Related papers: Nilpotent Centers in $\mathbb{R}^3$

Consider analytical three-dimensional differential systems having a singular point at the origin such that its linear part is $y\partial_x-\lambda z\partial_z$ for some $\lambda\neq 0$. The restriction of such systems to a Center Manifold…

Given a nilpotent singular point of a planar vector field, its monodromy is associated with its Andreev number $n$. The parity of $n$ determines whether the existence of an inverse integrating factor implies that the singular point is a…

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincar\'e--Liapunov method to find…

In this work we deal with analytic families of real planar vector fields $\mathcal{X}_\lambda$ having a monodromic singularity at the origin for any $\lambda \in \Lambda \subset \mathbb{R}^p$ and depending analytically on the parameters…

In this work it is solved the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions.

The aim of this paper is to investigate two classical problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching…

Classifications of irreducible components of the set of polynomial differential equations with a fixed degree and with at least one center singularity lead to some other new problems on Picard-Lefschetz theory and Brieskorn modules of…

In this paper, we give a direct method to study the isochronous centers on center manifolds of three dimensional polynomial differential systems. Firstly, the isochronous constants of the three dimensional system are defined and its…

We give a proof of existence of centre manifolds within large domains for systems with an integral of motion. The proof is based on a combination of topological tools, normal forms and rigorous-computer-assisted computations. We apply our…

Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this…

In this paper, we generalize the Poincar\'e-Lyapunov method for systems with linear type centers to study nilpotent centers in switching polynomial systems and use it to investigate the bi-center problem of planar $Z_2$-equivariant cubic…

Consider a family of planar polynomial systems $\dot x = y^{2l-1} - x^{2k+1}, \dot y =-x +m y^{2s+1},$ where $l,k,s\in\mathbb{N^*},$ $2\le l \le 2s$ and $m\in\mathbb{R}.$ We study the center-focus problem on its origin which is a monodromic…

A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with anisotropic planes in the…

Using a new compactification (toroidal compactification) and desingularization, we obtain a complete characterization of monodromy at infinity for polynomial Newton system of arbitrary degree, in which we establish an equivalence between…

We work with polynomial three-dimensional rigid differential systems. Using the Lyapunov constants, we obtain lower bounds for the cyclicity of the known rigid centers on their center manifolds. Moreover, we obtain an example of a quadratic…

We establish a sharpening of Kirillov's lemma on nilpotent Lie algebras with 1-dimensional center and use it to study the structure of 3-step nilpotent Lie algebras.

The aim of this paper is to give a sufficient and necessary condition of the generalized polynomial Li\'enard system with a global center (including linear typer and nilpotent type). Recently, Llibre and Valls [J. Differential Equations,…

We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center focus problem asks for the conditions under which these integrals vanish identically. The problem is closely related…

Using a multiple critical points theorem for locally Lipschitz continuous functionals, we establish the existence of at least three distinct solutions for a parametric discrete differential inclusion problem involving a real symmetric and…

This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of…