Estimates for the number of limit cycles of the planar polynomial differential systems with homogeneous nonlinearities

Abstract

This paper devotes to the study of planar polynomial differential systems with homogeneous nonlinearities of degree n>1. We are concerned with the maximum number of limit cycles surrounding the origin of such systems, denoted by Ho(n). By means of the second order analysis using the theories of Melnikov functions, we provide new estimates for Ho(n) restricted to the cases where the origin is a focus, a node, a saddle or a nilpotent singularity. In particular, Ho(n)≥n for each n in the case of focus. To the best of our knowledge, this improves the previous works in the literature.