Quadratic Differential Systems Research Articles

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

Global Phase Portraits of Piecewise Quadratic Differential Systems with a Pseudo-Center

This paper deals with the global dynamics of planar piecewise smooth differential systems constituted by two different vector fields separated by one straight line that passes through the origin. From a quasi-canonical family of piecewise quadratic differential systems with a pseudo-focus point at the origin, which has six parameters, we investigate the subfamilies where the origin is indeed a pseudo-center. For such subfamilies, we classify its global phase portraits in the Poincaré disk and the associated bifurcation sets.

New Insights on Non-integrability and Dynamics in a Simple Quadratic Differential System

This study focuses on the integrability and qualitative behaviors of a quadratic differential system x˙=a+yz,y˙=-y+x2,z˙=b-4x.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{x}=a+yz,\\quad\\dot{y}=-y + x^{2},\\quad\\dot{z}=b-4x.$$\\end{document}We provide some new perspectives on the system and reveal its diverse properties, including non-integrability in the sense of absence of first integrals, bifurcations of co-dimension one or two, Jacobi instability and dynamics at infinity.

Limit Cycles of the Three-dimensional Quadratic Differential System via Hopf Bifurcation

In this study, the quadratic 3-dimensional differential system is considered, in which the origin of the coordinate becomes the Hopf equilibrium point. The existence and stability of limit cycles that emerge from the Hopf point are being investigated. The Lyapunov coefficients connected to the Hopf point are calculated using the projection method. First, four families of parameter conditions are driven by which the quadratic 3-dimensional differential system can exhibit codimension three of the Hopf bifurcation. The analytical proof of each parameter family of conditions is given by calculating the Lyapunov coefficients, the vanishing of the first, sconed Lyapunov, and the non-zero of the third Lyapunov coefficients. The explicit conditions are presented for the existence and stability of three limit cycles arising from each family of Hopf bifurcations. The output of existence displays a stable (unstable) Hopf point, accompanied by the emergence of two stable (unstable) limit cycles alongside one unstable (stable) limit cycle in the neighborhood of the unstable (stable) origin of the coordinated 3-dimensional quadratic system. In addition, the outcome is utilized for exploring the limit cycles of the n-scroll chaotic attractor system, which has many practical uses, including secure communication, encryption, random number generation, and autonomous mobile robots. The conditions are derived under which the origin point of this system becomes the Hopf point, and three limit cycles can exist around the Hopf point. Finally, the numerical demonstrations show that the system undergoes a supercritical Hopf bifurcation, resulting in two stable and one unstable limit cycle. Furthermore, all results are verified.

О предельных циклах квадратичных дифференциальных систем, имеющих два фокуса*

Получены достаточные условия отсутствия предельных циклов у квадратичных систем, имеющих в фазовой плоскости два и только два простых состо-яния равновесия, если эти состояния равновесия – фокусы. С помощью аффинных преоб-разований фазовых переменных определены критерии, которые позволяют из множе-ства всех квадратичных систем выделить класс систем, имеющих два и только два про-стых состояния равновесия. Приведены примеры, подтверждающие сформулированные утверждения. Sufficient conditions are obtained for the absence of limit cycles in quadratic systems having two and only two simple equilibrium states in the phase plane, if these equilibri-um states are focuses. With the help of affine transformations of phase variables, criteria are determined that allow us to distinguish from the set of all quadratic systems a class of systems having two and only two simple equilibrium states. Examples confirming the formulated state-ments are given.

Dynamics of the Isotropic Star Differential System from the Mathematical and Physical Point of Views

The following differential quadratic polynomial differential system dxdt=y−x, dydt=2y−yγ−12−γy−5γ−4γ−1x, when the parameter γ∈(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)≥0 is the mass inside the sphere of radius r of the star, y=4πr2ρ where ρ is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ∈R∖{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4πr2ρ of an isotropic star when r decreases.

Global Analysis of Riccati Quadratic Differential Systems

In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait.

Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing One Finite Saddle-Node and a Separatrix Connection

This paper is part of a series of works whose ultimate goal is the complete classification of phase portraits of quadratic differential systems in the plane modulo limit cycles. It is estimated that the total number may be around 2000, so the work to find them all must be split in different papers in a systematic way so to assure the completeness of the study and also the non intersection among them. In this paper we classify the family of phase portraits possessing one finite saddle-node and a separatrix connection and determine that there are a minimum of 77 topologically different phase portraits plus at most 16 other phase portraits which we conjecture to be impossible. Along this paper we also deploy a mistake in the book (Artés et al. in Structurally unstable quadratic vector fields of codimension one, Birkhäuser/Springer, Cham, 2018) linked to a mistake in Reyn and Huang (Separatrix configuration of quadratic systems with finite multiplicity three and a M1,10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^0_{1,1}$$\\end{document} type of critical point at infinity. Report Technische Universiteit Delft, pp 95–115, 1995).

Lower bounds for the cyclicity of centers of quadratic three-dimensional systems

We consider quadratic three-dimensional differential systems having a Hopf singular point. We study their cyclicity when the singular point is a center on the center manifold using higher-order developments of the Lyapunov constants. As a result, we make a chart of the cyclicity by establishing the lower bounds for several known systems in the literature, including the Rössler, Lorenz, and Moon-Rand systems. Moreover, we construct an example of a jerk system to obtain 12 limit cycles bifurcating from the center, which is a new lower bound for three-dimensional quadratic systems.

Phase Portraits of Families VII and VIII of the Quadratic Systems

The quadratic polynomial differential systems in a plane are the easiest nonlinear differential systems. They have been studied intensively due to their nonlinearity and the large number of applications. These systems can be classified into ten classes. Here, we provide all topologically different phase portraits in the Poincaré disc of two of these classes.

The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields

We denote by ${\mbox{\boldmath $QSL$}}_3$ the family of quadratic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly three. In a sequence of papers the complete study of quadratic systems with invariant lines of total multiplicity at least four was achieved. In addition three more families of quadratic systems possessing invariant lines of total multiplicity at least three were also studied, among them the Lotka-Volterra family. However there were still systems in ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ missing from all these studies. The goals of this article are: to complete the study of the geometric configurations of invariant lines of ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ by studying all the remaining cases and to give the full classification of this family modulo their configurations of invariant lines together with their bifurcation diagram. The family ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ has a total of 81 distinct configurations of invariant lines. This classification is done in affine invariant terms and we also present the bifurcation diagram of these configurations in the 12-parameter space of coefficients of the systems. This diagram provides an algorithm for deciding for any given system whether it belongs to ${\mbox{\boldmath $QSL$}}_3$ and in case it does, by producing its configuration of invariant straight lines.

Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves

Abstract The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R 2 {{\mathbb{R}}}^{2} , particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.

Global Phase Portraits of the Quadratic Systems Having a Singular and Irreducible Invariant Curve of Degree 3

Any singular irreducible cubic curve (or simply, cubic) after an affine transformation can be written as either [Formula: see text], or [Formula: see text], or [Formula: see text]. We classify the phase portraits of all quadratic polynomial differential systems having the invariant cubic [Formula: see text]. We prove that there are [Formula: see text] different topological phase portraits for such quadratic polynomial differential systems. We control all the bifurcations among these distinct topological phase portraits. These systems have no limit cycles. Only three phase portraits have a center, 19 of these phase portraits have one polycycle, three of these phase portraits have two polycycles. The maximum number of separartices that have these phase portraits is 26 and the minimum number is nine, the maximum number of canonical regions of these phase portraits is seven and the minimum is three.

Global Phase Portraits of Quadratic Polynomial Differential Systems Having as Solution Some Classical Planar Algebraic Curves of Degree 6

Abstract The main goal of this paper is to classify the global phase portraits of seven quadratic polynomial differential systems, exhibiting as invariant algebraic curves seven well-known algebraic curves of degree six. We prove that these systems have five topologically different phase portraits in the Poincarédisc.

Limit Cycles of a Class of Discontinuous Piecewise Differential Systems Separated by the Curve y = xn Via Averaging Theory

Recently there is increasing interest in studying the limit cycles of the piecewise differential systems due to their many applications. In this paper we prove that the linear system [Formula: see text], [Formula: see text], can produce at most seven crossing limit cycles for [Formula: see text] using the averaging theory of first order, where the bounds [Formula: see text] for [Formula: see text] even and the bounds [Formula: see text] for [Formula: see text] odd are reachable, when it is perturbed by discontinuous piecewise polynomials formed by two pieces separated by the curve [Formula: see text] ([Formula: see text]), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has [Formula: see text] or [Formula: see text] crossing limit cycles for [Formula: see text] even and, furthermore, under a particular condition we prove that the number of crossing limit cycles does not exceed [Formula: see text] (resp., [Formula: see text]) for [Formula: see text] even (resp., [Formula: see text] even). The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if [Formula: see text] is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems.

On the limit cycles of the piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center

While the limit cycles of the discontinuous piecewise differential systems formed by two linear differential systems separated by one straight line have been studied intensively, and up to now there are examples of these systems with at most 3 limit cycles. There are almost no works studying the limit cycles of the discontinuous piecewise differential systems formed by one linear differential system and a quadratic polynomial differential system separated by one straight line.In this paper using the averaging theory up to seven order we prove that the discontinuous piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center separated by one straight line can have 8 limit cycles. More precisely, at every order of the averaging theory from order one to order seven we provide the maximum number of limit cycles that can be obtained using the averaging theory. 2010 mathematics subject classificationPrimary 34C05, 34A34.

Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line

We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being 3 such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of 7 limit cycles of small amplitude.

Geometric analysis of quadratic differential systems with invariant ellipses

Geometric analysis of quadratic differential systems with invariant ellipses

Bifurcation of Limit Cycles for a Perturbed Piecewise Quadratic Differential Systems

Bifurcation of Limit Cycles for a Perturbed Piecewise Quadratic Differential Systems

EXISTENCE AND COMPUTATION OF INVARIANT ALGEBRAIC CURVES FOR PLANAR QUADRATIC DIFFERENTIAL SYSTEMS

<abstract> Some necessary conditions are given for the existence of invariant algebraic curves for planar quadratic differential systems in a special canonical form. An efficient algorithm is then designed for computations of invariant algebraic curves. From the algorithm, a quadratic differential system is found with two Hopf bifurcations as the parameter varies, each leading to an invariant algebraic limit cycle of degree 5. A family of degree 6 invariant algebraic limit cycles is also produced. To further demonstrate the capability of the algorithm, we provide a quadratic system with a family of degree 7 invariant algebraic curves enclosing one or two centers, and a system possessing a degree 16 irreducible invariant algebraic curve with a singular point of multiplicity 8 on the curve. </abstract>