In this paper we illustrate the explicit implementation of a method for computing limit cycles which bifurcate from a 2-dimensional isochronous set contained in ℝ

We consider the polynomial vector fields of arbitrary degree in R 3 having the 2-dimensional algebraic toruswhere l, m and n positive integers, and r ∈ (1, ∞), invariant by their flow. We study the possible configurations of invariant meridians and parallels that these vector fields can exhibit on T 2 (l, m, n). Furthermore we analyze when these invariant meridians or parallels are limit cycles.

No 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.

We will consider two special families of polynomial perturbations of the linear center. For the resulting perturbed systems, which are generalized Liénard systems, we provide the exact upper bound for the number of limit cycles that bifurcate from the periodic orbits of the linear center.