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 $h_j$ 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 $Psi(x,y,mu)=0.$