Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.     

Controllability and stabilizability of linear time‐varying distributed hereditary control systems

This paper is concerned with the controllability and stabilizability problem for control systems described by a time‐varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite difference solution of the third boundary problem in elliptic and parabolic equations

Summary Finite difference methods (including the Peaceman-Rachford method) are considered for the solution of the third boundary value problem for parabolic and elliptic equations. Conditions on the coefficients involved in the boundary conditions are obtained from the stability requirements of the difference methods and shown to coincide with those necessary for asymptotic stability of the differential system.     

Approximate solution to a boundary value problem for the Lyapunov differential equation with a parameter

A special boundary value problem is studied for the Lyapunov differential equation which is used for investigation of the asymptotic properties of solutions to systems of periodic differential equations with a parameter. An algorithm is proposed for constructing an approximate solution to this boundary value problem, and conditions on the parameter are found under which the zero solution to the system is asymptotically stable.     

On numerical study of periodic solutions of a delay equation in biological models

Some results are presented of the numerical study of periodic solutions of a nonlinear equation with a delayed argument in connection with themathematical models having real biological prototypes. The problem is formulated as a boundary value problem for a delay equation with the conditions of periodicity and transversality. A spline-collocation finite-difference scheme of the boundary value problem using a Hermitian interpolation cubic spline of the class C ¹ with fourth order error is proposed. For the numerical study of the system of nonlinear equations of the finitedifference scheme, the parameter continuation method is used, which allows us to identify possible nonuniqueness of the solution of the boundary value problem and, hence, the nonuniqueness of periodic solutions regardless of their stability. By examples it is shown that the periodic oscillations occur for the parameter values specific to the real molecular-genetic systems of higher species, for which the principle of delay is quite easy to implement.     

Numerical approximation of connecting orbits with asymptotic rate

Summary. In this paper we set up and analyze a numerical method for so called {\bf connecting orbits with asymptotic rate } in parameterized dynamical systems. A connecting orbit with asymptotic rate has its initial value in a given submanifold of the phase space (or its cross product with parameter space) and it converges with an exponential rate to a given orbit, e. g. a steady state or a periodic orbit. It is well known that orbits with asymptotic rate can be used to foliate stable or strong stable manifolds of invariant sets. We show that the problem of determining a connecting orbit with asymptotic rate is well-posed if a certain transversality condition is made and a specific relation between the number of stable dimensions and the number of parameters holds. For the proof we employ the implicit function theorem in spaces of exponentially decaying functions. Using asymptotic boundary conditions we truncate the original problem to a finite interval and show that the error decays exponentially. Typically the asymptotic boundary conditions by themselves are the result of a boundary value problem, e. g. if the limiting orbit is periodic. Thus it is expensive to calculate them in a parameter dependent way during the approximation procedure. To avoid this we develop a boundary corrector method which turns out to be nearly optimal after very few steps. Received April 28, 2000 / Revised version received December 18, 2000 / Published online May 30, 2001     

Stability of a critical nonlinear neutral delay differential equation

This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. An analysis based on the energy method provides new results about the asymptotic stability of constant and periodic solutions. A complete analysis of the stability diagram is given in the linear homogeneous case. Under periodic forcing, existence of periodic solutions is discussed, involving a Diophantine condition on the period of the source.     

Global asymptotic stability for a periodic delay hematopoiesis model with impulses

In this paper, sufficient conditions for the global asymptotic stability of a broad family of periodic impulsive scalar delay differential equations are obtained. These conditions are applied to a periodic hematopoiesis model with multiple time-dependent delays and linear impulses, in order to establish criteria for the global asymptotic stability of a positive periodic solution. The present results are discussed within the context of recent literature. In conclusion, when compared with previous works, not only sharper stability criteria are obtained here, even for models without impulses, but also the usual constraints imposed on the linear impulses are relaxed.     

On the existence,uniqueness and stability of periodic solutions of large scale systems with unbounded delay

In this paper, we consider the periodic solution problems for the systems with unbounded delay, and the existence, uniqueness and stability of the periodic solution are dealt with unitedly. First we establish the suitable delay-differential inequality, then study seperately the problems of periodic solution for the systems with bounded delay, with unbounded delay and the Volterra integral-differential systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré's periodic operator in the dfferent phase spaces. A series of simple criteria for the existence, uniqueness and stability of these systems are obtained.     

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.     

On periodic solutions of nonlinear second order differential systems

The paper deals with the second order differential systems with periodic boundary conditions. In the first part of the paper four different methods are employed to find the unique solution of the linear systems. One of the methods is a shooting type which converts the periodic boundary value problem into its equivalent initial value problem. In the second part of the paper several sufficient conditions are provided for the existence and uniqueness for the nonlinear systems. The technique developed for the linear systems to convert into its equivalent initial value problem is used in an iterative way for the nonlinear systems. It is shown that the iterations converge quadratically.     

Estimates of solutions to a class of systems of nonlinear delay differential equations

Under study are the systems of nonlinear delay differential equations with periodic coefficients of the linear terms. Some sufficient conditions for the asymptotic stability of the zero solution are established. We obtain the estimates that characterize the decay rate of solutions at infinity and describe the attraction sets of the zero solution.     

Asymptotics for solutions of periodic difference systemsSupported in part by Fondecyt 1030535, 1080034, DGI-MATH UNAP 2008 and Diufro 120521.

Using dichotomies and periodic conditions, we obtain asymptotic formulas for solutions of a difference system of Poincaré type with periodic coefficients. Some results about the theory of existence of periodic solutions for linear difference systems are presented. At the end, an open problem on the asymptotic spectral representation is proposed.     

Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices

In this paper a generalization of the delayed exponential defined by Khusainov and Shuklin (2003) [1] for autonomous linear delay systems with one delay defined by permutable matrices is given for delay systems with multiple delays and pairwise permutable matrices. Using this multidelay-exponential a solution of a Cauchy initial value problem is represented. By an application of this representation and using Pinto’s integral inequality an asymptotic stability results for some classes of nonlinear multidelay differential equations are proved.     

Local bifurcations of plane running waves for the generalized cubic Schrödinger equation

For the generalized cubic Schrödinger equation, we consider a periodic boundary value problem in the case of n independent space variables. For this boundary value problem, there exists a countable set of plane running waves periodic with respect to the time variable. We analyze their stability and local bifurcations under the change of stability. We show that invariant tori of dimension 2, ..., n + 1 can bifurcate from each of them. We obtain asymptotic formulas for the solutions on invariant tori and stability conditions for bifurcating tori as well as parameter ranges in which, starting from n = 3, a subcritical (stiff) bifurcation of invariant tori is possible.     

Discrete approximations and periodic solutions of differential inclusions

We consider two numerical methods for solving a periodic boundary value problem for a system of differential inclusions, the Galerkin method and the polygon method. To the original problem, we assign a sequence of its discretizations. Conditions under which the existence of solutions of the periodic boundary value problem implies the solvability of its discrete versions are presented. The convergence of the sequence of approximate solutions is analyzed.     

On the Riesz basis property of the root functions of a discontinuous boundary problem

In this paper, we consider the nonself‐adjoint discontinuous Sturm Liouville operator with periodic (antiperiodic) boundary condition and compatibility conditions. Asymptotic formulas of eigenvalues and eigenfunctions of the operator are obtained. Using these accurate asymptotic formulas for eigenvalues and eigenfunctions, we prove the basisness of the root functions of the boundary value problem.     

Asymptotic behavior of Timoshenko beam with dissipative boundary feedback

This paper is concerned with the stabilization problem of Timoshenko beam in the presence of linear dissipative boundary feedback controls. Using C₀-semigroups theory we establish the existence and the uniqueness of solution of the proposed closed loop system. In order to consider the asymptotic behavior of the closed loop system, we first discuss the existence of nonzero solution of a closely related boundary value problem. Then we derive various necessary and sufficient conditions for the system to be asymptotically stable. Finally, we prove the equivalence between the exponential stability and the asymptotic stability for the closed loop system.     

Stability with probability 1 of solutions of systems of linear stochastic differential-difference Ito equations

Conditions for absolute (independent of lags) asymptotic stability with probability 1 of systems of stochastic equations cited in the title of this paper are obtained. The proposed approach allows us to reduce the problem of analyzing stability to determination of the conditions for the existence of a positively determined solution of a linear matrix equation. These conditions are stated in terms of locations of eigenvalues of the matrix constructed from the elements of the matrices of the initial system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 147–151, February, 1991.     

On the asymptotics of solutions to the second initial boundary value problem for Schrödinger systems in domains with conical points

In this paper, for the second initial boundary value problem for Schrödinger systems, we obtain a performance of generalized solutions in a neighborhood of conical points on the boundary of the base of infinite cylinders. The main result are asymptotic formulas for generalized solutions in case the associated spectrum problem has more than one eigenvalue in the strip considered.