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

Consider the class of systems dxdt=y,dydt=−x+μ∑j=03hj(x,μ)yj depending on the real parameter μ. 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 μ belonging to some (global) interval. Our approach to treat this problem is based on the construction of suitable Dulac-Cherkas functions Ψ(x,y,μ) 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 Ψ(x,y,μ)=0.