Geometry of quadratic differential systems in the neighborhood of infinity

Schlomiuk, Dana; Vulpe, Nicolae

doi:10.1016/j.jde.2004.11.001

Cited by 44 publications

(125 citation statements)

References 15 publications

Supporting

Mentioning

123

Contrasting

Unclassified

Order By: Relevance

“…. , 4 is revealed by the next two lemmas (see [55]). Using the transvectant differential operator (3.4) and the invariant polynomials (3.2), (3.5) and (3.7) constructed earlier, we can define the following invariant polynomials which were used for example in [57], [58]:…”

Section: Let Us Consider the Following Gl-comitants Of Systems (S)mentioning

confidence: 83%

“…This resulted in a collaboration of the author with Vulpe who combined in [55] the two approaches: of global geometrical concepts such as for example multiplicity divisors on the plane or on the line at infinity in [53], and invariant polynomials such as they were used in [55]. Due to the geometry introduced in [53], Vulpe and the author simplified in [55] the invariants used in [45] so that the final classification in terms of invariant polynomials became more transparent.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Schlomiuk¹

2014

Publicacions Matemàtiques

View full text Add to dashboard Cite

Abstract:We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial vector fields. The concept of moduli space is discussed in the last section and we indicate its value in understanding the dynamics of families of such systems. Our interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated in the proof of our theorem in the section next to last. These concepts have proven their worth in a number of classification results, among them the most recent work on the geometric classification of the whole class of quadratic vector fields, according to their configurations of infinite singularities. An analog work including both finite and infinite singularities of the whole quadratic class, joint work with J. C. Artés, J. Llibre, and N. Vulpe, is in progress.

show abstract

Section: Let Us Consider the Following Gl-comitants Of Systems (S)mentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

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

Schlomiuk¹

2014

Publicacions Matemàtiques

View full text Add to dashboard Cite

show abstract

“…In this sense, we have to consider a bifurcation related to the existence of either the double infinite singularity 0 2 SN plus a simple one, or a triple one. This phenomenon is ruled by the T-comitant M as proved in [Schlomiuk et al, 2005, Artés et al, 2012. The equation of this surface is…”

Section: Bifurcation Surfaces Due To the Changes In The Nature Of Sinmentioning

confidence: 88%

“…7 of [Artés et al, 2008] we get the formulas which give the bifurcation surfaces of singularities in R 12 , produced by changes that may occur in the local nature of finite singularities. From [Schlomiuk et al, 2005] we get equivalent formulas for the infinite singular points. These bifurcation surfaces are all algebraic and they are the following:…”

Section: Bifurcation Surfaces Due To the Changes In The Nature Of Sinmentioning

confidence: 99%

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

Artés

Rezende

Oliveira

2014

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, Hilbert, 1902, are still open for this family. Our goal is to make a global study of the family QsnSN of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the collision of two infinite singular points. This family can be divided into three different subfamilies, all of them with the finite saddle-node in the origin of the plane with the eigenvectors on the axes and (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and time homotheties are three-dimensional and we give their bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of these forms. In this paper we provide the complete study of the geometry of the first two families, (A) and (B). The bifurcation diagram for the subfamily (A) yields 29 phase portraits for systems in QsnSN (A) counting phase portraits with and without limit cycles, while the bifurcation diagram for the subfamily (B) yields 16 phase portraits for systems in QsnSN (B) under the same conditions. Case (C) will yield quite more cases and will have an independent paper in short. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of QsnSN (A) is not only algebraic due to the presence of a surface found numerically. All points in this surface correspond to connections of separatrices.

show abstract

“…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). …”

Section: Large and Small Limit Cyclesmentioning

confidence: 99%

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

Леонов

Kuznetsov

2013

Int. J. Bifurcation Chaos

770

414

View full text Add to dashboard Cite

From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors.At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50-60s of the last century, the investigations of widely known Markus-Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes.Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit.This survey is dedicated to efficient analytical-numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods. †

show abstract

Geometry of quadratic differential systems in the neighborhood of infinity

Cited by 44 publications

References 15 publications

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

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

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

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

Contact Info

Product

Resources

About