A cubic polynomial system with seven limit cycles at infinity

Abstract

In this paper, we study bifurcation of limit cycles at infinity for a class of cubic polynomial system with no singular points at infinity, in which the problem for bifurcation of limit cycles from infinity be transferred into that from the origin. By computation of singular point values, the conditions of the origin (correspondingly, infinity) to be a center and the highest degree fine focus are derived. Consequently, we construct a cubic system which can bifurcate seven limit cycles from infinity when let normal parameters be suitable values. The positions of these limit cycles without constructing Poincaré cycle fields can be pointed out exactly.