The center problem for Z2-symmetric nilpotent vector fields

Abstract

We say that a polynomial differential systemẋ = P (x, y),ẏ = Q(x, y) having the origin as a singular point is Z 2-symmetric if P (−x, −y) = −P (x, y) and Q(−x, −y) = −Q(x, y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C ∞ first integral. But up to know there are no characterized these two kinks of nilpotent centers. Here we prove that the origin of any Z 2-symmetric is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x, y) = y 2 + • • •, where the dots denote terms of degree higher than two. 2010 Mathematics Subject classification. 34C05, 34A34, 34C14. Keywords and phrases. Z 2-symmetric differential systems, center problem, nilpotent singularity.

A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

“…However, in general, it is not analytically integrable around the singularity. Therefore, it is interesting to distinguish when a vector field with a center has an analytical first integral, see [4,5,11,25,29,32,39,40].…”

“…However, this procedure is difficult to implement and with a high cost of computations to eliminate all the terms that not appear in the final expression of the normal form. Is for this reason that there are few families of nilpotent vector fields where the center conditions are known, see, for instance, [7,11,12,20,27]. Moreover, alternative methods are used to find such centers, as the generalized polar coordinates, Cherkas' method, approximation by nondegenerate systems, see, for instance, [6,16,17,[28][29][30]34,35].…”

Cyclicity of nilpotent centers with minimum Andreev number

“…Notice that this restriction is never satisfied by our systems (2) or (3). As far as I know this is the first work where a general method for computing global upper bound on the nilpotent center cyclicity is obtained for the monodromic class (n,β) = (2,1). By the way we will observe the troublesome role that the parameter ω plays, particularly when we need to complexify the parameter space in order to analyze the cyclicity problem.…”

“…In some papers like in [33] the Hopf bifurcation at nilpotent center is analyzed under the restriction that the family is close to a Hamiltonian center by introducing a small perturbation parameter. Notice that this restriction is never satisfied by our systems (2) or (3). As far as I know this is the first work where a general method for computing global upper bound on the nilpotent center cyclicity is obtained for the monodromic class (n,β) = (2,1).…”

Simultaneity of centres in ℤq-equivariant systems

“…Z 2 -symmetric systems are of special interest. Recently, the centre problem for such systems was solved in [27]. The bifurcations of the limit cycles for Z 2 -symmetric cubic systems have been studied in [28], and for Z 2 -symmetric Liénard systems in [29].…”