Theory Of Differential Systems Research Articles

Cluster synchronization of coupled complex-valued neural networks with leakage and time-varying delays in finite-time

<abstract><p>In cluster synchronization (CS), the constituents (i.e., multiple agents) are grouped into a number of clusters in accordance with a function of nodes pertaining to a network structure. By designing an appropriate algorithm, the cluster can be manipulated to attain synchronization with respect to a certain value or an isolated node. Moreover, the synchronization values among various clusters vary. The main aim of this study is to investigate the asymptotic and CS problem of coupled delayed complex-valued neural network (CCVNN) models along with leakage delay in finite-time (FT). In this paper, we describe several sufficient conditions for asymptotic synchronization by utilizing the Lyapunov theory for differential systems and the Filippov regularization framework for the realization of finite-time synchronization of CCVNNs with leakage delay. We also propose sufficient conditions for CS of the system under scrutiny. A synchronization algorithm is developed to indicate the usefulness of the theoretical results in case studies.</p></abstract>

Problems of the theory of invariants and Lie algebras applied in the qualitative theory of differential systems

In this work there were formulated 18 problems from the theory of invariant processes, Lie algebras, commutative graded algebras, generating functions and Hilbert series, orbit theory and Lyapunov stability theory that are important to be solved. There was substantiated the necessity of using the solutions of these problems in the qualitative theory of differential systems.

Some notes on the dynamic behavior of COVID-19 epidemic disease governed by modified SEIR epidemic model with saturated treatment function

In this work, based on theory of differential systems and stability theory, we are in the first stage to study some applications of differential systems in real-world modelling problems. Here, we propose a mathematical epidemic model, namely SEIR epidemic model with saturated treatment to describe the COVID-19 infection. Based on the next–generation matrix method and the theory of Lyapunov stability, we evaluate the basic reproduction number and investigate the asymptotic behavior of disease-free equilibrium. Additionally, in the case , we also prove the existence of a unique endemic equilibrium. Finally, in order to dicuss the effect of isolation control strategies, we study the optimal control problem for this epidemic model.

New Outcomes Regarding the Existence of Hilfer Fractional Stochastic Differential Systems via Almost Sectorial Operators

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using the concepts and ideas from fractional calculus, multivalued maps, semigroup theory, sectorial operators, and the fixed-point technique. We start by confirming the existence of the mild solution by using Dhage’s fixed-point theorem. Finally, an example is provided to demonstrate the considered Hilferr fractional stochastic differential systems theory.

The inverse problem in the calculus of variations: new developments

Abstract We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of n second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for n = 2. We then examine a new class of solutions in arbitrary dimension n and give some non-trivial examples in dimension 3.

New progress in the inverse problem in the calculus of variations

We present a new class of solutions for the inverse problem in the calculus of variations in arbitrary dimension n. This is the problem of determining the existence and uniqueness of Lagrangians for systems of n second order ordinary differential equations. We also provide a number of new theorems concerning the inverse problem using exterior differential systems theory (EDS). Concentrating on the differential step of the EDS process, our new results provide a significant advance in the understanding of the inverse problem in arbitrary dimension. In particular, we indicate how to generalise Jesse Douglas's famous solution for n=2. We give some non-trivial examples in dimensions 2, 3 and 4. We finish with a new classification scheme for the inverse problem in arbitrary dimension.

Generalized Convolution Behaviors and Topological Algebra

We investigate one-dimensional ‘generalized convolution behaviors’ (gen. beh.) that comprise differential and delay-differential behaviors in particular. We thus continue work of, for instance, Brethé, van Eijndhoven, Fliess, Gluesing-Luerssen, Habets, Loiseau, Mounier, Rocha, Vettori, Willems, Yamamoto, Zampieri of the last twenty-five years. The signal space for these behaviors is the space E of smooth complex-valued functions on the real line. The ring of operators is the commutative integral domain E′ of distributions with compact support with its convolution product that acts on E by a variant of the convolution product and makes it an E′-module. Both E and E′ carry their standard topologies. Closed E′-submodules of finite powers of E were introduced and studied by Schwartz already in 1947 under the name ‘invariant varieties’ and are called gen. beh. here. A gen. beh. is called a behavior if it can be described by finitely many convolution equations. The ring E′ is not noetherian and therefore the standard algebraic arguments from one-dimensional differential systems theory have to be completed by methods of topological algebra. Standard constructions like elimination or taking (closed) images of behaviors may lead to gen. beh. and therefore the consideration of the latter is mandatory. It is not known whether all gen. beh. are indeed behaviors, but we show that many of them are, in particular all autonomous ones. The E′-module E is neither injective nor a cogenerator and, in particular, does not admit elimination in Willems’ sense. But the signal submodule PE of all polynomial-exponential signals is injective for finitely generated modules and thus admits elimination. This is a useful replacement and approximation of the injectivity of E since the polynomial-exponential part of any gen. beh. is dense in it. We also describe a useful replacement of the cogenerator property and thus establish a strong relation between convolution equations and their solution spaces. Input/output structures of gen. beh. exist and are used to prove that also many nonautonomous generalized behaviors are indeed behaviors. The E′-torsion elements of E, i.e., the smooth functions which satisfy at least one nonzero convolution equation, are called ‘mean-periodic functions’ and were studied by many outstanding analysts. Their results are significant for gen. beh.

Weyl metrisability of two-dimensional projective structures

AbstractWe show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the ‘twistor’ bundle of conformal inner products having holomorphic image. The second solution allows to use standard results in algebraic geometry to show that the Weyl connections on the two-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane.

ON IMPLICIT SECOND-ORDER PDE OF A SCALAR FUNCTION ON A PLANE VIA DIFFERENTIAL SYSTEMS

In the present paper, we study implicit second-order PDEs (i.e. partial differential equations) of single type for one unknown function of two variables. In particular, by using the theory of differential systems, we give a geometric characterization of PDEs which have a certain singularity. Moreover, we provide a new invariant of PDEs under contact transformations.

Geometric theory of differential systems: Linearization criterion for systems of second-order ordinary differential equations with a 4-dimensional solvable symmetry group of the Lie–Petrov type VI 1

In the framework of projective-geometric theory of systems of differential equations developed by the authors, this paper studies the group properties of systems of two (resolved with respect to the second derivatives) second-order ordinary differential equations whose right-hand sides are polynomials of the third degree with respect to the derivatives of the unknown functions. A classification of such systems admitting four-dimensional symmetry group of the Lie–Petrov type VI 1 is given. For each of the systems, a necessary and sufficient linearization criterion is obtained, i.e., the authors find the necessary and sufficient conditions under which, by a change of variables, the system can be reduced to a differential system whose integral curves are straight lines and are expressed by three linear parametric equations or two linear equations with constant coefficients. For all linearizable systems, the linearizing changes of variables are indicated.

The variational Lyapunov function and strict stability theory for differential systems

In this paper, we investigate strict stability of differential systems by variational Lyapunov function. We obtain some sufficient conditions and comparison theorems.

On the Lagrangian and Hamiltonian Aspects of Infinite-Dimensional Dynamical Systems and their Finite-Dimensional Reductions

Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the invariance structure of given nonlinear dynamical systems on an infinite-dimensional functional manifold are presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan theory of differential systems on associated jet-manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler-Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. A relationship between this hierarchy and Lax-integrable spatially two-dimensional systems is studied.

Asymptotic Solution of a Differential System Related to the Generalized Hypergeometric Equation

We introduce a first-order differential system Y′(x) =A(x)Y(x) on [a, ∞) particular cases of which are equivalent to standard forms of the generalized hypergeometric equation. Our purpose is to obtain the asymptotic solution of the system as x → ∞ by defining suitable transformations of the solution vector Y and using ideas from a unified asymptotic theory of differential systems. Thus our methods place the system within the scope of this unified theory, and they are independent of specialized properties of the Meijer G-function solutions of generalized hypergeometric equations. As such, our methods are also capable of extension to other situations not covered by these special functions.

EDS: A reduced package for exterior differential systems

The EDS package for symbolic analysis of differential equations using the geometrical approach of exterior differential systems is described. The package implements much of exterior differential systems theory, including prolongation and involution analysis, and has been optimised for large, nonlinear problems. Besides the description of the facilities in the package, two nontrivial sample calculations arising from relativity are presented.

Real and complex homogeneous polynomial ordinary differential equations in $n$-space and $m$-ary real and complex non-associative algebras in $n$-space

There is a long-standing attempt to extend the theory of linear differential systems that are additively perturbed by higher-order terms to differential systems whose lowest degree terms are homogeneous forms of degree m with additive perturbations of degree greater than m. In order to do this, the first step must be the construction of a complete theory of differential systems whose rate functions are homogeneous of degreem (and no higher order perturbations). This will depend upon a full understanding of m-ary algebras over real or complex field, algebras which are commutative, but in general non-associative. The purpose of this work is to give some contributions to this problem. We will generalize results of C. Coleman [C1] and L. Markus [Ma]. Namely, the two results together say:

Some Explicit Conditions for a Control System to be Feedback Equivalent to Extended Goursat Form

The goal of this paper is to derive necessary and sufficient conditions under which a driftless system is feedback equivalent to a system in extended Goursat form. Our results are formulated using tools from exterior differential systems theory and improve the results previously reported in the literature in two ways. First, an algorithmic procedure is given which allows to decide in a finite number of steps whether these conditions are met. Second, up to integration of a Pfaffian system, we are able to explicitly compute the transformation which brings the system into extended Goursat form.

Formal and p-adic theory of differential systems with logarithmic singularities depending upon parameters

Formal and p-adic theory of differential systems with logarithmic singularities depending upon parameters

Perturbation theory for nonlinear feedback control systems and spencer-goldschmidt integrability of linear partial differential equations

This paper aims to develop the differential-geometric and Lie-theoretic foundations of perturbation theory for control systems, extending the classical methods of Poincare from the differential equation-dynamical system level where they are traditionally considered, to the situation where the element of control is added. It will be guided by general geometric principles of the theory of differential systems, seeking approximate solutions of the feedback linearization equations for nonlinear affine control systems. In this study, certain algebraic problems of compatibility of prolonged differential systems are encountered. The methods developed by D. C. Spencer and H. Goldschmidt for studying over-determined systems of partial differential equations are needed. Work in the direction of applying theio theory is presented.

Nonlinear feedback control and systems of partial differential equations

Finding an equivalence between two feedback control systems is treated as a problem in the theory of partial differential equation systems. The mathematical aim is to embed the Jakubzyk-Respondek, Hunt-Meyer-Su work on feedback linearization in the general theory of differential systems due to Lie, Cartan, Vessiot, Spencer, and Goldschmidt. We do this by using the functor taking control systems into differential systems, and studying the equivalence invariants of such differential systems. After discussing the general case, attention is focussed on the special situation of most immediate practical importance, the theory of feedback linearization. In this case, the general system for feedback equivalence becomes a system of linear partial differential equations. Conditions are found that the general solution of this system may be described in terms of a Frobenius system and certain differential-algebraic operations.

Spectral theory for linearizations of dynamical systems

The description of the asymptotic growth rates for the solution trajectories of a system of differential equations is a central problem in the qualitative theory of dynamical systems. These growth rates are described by the eigenvalues of the Jacobian matrix for trajectories near a critical point, by the Floquet exponents for trajectories close to a periodic orbit and, generally, by the Lyapunov characteristic exponents in a neighborhood of an arbitrary trajectory. In the examples, the asymptotic behavior of the solution trajectories is given by the association of a “spectrum,” and this is a recurrent theme in the method of linearization. Sacker and Sell [IO] recently introduced a spectral theory for differential systems of quite general type: the (linear) skew-product flows (see Section 3). Their work is in the spirit of the classical linearization theory and is closely tied to the idea of the Lyapunov exponents. We prefer to use the word “spectrum” literally, identifying it with the spectrum of an operator on a Banach space. In general, we consider a one-parameter group 4’ of homeomorphisms of a compact metric space M which, for example, may be the solution flow of a differential equation on a smooth manifold. Second, we assume that a continuous vector bundle (E, M) is defined, i.e., a locally trivial continuous map n: E -+ M with linear fibers which are isomorphic to either real n-dimensional space R” or complex n-dimensional space C”. In case #’ is the flow of a vector field X on M, this vector bundle could be the tangent bundle TM. Finally, we assume that there is group of homeomorphisms @’ on E with n# = 4% and such that @i: E, --t E,,c,, is a linear isomorphism for each x E M. For instance, in case 4’ is smooth, Cp’ could be the tangent flow on TM given by the derivative of 4’. Using the compactness of M we define a Finsler on E which continuously assigns a norm, denoted / I, to each fiber E,. With this assignment, the space r(E) of continuous sections of E (functions q: M + E such that 7~ o q(x) = x) forms a Banach space with norm IIqll= supXEM u x 155 1 ( )I. The topology of T(E) is