The Geometry of Quadratic Differential Systems With a Weak Focus of Second Order

Artés, Joan C.; Llibre, Jaume; Schlomiuk, Dana

doi:10.1142/s0218127406016720

Cited by 51 publications

(66 citation statements)

References 30 publications

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“…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

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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. †

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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

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“…For these as well as for other families, the polynomial invariants were sufficient to obtain the topological classification. In general, the polynomial invariants need however to be used in conjunction with other methods, analytical, geometric and numerical, to obtain full results (see for example [5]). …”

Section: Interaction Between Topological and Polynomial Invariantsmentioning

confidence: 99%

“…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]). …”

Section: Normal Forms and Bifurcation Diagramsmentioning

confidence: 99%

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

Schlomiuk¹

2014

Publicacions Matemàtiques

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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.

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“…Recently related analysis are done in [3,4,5,7,11,12]. In particular, in [12], the so-called SIS-model is considered, that is used in the study of infectious diseases.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, in [12], the so-called SIS-model is considered, that is used in the study of infectious diseases. The papers [3,4,5,7,12] contribute to the classification of planar quadratic differential systems; due to the 6-dimensional parameter and the richness of phase portraits, its bifurcation diagram also is studied for intrinsic subclasses reducing its dimension, and similarly these sub-bifurcation diagrams then are analyzed by slicing and imbedding in projective planes. Furthermore another generalization can be found in [11], where a global topological classification is studied for a 1-parameter cubic Hamiltonian planar differential system in which a finite center is linked to singularities at infinity.…”

Section: Introductionmentioning

confidence: 99%

Global Phase Portraits of Some Reversible Cubic Centers With Noncollinear Singularities

Caubergh

Torregrosa

2013

Int. J. Bifurcation Chaos

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Abstract. The results in this paper show that the cubic vector fieldsẋ = −y + M (x, y) − y(x 2 + y 2 ),ẏ = x + N (x, y) + x(x 2 + y 2 ), where M, N are quadratic homogeneous polynomials, having simultaneously a center at the origin and at infinity, have at least 61 and at most 68 topologically different phase portraits. To this end the reversible subfamily defined by M (x, y) = −γxy, N (x, y) = (γ − λ)x 2 + α 2 λy 2 with α, γ ∈ R and λ = 0, is studied in detail and it is shown to have at least 48 and at most 55 topologically different phase portraits. In particular, there are exactly 5 for γλ < 0 and at least 46 for γλ > 0. Furthermore, the global bifurcation diagram is analyzed.

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The Geometry of Quadratic Differential Systems With a Weak Focus of Second Order

Cited by 51 publications

References 30 publications

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

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

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

Global Phase Portraits of Some Reversible Cubic Centers With Noncollinear Singularities

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