Center cyclicity for some nilpotent singularities including the ℤ2-equivariant class

This work concerns with polynomial families of real planar vector fields having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal inverse integrating factor vanishing at the singularity with a leading term of minimum [Formula: see text]-quasihomogeneous weighted degree, being [Formula: see text] the Andreev number of the singularity. These families strictly include the case [Formula: see text] and also the [Formula: see text]-equivariant families. In some cases for such families we solve, under additional assumptions, the local Hilbert 16th problem giving global bounds on the maximum number of limit cycles that can bifurcate from the singularity under perturbations within the family. Several examples are given.

Polynomial Inverse Integrating Factors, First Integral and Non-Existence of Limit Cycles in the Plane for Quadratic Systems

The main purpose of this paper is to study the existence of polynomial inverse integrating factor and first integral, and non-existence of limit cycles for all systems. Furthermore, we consider some applications.

Nilpotent centres via inverse integrating factors

In this paper, we are interested in the nilpotent centre problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centres. More general objects are considered as the formal inverse integrating factors. However, the existence of a formal inverse integrating factor is not sufficient to describe all the nilpotent centres. For the family studied in this paper, it is enough.

Analytic Integrability of Two Lopsided Systems

In this paper, we present two classes of lopsided systems and discuss their analytic integrability. The analytic integrable conditions are obtained by using the method of inverse integrating factor and theory of rotated vector field. For the first class of systems, we show that there are [Formula: see text] small-amplitude limit cycles enclosing the origin of the systems for [Formula: see text], and ten limit cycles for [Formula: see text]. For the second class of systems, we prove that there exist [Formula: see text] small-amplitude limit cycles around the origin of the systems for [Formula: see text], and nine limit cycles for [Formula: see text].

On the Shape of Limit Cycles That Bifurcate from Isochronous Center

New idea and algorithm are proposed to compute asymptotic expression of limit cycles bifurcated from the isochronous center. Compared with known inverse integrating factor method, new algorithm to analytically computing shape of limit cycle proposed in this paper is simple and easy to apply. The applications of new algorithm to some examples are also given.