From topological to geometric equivalence in the classification of singularities at infinity for quadratic vector fields

Abstract

In the topological classification of phase portraits no distinctions are made between a focus and a node and neither are they made between a strong and a weak focus or between foci of different orders.These distinction are however important in the production of limit cycles close to the foci in perturbations of the systems. The distinction between the one direction node and the two directions node, which plays a role in understanding the behavior of solution curves around the singularities at infinity, is also missing in the topological classification.In this work we introduce the notion of geometric equivalence relation of singularities which incorporates these important purely algebraic features. The geometric equivalence relation is finer than the topological one and also finer than the qualitative equivalence relation introduced in [19]. We also list all possibilities we have for singularities finite and infinite taking into consideration these finer distinctions and introduce notations for each one of them. Our long term goal is to use this finer equivalence relation to classify the quadratic family according to their different geometric configurations of singularities, finite and infinite.In this work we accomplish a first step of this larger project. We give a complete global classification, using the geometric equivalence relation, of the whole quadratic class according to the configuration of singularities at infinity of the systems. Our classification theorem is stated in terms of invariant polynomials and hence it can be applied to any family of quadratic systems with respect to any particular normal form. The theorem wegive also contains the bifurcation diagram, done in the 12-parameter space, of the geometric configurations of singularities at infinity, and this bifurcation set is algebraic in the parameter space. To determine the bifurcation diagram of configurations of singularities at infinity for any family of quadratic systems, given in any normal form, becomes thus a simple task using computer algebra calculations.1991 Mathematics Subject Classification. Primary 58K45, 34C05, 34A34.

The Geometry of Quadratic Polynomial Differential Systems with a Finite and an Infinite Saddle-Node (A, B)

“…By Proposition 2.1, if g = 0, then we have a double saddle-node sn (2) , using the notation introduced in [Artés et al, 2012].…”

“…By a semi-elemental point we understand a point with zero determinant of its Jacobian, but only one eigenvalue zero. These points are known in classical literature as semi-elementary, but we use the term semi-elemental introduced in [Artés et al, 2012] as part of a set of new definitions more deeply related to singular points, their multiplicities and, specially, their Jacobian matrices. In addition, an infinite saddle-node of type 0 2 SN is obtained by the collision of an infinite saddle with an infinite node.…”

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

“…• The geometrical equivalence of two quadratic systems according to their singularities at infinity, [6]. The geometrical equivalence relation is finer than the topological one, i.e.…”

“…For example in the case of the whole family QS we can completely describe the global behavior of the configurations of singularities at infinity in terms of polynomial invariants (see [6]). For understanding other features of the family QS we would need to combine the use of polynomial invariants with other methods such as for example geometric methods or methods of numerical analysis (see [5]).…”

“…Much work on geometric classifications of families of polynomial vector fields followed, some done by the author in collaboration with Vulpe ([54], [56], [57], [58], [59], [61]), other work done by Vulpe together with Artés and Llibre ([7], [9]) or by the author together with Artés, Llibre, and Vulpe ( [6]). …”