On the construction of a class of generalized Kukles systems having at most one limit cycle

Abstract

Consider the class of planar systems fracdxdt=y,quadfracdydt=−x+musumj=03hj(x,mu)yj depending on the real parameter mu. We are concerned with the inverse problem: How to construct the functions hj such that the system has not more than a given number of limit cycles for mu belonging to some (global) interval. Our approach to treat this problem is based on the construction of suitable Dulac-Cherkas functions Psi(x,y,mu) and exploiting the fact that in a simply connected region the number of limit cycles is not greater than the number of ovals contained in the set defined by Psi(x,y,mu)=0.