The integrability problem for a class of planar systems

Abstract

In this paper we consider perturbations of quasi-homogeneous planar Hamiltonian systems, where the Hamiltonian function does not contain multiple factors. It is important to note that the most interesting cases (linear saddle, linear centre, nilpotent case, etc) fall into this category. For such kinds of systems, we characterize the integrability problem, by connecting it with the normal form theory.

Nilpotent Systems Admitting an Algebraic Inverse Integrating Factor over $${{\mathbb{C}}((x,y))}$$

“…, where D 0 ∧F k ∈ P t k+|t| is the product wedge of both vector fields and div(F k ) ∈ P t k is the divergence of F k , see [2]. This sum is known as the conservative-dissipative splitting of a quasi-homogeneous vector field.…”

Application of Power Geometry and Normal Form Methods to the Study of Nonlinear ODEs

“…This system is a subfamily of the system introduced in [6]. Systems with a nilpotent matrix of the linear part were thoroughly studied by Lyapunov and others.…”

“…In the general case, such problems have not been studied yet. However, a particular case of the system of this type was considered in [6], where the authors put −α = δ = 1 and 3 β + 2 γ = 0. Further, the authors of [6] studied the Hamiltonian subcase of this system under the additional assumption that the Hamiltonian function is expandable into the product of only square-free factors.…”

On the Formal Integrability Problem for Planar Differential Systems

“…One of the main open problems in the qualitative theory of differential systems in R 2 is the distinction between a center and a focus, called the center problem, and its relation with the integrability problem; see, for instance, [1][2][3][4][5]. The notion of center can be extended to the case of a : − resonant singular point of a polynomial vector field in C 2 and to some other situations (resonant node, saddle node, and nonelementary singular points); see [6].…”

“…There exist several methods to find necessary conditions; see [1,18]. However, there is no general method to provide the sufficiency for each family that satisfies some necessary conditions.…”