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

Consider the class of systems dx/dt = y, dy/dt = -x + mu Sigma(3)(j=0) h(j)(x, mu)y(j) 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. (C) 2013 Elsevier Inc. All rights reserved.