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

Consider the class of systems d x d t = y , d y d t = − x + μ ∑ j = 0 3 h j ( x , μ ) y j depending on the real parameter μ . 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 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 .