Solutions Of Linear Differential Systems Research Articles

(ω, c)-Almost Periodic Distributions

The aim of this work is the introduction of (w,c)-almost periodicity (resp. asymptotic (w,c)-almost periodicity) in distributions spaces. The characterizations and main properties of these distributions are given. We also study the existence of distributional (w,c)-almost periodic solutions of linear differential systems.

Boundedness and almost periodicity of solutions of linear differential systems

Abstract In this paper, we study the following linear differential system (1) x ′ ( t ) = A ( t ) x ( t ) , x ( t ) ∈ ℝ n , t ∈ ℝ , $${{x}^{\prime }}(t)=A(t)x(t),\,\,\,\,x(t)\in {{\mathbb{R}}^{n}},\quad t\in \mathbb{R},$$ where t ↦ A(t) is a matrix valued almost periodic function. We prove that if all the solutions of the above system are almost periodic, there exists an almost periodic function b : R → R n such that the following differential equation (2) x ′ ( t ) = A ( t ) x ( t ) + b ( t ) , x ( t ) ∈ ℝ n , t ∈ ℝ $${{x}^{\prime }}(t)=A(t)x(t)+b(t),\,\,\,\,x(t)\in {{\mathbb{R}}^{n}},\quad t\in \mathbb{R}$$ has no bounded solution. In particular, if for each almost periodic function b there exists a bounded solution to (2), there exists at least one solution for (1) that is not almost periodic.

Asymptotic solutions of (q(qy′)′)′+py′+ry=0

We are concerned with the third-order linear differential equation (q(qy′)′)′+py′+ry=0 defined on an interval [t0,∞). We derive asymptotic solutions for cases characterized by p being “small” and “large”, respectively. Our main tool is a result by Eastham on the asymptotic solution of linear differential systems.

Representation of solutions of linear differential systems with pure delay and multiple delays with linear parts given by non-permutable matrices

Nonhomogeneous linear systems of second order differential equations with pure delay and multiple delays are considered. Representations of their solutions without a commutativity assumption on the matrix coefficients are obtained by using delayed matrix functions and Laplace transform. Our results improve and extend previous results.

On Construction of Solutions of Linear Differential Systems with Argument Deviations of Mixed Type

We show the use of parametrization techniques and successive approximations for the effective construction of solutions of linear boundary value problems for differential systems with multiple argument deviations. The approach is illustrated with a numerical example.

(ω, c)- Pseudo almost periodic distributions

Abstract The paper is a study of the (w, c) −pseudo almost periodicity in the setting of Sobolev-Schwartz distributions. We introduce the space of (w, c) −pseudo almost periodic distributions and give their principal properties. Some results about the existence of distributional (w, c) −pseudo almost periodic solutions of linear differential systems are proposed.

Another view of the maximum principle for infinite-horizon optimal control problems in economics

The authors present their recently developed complete version of the Pontryagin maximum principle for a class of infinite-horizon optimal control problems arising in economics. The main distinguishing feature of the result is that the adjoint variable is explicitly specified by a formula analogous to the Cauchy formula for solutions of linear differential systems. In certain situations this formula implies the ‘standard’ transversality conditions at infinity. Moreover, it can serve as an alternative to them. Examples demonstrate the advantages of the proposed version of the maximum principle. In particular, its applications are considered to Halkin’s example, to Ramsey’s optimal economic growth model, and to a basic model for optimal extraction of a non-renewable resource. Also presented is an economic interpretation of the characterization obtained for the adjoint variable. Bibliography: 62 titles.

Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous n -dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold \mathcal{Z}\subset\mathbb{R}^n of periodic solutions satisfying \dim(\mathcal{Z}) < n. Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, x'=Mx , when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system: x'=Mx+ \varepsilon F_1^n(x)+\varepsilon^2 F_2^n(x), in \mathbb{R}^{d+2} , where \varepsilon is a small parameter, M is a (d+2)\times(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d-m non-zero real eigenvalues.

Dynamic Evaluation of Exponential Polynomial Curves and Surfaces via Basis Transformation

It is shown in "SIAM J. Sci. Comput. 39 (2017):B424-B441" that free-form curves used in computer aided geometric design can usually be represented as the solutions of linear differential systems and points and derivatives on the curves can be evaluated dynamically by solving the differential systems numerically. In this paper we present an even more robust and efficient algorithm for dynamic evaluation of exponential polynomial curves and surfaces. Based on properties that spaces spanned by general exponential polynomials are translation invariant and polynomial spaces are invariant with respect to a linear transform of the parameter, the transformation matrices between bases with or without translated or linearly transformed parameters are explicitly computed. Points on curves or surfaces with equal or changing parameter steps can then be evaluated dynamically from a start point using a pre-computed matrix. Like former dynamic evaluation algorithms, the newly proposed approach needs only arithmetic operations for evaluating exponential polynomial curves and surfaces. Unlike conventional numerical methods that solve a linear differential system, the new method can give robust and accurate evaluation results for any chosen parameter steps. Basis transformation technique also enables dynamic evaluation of polynomial curves with changing parameter steps using a constant matrix, which reduces time costs significantly than computing each point individually by classical algorithms.

Representation of Solutions of Linear Differential Systems of the Second Order with Constant Delays

We deduce representations for the solutions of initial-value problems for n-dimensional differential equations of the second order with delays: $$ x^{{\prime\prime} }(t)=2 Ax^{\prime}\left( t-\tau \right)-\left({A}^2+{B}^2\right) x\left( t-2\tau \right) $$ and $$ x^{{\prime\prime} }(t)=\left( A+ B\right) x^{\prime}\left( t-\tau \right)- A B x\left( t-2\tau \right) $$ by using special delay matrix functions. Here, A and B are commuting (n × n)-matrices and τ > 0. Moreover, a formula connecting the delay matrix exponential function with delayed matrix sine and delayed matrix cosine is obtained. We also discuss common features of the considered equations.

Dynamic Evaluation of Free-Form Curves and Surfaces

Free-form curves and surfaces represented by control points together with well-defined basis functions are widely used in computer aided geometric design. Efficient evaluation of points and derivatives of free-form curves and surfaces plays an important role in interactive rendering or CNC machining. In this paper we show that free-form curves with properly defined basis functions are the solutions of linear differential systems. By employing typical numerical methods for solving the differential systems, points and derivatives of free-form curves and surfaces can be computed in a dynamical way. Compared with traditional methods for evaluating free-form curves and surfaces there are two advantages of the proposed technique. First, the proposed method is universal and efficient for evaluating a large class of free-form curves and surfaces. Second, the evaluation needs only arithmetic operations even when the free-form curves and surfaces are defined using some transcendental functions.

Procedures for searching local solutions of linear differential systems with infinite power series in the role of coefficients

Construction of Laurent, regular, and formal (exponential---logarithmic) solutions of full-rank linear ordinary differential systems is discussed. The systems may have an arbitrary order, and their coefficients are formal power series given algorithmically. It has been established earlier that the first two problems are algorithmically decidable and the third problem is not decidable. A restricted variant of the third problem was suggested for which the desired algorithm exists. In the paper, a brief survey of algorithms for the abovementioned decidable problems is given. Implementations of these algorithms in the form of Maple procedures with a uniform interface and data representation are suggested.

On exponential-logarithmic solutions of linear differential systems with power series coefficients

The paper studies construction of formal exponential-logarithmic solutions for a system of ordinary linear differential equations the coefficients of which are algorithmically defined formal power series. The construction of all such solutions is generally algorithmically undecidable problem. We propose an algorithm and its implementation in Maple that make it possible to construct a basis of the space of solutions of the full-rank system if the dimension of this space is known.

Oscillation and rotability characteristics of solutions of linear differential systems

We present examples illustrating the realizability of all admissible relations between the frequencies and the oriented rotability exponents of solutions of two- and three-dimensional linear differential systems.

Local improvements of lower bounds for valuations of solutions of linear difference systems with meromorphic coefficients

An algorithm for improvement of lower bounds for valuations of components of meromorphic solutions of linear difference systems with meromorphic coefficients is proposed. The system may have an arbitrary order. The algorithm is based on comparison of valuations of the equation coefficients with the given bounds of solution valuations.

Regular solutions of linear differential systems with power series coefficients

The following problem is considered: given a system of linear ordinary differential equations of arbitrary order with power series coefficients, to recognize whether it has regular solutions at point 0 and, if it does, to find them. An algorithm for solving this problem is proposed. Each power series that is a coefficient of the original system is specified by a procedure that computes the series coefficient by the index of this coefficient. The original system is assumed to have full rank; i.e., the equations of the system are independent. The algorithm is implemented in Maple.

On singular points of solutions of linear differential systems with polynomial coefficients

We consider systems of linear ordinary differential equations containing m unknown functions of a single variable x. The coefficients of the systems are polynomials over a field k of characteristic 0. Each of the systems consists of m equations independent over k[x, d/dx]. The equations are of arbitrary orders. We propose a computer algebra algorithm that, given a system S of this form, constructs a polynomial d(x) ∈ k[x] \ {0} such that if S possesses a solution in \( \bar{k}{\left( {\left( {x - \alpha } \right)} \right)^m} \) for some \( \alpha \in \bar{k} \) and a component of this solution has a nonzero polar part, then d(α) = 0. In the case where k ⊆ ℂ and S possesses an analytic solution having a singularity of an arbitrary type (not necessarily a pole) at α, the equality d(α) = 0 is also satisfied.

Denominators of rational solutions of linear difference systems of an arbitrary order

An algorithm for finding a universal denominator of rational solutions of a system of linear difference equations with polynomial coefficients is proposed. The equations may have arbitrary orders.

Oscillation and wandering characteristics of solutions of a linear differential system

We introduce new Lyapunov characteristics for the oscillation and wandering of solutions of linear differential equations or systems, namely, the frequency of a solution (the mean number of zeros on the time axis), of some coordinate of the solution, or of all possible linear combinations of these coordinates, and also the mean angular velocity of the rotation of a solution (about the origin in the phase space) and various wandering exponents (derived from the mean angular velocity). We shall show that the sets of values of all these quantities on the solutions of a linear autonomous system coincide with the set of absolute values of the imaginary parts of eigenvalues of the matrix of the system. We shall see that the frequencies of solutions are bounded above by their wandering exponents, and the frequencies and wandering exponents of all solutions of an arbitrary second-order equation coincide.

Exponential Stability and Estimation of Solutions of Linear Differential Systems of Neutral Type with Constant Coefficients

This paper investigates the exponential-type stability of linear neutral delay differential systems with constant coefficients using Lyapunov-Krasovskii type functionals, more general than those reported in the literature. Delay-dependent conditions sufficient for the stability are formulated in terms of positivity of auxiliary matrices. The approach developed is used to characterize the decay of solutions (by inequalities for the norm of an arbitrary solution and its derivative) in the case of stability, as well as in a general case. Illustrative examples are shown and comparisons with known results are given.