Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields

Abstract

In this paper we consider the limit cycles of the planar system d dt (x, y) = Xn + Xm,where Xn and Xm are quasi-homogeneous vector fields of degree n and m respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is 1. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to be dealt with separately, and for both equations, an efficient approach to estimate the number of periodic solutions is constructing suitable auxiliary functions. In the present paper we introduce a formula on the divergence, which allows us to construct an auxiliary function of one equation with the auxiliary function of the other equation, and vice versa. Contents 1. Introduction and statements of main results 862 2. Preliminaries 864 2.1. A basic result for 1-dimensional non-autonomous differential equation 864 2.2. An estimate for the number of limit cycles of Abel equation 865 2.3. A formula on the divergence 867 2.4. The distribution of limit cycles of system (1) 869