SINGULAR POINTS OF QUADRATIC SYSTEMS: A COMPLETE CLASSIFICATION IN THE COEFFICIENT SPACE ℝ¹²

Abstract

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert,1900], are still open for this class. Even when not dealing with limit cycles, still some problems have remained unsolved like a complete classification of different phase portraits without limit cycles. For some time it was thought (see [Coppel, 1966]) there could exist a set of algebraic functions whose signs would completely determine the phase portrait of a quadratic system. Nowadays we already know that this is not so, and that there are some analytical, nonalgebraic functions that also play a role when dealing with limit cycles and separatrix connections. However, it is possible to find out a set of algebraic functions whose signs determine the characteristics of all finite and infinite singular points. Most of the work up to now has dealt with this problem studying it using different normal forms adapted to some subclasses of quadratic systems. A general work useful for any quadratic system regardless of affine changes has only been done for the study of infinite singular points [Schlomiuk et al., 2005]. In this paper, we give a complete global classification of quadratic differential systems according to their topological behavior in the vicinity of the finite singular points. Our classification Main Theorem gives us a complete dictionary describing the local behavior of finite singular points using algebraic invariants and comitants which are a powerful tool for algebraic computations. Linking the result of this paper with the main one of [Schlomiuk et al., 2005] which uses the same algebraic invariants, it is possible to complete the algebraic classification of singular points (finite and infinite) for quadratic differential systems.

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

“…Since our normal form does not allow the existence of a finite singular point of multiplicity 3, the only possible bifurcation related to collisions of finite singularities with themselves is whether the other two finite singularities are either real, or complex, or form a double one. This phenomenon is captured by the T-comitant T as proved in [Artés et al, 2008]. The equation of this surface is…”

“…However, this surface is relevant for isolating the regions where a limit cycle surrounding an antisaddle cannot exist. Using the results of [Artés et al, 2008], the equation of this surface is given by W 4 = 0, where…”

More Compactification for Differential Systems

“…(50) can be shown by induction. The conclusions in (51) and (52) follow from (50). The key to the induction is the useful result that…”

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

“…Asymptotic analysis, of the existence of separating solutions in the bands −2 < x < −1 and −1 < x < 0, is an extension of the approach to the study of critical saddle point at infinity under the mapping on the Poincaré sphere [Artes et al, 2008]. The ideas used were proposed in [Leonov, 2009a[Leonov, , 2010a and further developed in .…”

“…At present, in the frame of the proof of this hypothesis, investigations are performed concerning the systematization and analysis of various cases of qualitative behavior in quadratic systems, but these investigations are yet to be completed (see [Artes & Llibre, 1997;Schlomiuk & Pal, 2001;Schlomiuk & Vulpe, 2005;Artes et al, 2006Artes et al, , 2008). …”