Some problems of the linear theory of systems of ordinary differential equations

We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet–Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula to the introduction of amplitude–phase coordinates in a neighborhood of a periodic trajectory of a dynamical system.     

Versions of the Feynman-Kac formula

Some new versions of the Feynman-Kac formula for Brownian motion are considered. An interesting generalization of the formula is related to solutions of systems of linear differential equations. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 46–60. Translated by T. Safonova     

Picone type formula for half‐linear impulsive differential equations with discontinuous solutions

Picone type formula for half‐linear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions is derived. Employing the formula, Leighton and Sturm–Picone type comparison theorems as well as several oscillation criteria for impulsive differential equations are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Systems of essentially infinite-dimensional differential equations

We investigate systems of differential equations with essentially infinite-dimensional elliptic operators (of the Laplace–Lévy type). For nonlinear systems, we prove theorems on the existence and uniqueness of solutions. For a linear system, we give an explicit formula for the solution.     

On Periodic Solutions of One Class of Systems of Differential Equations

We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 483–495, April, 2005.     

Linear systems on curves with no Weierstrass points

We study order-sequences of linear systems on smooth curves and establish the formula:b _j +b _N−j ≤b _N for allj, where {b ₀<b ₁<...<b _N } is the order-sequence of a linear system on a curve. As an application of the formula, we describe all linear systems on curves which have no Weierstrass points.     

The tangent direction bundle of an algebraic variety and generalized Jacobians of linear systems

Summary It is well-known that, on an algebraic variety V of dimension d, there is associated with a set of linear systems whose total dimension is d a Jacobian variety (of dimension d−1) at any point of which (other than base points of the linear systems) there is at least one line (formally) tangent to every variety of each system which passes through the point. This notion generalizes to a set of linear systems of total dimension d+r (0≤r<d), the generalized Jacobian being then of dimension d−r−1. The final aim of this paper is to obtain a general formula (Theorem 5.2) for the homology class of this generalized Jacobian. The proof is derived with the aid of cohomological and bundle-theoretic methods from the study of the tangent direction bundle of V, and the earlier part of the paper establishes the necessary techniques (which are not without their independent interest) for our purposes.     

On the nonresonance property of linear differential-algebraic systems

We consider a system of linear ordinary differential equations in which the coefficient matrix multiplying the derivative of the unknown vector function is identically singular. For systems with constant and variable coefficients, we obtain nonresonance criteria (criteria for bounded-input bounded-output stability). For single-input control systems, we consider the problem of synthesizing a nonresonant system in the stationary and nonstationary cases. An arbitrarily high unsolvability index is admitted. The analysis is carried out under assumptions providing the existence of a so-called “equivalent form” with separated “algebraic” and “differential” components.     

时不变和时变的Caputo分数阶时滞微分系统的常数变易公式

This paper studies the variation of constant formulae for linear Caputo fractional delay differential systems. We discuss the exponential estimates of the solutions for linear time invariant fractional delay differential systems by using the Gronwall＇s integral inequality. The variation of constant formula for linear time invariant fractional delay differential systems is obtained by using the Laplace transform method. In terms of the superposition principle of linear systems and fundamental solution matrix, we also establish the variation of constant formula for linear time varying fractional delay differential systems. The obtained results generalize the corresponding ones of integer-order delayed differential equations.     

Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω ^n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω ^d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.     

Approximated solutions in rational form for systems of differential equations

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Padé approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.     

Entropy Theory from the Orbital Point of View

 Inspired by [17], we develop an orbital approach to the entropy theory for actions of countable amenable groups. This is applied to extend – with new short proofs – the recent results about uniform mixing of actions with completely positive entropy [17], Pinsker factors and the relative disjointness problems [10], Abramov–Rokhlin entropy addition formula [19], etc. Unlike the cited papers our work is independent of the standard machinery developed by Ornstein–Weiss [14] or Kieffer [12]. We do not use non-orbital tools like the Rokhlin lemma, the Shannon–McMillan theorem, castle analysis, joining techniques for amenable actions, etc. which play an essential role in [17], [19] and [10]. (Received 23 October 2000)     

Filtration and prediction of random solutions of a system of linear differential equations with coefficients depending on a finite-valued Markov process

We obtain an equation of optimal filtration for processes of Markov random evolution, which is a solution of systems of linear differential equations with Markov switchings. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 997–1000, July, 1998.     

Comparison theorems for symplectic systems of difference equations

In the present paper, we prove comparison theorems for symplectic systems of difference equations, which generalize difference analogs of canonical systems of differential equations. We obtain general relations between the number of focal points of conjoined bases of two symplectic systems with matrices W _i and $ \hat W_i $ \hat W_i as well as their corollaries, which generalize well-known comparison theorems for Hamiltonian difference systems. We consider applications of comparison theorems to spectral theory and in the theory of transformations. We obtain a formula for the number of eigenvalues λ of a symplectic boundary value problem on the interval (λ ₁, λ ₂]. For an arbitrary symplectic transformation, we prove a relationship between the numbers of focal points of the conjoined bases of the original and transformed systems. In the case of a constant transformation, we prove a theorem that generalizes the well-known reciprocity principle for discrete Hamiltonian systems.     

Invariant cones and stability of linear dynamical systems

We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1446–1461, November, 2006.     

Analytic representation of periodic solutions of linear differential systems

We study the problem of periodic solutions of linear differential systems with small parameter. We establish new conditions for the existence and uniqueness of periodic solutions of these systems, which can be efficiently verified. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 731–735, May, 1997.     

Green’s formula for difference operators

We construct an analogue of the classic Green’s formula for linear partial differential operators for difference operators on a multidimensional integer lattice.     

Distance to Ill-Posedness in Linear Optimization via the Fenchel-Legendre Conjugate

We consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space and with a fixed index set, endowed with the topology of the uniform convergence of the coefficient vectors. A system is ill-posed with respect to the consistency when arbitrarily small perturbations yield both consistent and inconsistent systems. In this paper, we establish a formula for measuring the distance from the nominal system to the set of ill-posed systems. To this aim, we use the Fenchel-Legendre conjugation theory and prove a refinement of the formula in Ref. 1 for the distance from any point to the boundary of a convex set.This research has been partially supported by grants BFM2002–04114-C02 (01–02) from MEC (Spain) and FEDER (EU) and by grants GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain).     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.