scholarly journals Research of Chaotic Dynamics of 3D Autonomous Quadratic Systems by Their Reduction to Special 2D Quadratic Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Vasiliy Belozyorov
Keyword(s):  
Differential Equations ◽  
Quadratic Differential ◽  
Chaotic Dynamics ◽  
Quadratic System ◽  
System Of Differential Equations ◽  
Quadratic Systems

New results about the existence of chaotic dynamics in the quadratic 3D systems are derived. These results are based on the method allowing studying dynamics of 3D system of autonomous quadratic differential equations with the help of reduction of this system to the special 2D quadratic system of differential equations.

scholarly journals General Method of Construction of Implicit Discrete Maps Generating Chaos in 3D Quadratic Systems of Differential Equations

2014 ◽  
Vol 24 (02) ◽  
pp. 1450025 ◽  
Author(s):  
Vasiliy Ye. Belozyorov
Keyword(s):  
Differential Equations ◽  
Quadratic Differential ◽  
Chaotic Dynamics ◽  
Autonomous Systems ◽  
2D Systems ◽  
Lambert Function ◽  
Quadratic Systems ◽  
Discrete Maps ◽  
Systems Of Differential Equations ◽  
General Method

A method allowing to study the dynamics of 3D systems of quadratic differential equations by the reduction of these systems to the special 2D systems is presented. The mentioned 2D systems are used for the construction of new types of discrete maps generating the chaotic dynamics in some 3D autonomous systems of quadratic differential equations. Strong simplification of all results gives an introduction of the Lambert function. Due to this function some implicit discrete maps become explicit. Examples are given.

scholarly journals On acyclicity of quadratic system with two non-rough foci

2021 ◽  
pp. 41-46
Author(s):  
Адам Дамирович Ушхо
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Phase Plane ◽  
Second Order ◽  
Quadratic System ◽  
Equilibrium States ◽  
System Of Differential Equations ◽  
Right Hand ◽  
The Right

Доказывается, что система дифференциальных уравнений, правые части которой представляют собой полиномы второй степени, не имеет предельных циклов, если в ограниченной части фазовой плоскости она имеет только два состояния равновесия и при этом они являются состояниями равновесия второй группы. It is proved that a system of differential equations, the right-hand sides of which are second-order polynomials, has no limit cycles if it has only two equilibrium states in the bounded part of the phase plane, and they are the equilibrium states of the second group.

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.

scholarly journals GENERATING CHAOS IN 3D SYSTEMS OF QUADRATIC DIFFERENTIAL EQUATIONS WITH 1D EXPONENTIAL MAPS

2013 ◽  
Vol 23 (06) ◽  
pp. 1350105 ◽  
Author(s):  
VASILIY YE. BELOZYOROV ◽  
SERGEY V. CHERNYSHENKO
Keyword(s):  
Differential Equations ◽  
Quadratic Differential ◽  
Chaotic Behavior ◽  
Linear Part ◽  
Homoclinic Orbits ◽  
Chaotic Attractors ◽  
Quadratic Systems ◽  
Exponential Maps ◽  
Existence Conditions ◽  
Discrete Map

New existence conditions of homoclinic orbits for some systems of ordinary quadratic differential equations with singular linear part are found. A realization of these conditions guarantees the existence of chaotic attractors at 3D autonomous quadratic systems. In addition, a chaotic behavior of solutions of these systems is determined by the 1D discrete map [Formula: see text] at some values of parameters r > 0, p ∈ ℝ, and γ ∈ (-d, ∞), where d > 0; n = 0, 1, 2, …. Examples of chaotic attractors are given.

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.

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