scholarly journals The acyclicity of a quadratic differential system

2021 ◽  
pp. 32-41
Author(s):  
Адам Дамирович Ушхо ◽  
Вячеслав Бесланович Тлячев ◽  
Дамир Салихович Ушхо
Keyword(s):  
Differential Equations ◽  
Differential System ◽  
Limit Cycles ◽  
Quadratic Differential ◽  
Qualitative Theory ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Quadratic Differential System ◽  
Straight Line

Дан краткий обзор некоторых основных публикаций, посвященных исследованию вопроса о предельных циклах и сепаратрисах квадратичных дифференциальных систем. Рассмотрено наличие замкнутых траекторий для определенного класса автономных квадратичных систем на плоскости. Доказательство основано на применении теории прямых изоклин, признаков Дюлака и Бендиксона качественной теории дифференциальных уравнений. Предложенное доказательство покрывает результаты известной работы Л.А. Черкаса и Л.С. Жилевич. We now give a brief overview of some of the main publications devoted to the study of the question of limit cycles and separatrices of quadratic differential systems. In this paper, we consider the existence of closed trajectories for a certain class of autonomous quadratic systems on the plane. The proof is based on the application of the theory of straight line isoclines, Dulac and Bendixon criteria of the qualitative theory of differential equations. The proposed proof covers the results of the well-known work of L.A. Cherkas and L.S. Zhilevich.

SOME BIFURCATION DIAGRAMS FOR LIMIT CYCLES OF QUADRATIC DIFFERENTIAL SYSTEMS

2001 ◽  
Vol 11 (01) ◽  
pp. 197-206 ◽  
Author(s):  
H. S. Y. CHAN ◽  
K. W. CHUNG ◽  
DONGWEN QI
Keyword(s):  
Singular Point ◽  
Differential System ◽  
Limit Cycles ◽  
Quadratic Differential ◽  
Bifurcation Diagrams ◽  
Differential Systems ◽  
Numerical Examples ◽  
Quadratic Differential System ◽  
Realistic Parameter ◽  
Parameter Values

Concrete numerical examples of quadratic differential systems having three limit cycles surrounding one singular point are shown. In case another finite singular point also exists, a (3, 1) distribution of limit cycles is also obtained. This is the highest number of limit cycles known to occur in a quadratic differential system so far. Representative bifurcation diagrams are drawn for realistic parameter values.

The number of limit cycles of certain polynomial differential equations

1984 ◽  
Vol 98 (3-4) ◽  
pp. 215-239 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Equations ◽  
Image Position ◽  
Cubic Systems

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

2004 ◽  
Vol 56 (2) ◽  
pp. 310-343 ◽  
Author(s):  
Jaume Llibre ◽  
Dana Schlomiuk
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Topological Classification ◽  
Phase Portraits ◽  
Third Order ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Affine Invariants ◽  
Geometric Concepts

AbstractIn this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

THE GEOMETRY OF QUADRATIC DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS OF SECOND ORDER

2006 ◽  
Vol 16 (11) ◽  
pp. 3127-3194 ◽  
Author(s):  
JOAN C. ARTÉS ◽  
JAUME LLIBRE ◽  
DANA SCHLOMIUK
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Second Order ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Bifurcation Set ◽  
The Family

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, 1902], are still open for this class. In this article we make an interdisciplinary global study of the subclass [Formula: see text] which is the closure within real quadratic differential systems, of the family QW2 of all such systems which have a weak focus of second order. This class [Formula: see text] also includes the family of all quadratic differential systems possessing a weak focus of third order and topological equivalents of all quadratic systems with a center. The bifurcation diagram for this class, done in the adequate parameter space which is the three-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for [Formula: see text], 95 for QW2, 20 having limit cycles but only three with the maximum number of limit cycles (two) within this class. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of points which we suspect to be analytic corresponding to global separatrices which have connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. The global geometry of this class [Formula: see text] reveals interesting bifurcations phenomena; for example, all phase portraits with limit cycles in this class can be produced by perturbations of symmetric (reversible) quadratic systems with a center. Many other nonlinear phenomena displayed here form material for further studies.

Bifurcation Analysis in Planar Quadratic Differential Systems with Boundary

2020 ◽  
Vol 30 (07) ◽  
pp. 2030017
Author(s):  
Jocelyn A. Castro ◽  
Fernando Verduzco
Keyword(s):  
Quadratic Differential ◽  
Sufficient Conditions ◽  
The Other ◽  
Tangency Point ◽  
Differential Systems ◽  
Quadratic Differential System ◽  
Bifurcation Phenomena ◽  
Straight Line ◽  
Boundary Equilibrium ◽  
Parameterized Family

Given a planar quadratic differential system delimited by a straight line, we are interested in studying the bifurcation phenomena that can arise when the position on the boundary of two tangency points are considered as parameters of bifurcation. First, under generic conditions, we find a two-parametric family of quadratic differential systems with at least one tangency point. After that, we find a normal form for this parameterized family. Next, we study two subfamilies, one of them characterized by the existence of two fold points of different nature, and the other one, characterized by the existence of one fold point and one boundary equilibrium point. For the first family, we find sufficient conditions for the existence of stationary bifurcations: saddle-node, transcritical and pitchfork, while for the second family, the existence of the called transcritical Bogdanov–Takens bifurcation is proved. Finally, the results are illustrated with two examples.

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

2008 ◽  
Vol 18 (02) ◽  
pp. 313-362 ◽  
Author(s):  
JOAN C. ARTES ◽  
JAUME LLIBRE ◽  
NICOLAE VULPE
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Complete Classification ◽  
Quadratic System ◽  
Singular Points ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Algebraic Functions ◽  
Algebraic Invariants

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.

scholarly journals Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibria separated by reducible cubics

2021 ◽  
pp. 1-38
Author(s):  
Rebiha Benterki ◽  
Jeidy Jimenez ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian Systems ◽  
Differential System ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Main Role ◽  
Differential Systems ◽  
Cubic Curve ◽  
Straight Line ◽  
Linear Hamiltonian Systems ◽  
Linear Differential

Due to their applications to many physical phenomena during these last decades the interest for studying the discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system, but to determine the maximum number of limits cycles that a class of planar differential systems can have is one of the main problems in the qualitative theory of the planar differential systems. Thus in general to provide a sharp upper bound for the number of crossing limit cycles that a given class of piecewise linear differential system can have is a very difficult problem. In this paper we characterize the existence and the number of limit cycles for the piecewise linear differential systems formed by linear Hamiltonian systems without equilibria and separated by a reducible cubic curve, formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. Hence we have solved the extended 16th Hilbert problem to this class of piecewise differential systems.

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