Abstract
In this work we consider the problem of classifying all configurations of singularities, both finite and infinite of quadratic differential systems, with respect to the geometric equivalence relation defined in [2]. This relation is deeper than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddles) of different orders. Such distinctions are however important in the production of limit cycles close to the foci in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows to incorporate all these important geometric features which can be expressed in purely algebraic terms. This equivalence relation is also deeper than the qualitative equivalence relation introduced in [20]. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in [3] where the classification was done for systems with total multiplicity mf of finite singularities less than or equal to one. That work was continued in [4] where the geometric classification was done for the case mf = 2 and two more papers [5] and [6], which cover the case mf = 3. In this article we obtain the geometric classification of singularities, finite and infinite, for the three subclasses of quadratic differential systems with mf = 4 possessing exactly two finite singularities, namely: (i) systems with two double complex singularities (18 configurations); (ii) systems with two double real singularities (33 configurations) and (iii) systems with one triple and one simple real singularities (123 configurations). We also give here the global bifurcation diagrams of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for these subclasses of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, fact which gives an algorithm for determining the geometric configuration of singularities for any quadratic system.
Highlights
Introduction and statement of main resultsWe consider here differential systems of the form dx dt =p(x, y), q(x, y), (1.1)J
By a singular point at infinity of a planar polynomial vector field we mean a singular point of the vector field which is located on the equator of the sphere, located on the boundary circle of the Poincaré disk
If at a stage the coordinates are (x, y) and we do a blow-up of a singular point in y-direction, this means that we introduce a new variable z and consider the diffeomorphism of the (x, y) plane for x = 0 defined by φ(x, y) = (x, y, z) where y = xz
Summary
The geometric equivalence relation (see further below) for finite or infinite singularities, introduced in [6] and used in [2,3,4,5], takes into account such distinctions This equivalence relation is deeper than the qualitative equivalence relation introduced by Jiang and Llibre in [19] because it distinguishes among the foci (or saddles) of different orders and among the various types of nodes. These configurations are expressed using the notation described in Subsection 2.5
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