Bifurcation of limit cycles for a family of perturbed Kukles differential systems

Abstract

We consider an integrable non-Hamiltonian system, which belongs to the quadratic Kukles differential systems. It has a center surrounded by a bounded period annulus. We study polynomial perturbations of such a Kukles system inside the Kukles family. We apply averaging theory to study the limit cycles that bifurcate from the period annulus and from the center of the unperturbed system. First, we show that the periodic orbits of the period annulus can be parametrized explicitly through the Lambert function. Later, we prove that at most one limit cycle bifurcates from the period annulus, under quadratic perturbations. Moreover, we give conditions for the non-existence, existence, and stability of the bifurcated limit cycles. Finally, by using averaging theory of seventh order, we prove that there are cubic systems, close to the unperturbed system, with 1 and 2 small limit cycles.

Limit Cycles of a Class of Polynomial Differential Systems Bifurcating from the Periodic Orbits of a Linear Center

“…In [6], Chavarriga et al studied the maximum number of small amplitude limit cycles for Kukles systems which can coexist with some invariant algebraic curves. By averaging theory, bifurcation of limit cycles for a family of perturbed Kukles differential systems was studied in [7][8][9][10][11]. In [8], Llibre and Mereu studied the maximum number of limit cycles of the Kukles polynomial differential systems…”