On the center conditions for analytic monodromic degenerate singularities

In this paper we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields can be applied independently that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allows to study monodromic degenerate singularities with characteristic directions

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

In this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.The first and second authors are partially supported by Ministerio de Ciencia, Innovación y Universidades/ FEDER grant number PGC2018-096265- B-I00 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (projects P12-FQM-1658, FQM-276). The third author is partially supported by a MINECO/ FEDER grant number PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya) Grant number 2017SGR-1276

The center problem for Z_2-symmetric nilpotent vector fields

We say that a polynomial differential system ˙x = P(x, y), ˙y = Q(x, y) having the origin as a singular point is Z2-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 Z2-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.The first and second authors are partially supported by a MINECO/FEDER grant number MTM2014-56272-C2-2 and by the Consejer´ıa de Educaci´on y Ciencia de la Junta de Andaluc´ıa (projects P12-FQM-1658, FQM-276). The third author is partially supported by a MINECO/FEDER grant number MTM2017-84383-P and by an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The fourth author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568

Averaging methods of arbitrary order, periodic solutions and integrability

Agraïments: The third and fourth authors are partially supported by NNSF of China grant number 11271252, by RFDP of Higher Education of China grant number 20110073110054. The fourth author also is supported by the innovation program of Shanghai Municipal Education Commission grant 15ZZ012.In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods. Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability. For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number N such that if its first N averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for n = 2. Furthermore, in a punctured neighborhood of the origin, the system is C^ completely integrable for n > 2 provided that each periodic orbit has a trivial holonomy. Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities

Centers: their integrability and relations with the divergence

This is a brief survey on the centers of the analytic differential systems in R^2. First we consider the kind of integrability of the different types of centers, and after we analyze the focus--center problem, i.e. how to distinguish a center from a focus. This is a difficult problem which is not completely solved. We shall present some recent results using the divergence of the differential system