=3 and m=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.
引用
@article{arxiv.math/0505464,
title = {Explicit non-algebraic limit cycles for polynomial systems},
author = {Armengol Gasull and Hector Giacomini and Joan Torregrosa},
journal= {arXiv preprint arXiv:math/0505464},
year = {2007}
}