Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

Stability of solutions of nonlinear systems with unbounded perturbations

We study systems of differential equations with perturbations that are unbounded functions of time. We suggest a method for constructing Lyapunov functions to determine conditions under which the perturbations do not affect the asymptotic stability of the solutions.     

Partial asymptotic stability of abstract differential equations

We consider the problem of partial asymptotic stability with respect to a continuous functional for a class of abstract dynamical processes with multivalued solutions on a metric space. This class of processes includes finite-and infinite-dimensional dynamical systems, differential inclusions, and delay equations. We prove a generalization of the Barbashin-Krasovskii theorem and the LaSalle invariance principle under the conditions of the existence of a continuous Lyapunov functional. In the case of the existence of a differentiable Lyapunov functional, we obtain sufficient conditions for the partial stability of continuous semigroups in a Banach space. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 629–637, May, 2006.     

Surfaces of revolution associated with the kink-type solutions of the SIdV equation

In this paper, we study the evolution scenarios of surfaces of revolution associated with the kink-type solutions of an integrable equation, which is called the SIdV equation because of its scale-invariant property and relationship with the Korteweg-de Vries equation, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.     

Stabilization and Practical Asymptotic Stability of Abstract Differential Equations

This article studies the problem of stabilization of the infinite-dimension time-varying control systems in Hilbert spaces. We consider the problem of practical asymptotic stability with respect to a continuous functional for a class of abstract nonlinear infinite-dimensional processes with multivalued solutions on a metric space when the origin is not an equilibrium point. In the case of the existence of a differentiable Lyapunov functional, we obtain sufficient conditions for the practical stability of continuous semigroups in a Banach space.     

Stability of solutions of nonlinear systems with unbounded perturbations

We study systems of differential equations with perturbations that are unbounded functions of time. We suggest a method for constructing Lyapunov functions to determine conditions under which the perturbations do not affect the asymptotic stability of the solutions. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 3–8, January, 1998.     

Convergence and stability analysis of system of partial differential equations under Markovian structural perturbations–ii:vextor Lyapunov–like functionals

In this paper we employ the theory of dererministic ordinary differential inequalities together with the concept of vector Lyapunov–like functional to develop basic comparison theorems for system of partial differential equations of parabolic type under Markovian structural perturbations.These results will be utilized to give sufficient conditions for the convergence and stability of the solution process of the system.We also characterize the effects of the random structural perturbations on the qualitative properties of such system. Moreover,the Lyapunov–like functional approach provides a mechanism to characterize the diffusion effects on the qualitative properties of the system.     

Interaction of kinks for semilinear wave equations with a small parameter

We consider a class of semilinear wave equations with a small parameter and nonlinearities such that the equations have exact kink-type solutions. The main result consists in obtaining sufficient conditions for the nonlinearities under which the interaction of kinks preserves the sine-Gordon scenario. This means that the interaction occurs without changing the waves shape and with shifts of trajectories.     

Periodicity and stability in a single-species model governed by impulsive differential equation

A periodic single-species model with periodic impulsive perturbations was investigated. By using Brouwer’s fixed point theorem and the Lyapunov function, sufficient conditions for the existence and global asymptotic stability of positive periodic solutions of the system were derived. Numerical simulations were presented to verify the feasibilities of our main results.     

Almost periodic models in impulsive ecological systems with variable diffusion

By means of piecewise continuous functions Lyapunov’s functions we give new sufficient conditions for the global exponential stability of the unique positive almost periodic solutions of an non autonomous N-dimensional impulsive Lotka-Volterra diffusive competitive system with dispersion and fixed moments of impulsive perturbations.     

On the stability of jetlike magnetohydrodynamic flows

The linear stability problem is under study for steady axisymmetric translational flows of a density-homogeneous nonviscous incompressible ideal conducting fluid with free surface and “frozen-in” poloidal magnetic field. By the direct Lyapunov method, some sufficient conditions are obtained for the stability of these flows under small long-wave perturbations with the same symmetry. These stability conditions have partial converses; and, for unstable stationary flows, an a priori exponential lower bound is constructed on the growth of small perturbations under consideration, while the increment of the appearing exponent serves as an arbitrary positive parameter. An illustrative analytical example is given of steady flows with superimposed small long-wave axisymmetric perturbations growing in time in accordance with the estimate.     

The distance to strong stability

The notion of “strong stability” has been introduced in a recent paper [12]. This notion is relevant for state-space models described by physical variables and prohibits overshooting trajectories in the state-space transient response for arbitrary initial conditions. Thus, “strong stability” is a stronger notion compared to alternative definitions (e.g. stability in the sense of Lyapunov or asymptotic stability). This paper defines two distance measures to strong stability under absolute (additive) and relative (multiplicative) matrix perturbations, formulated in terms of the spectral and the Frobenius norm. Both symmetric and non-symmetric perturbations are considered. Closed-form or algorithmic solutions to these distance problems are derived and interesting connections are established with various areas in matrix theory, such as the field of values of a matrix, the cone of positive semi-definite matrices and the Lyapunov cone of Hurwitz matrices. The results of the paper are illustrated by numerous computational examples.     

The LMI method for stationary oscillation of interval neural networks with three neuron activations under impulsive effects

In this paper, we investigate the stationary oscillation of interval neural networks with three neuron activation functions and time-varying delays under impulsive perturbations. Several theorems are given which present some sufficient conditions to guarantee the existence, uniqueness, and global exponential stability of periodic solutions (i.e., stationary oscillation) based on Lyapunov functionals approach and inequality analysis techniques. The obtained results can be easily checked by the Linear Matrix Inequality control toolbox in MATLAB. Finally, an example is given to illustrate the advantage of the obtained results.     

Dynamics of multi-species competition-predator system with impulsive perturbations and Holling type III functional responses

We investigate the dynamics of a class of multi-species predator-prey interaction models with Holling type III functional responses based on systems of nonautonomous differential equations with impulsive perturbations. Sufficient conditions for existence of a positive periodic solution are investigated by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are established for the global stability and the globally exponential stability of the system by using the comparison principle and the Lyapunov method.     

Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

Stability of solutions of stochastic wave equations with the Bessel operator under Poisson perturbations

We obtain sufficient conditions for the stability of solutions of deterministic and the corresponding stochastic wave equations with the Bessel operator under Poisson perturbations. The sufficient conditions are expressed in terms of coefficients of the equations, which allows us to construct domains of stability in the parameter space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 974–978, July, 1990.     

Coefficient Stability of the Solution of a Difference Scheme Approximating a Mixed Problem for a Semilinear Parabolic Equation

We study the coefficient stability of a difference scheme approximating a mixed problem for a one-dimensional semilinear parabolic equation. We obtain sufficient conditions on the input data under which the solutions of the differential and difference problems are bounded. We also obtain estimates of perturbations of the solution of a linearized difference scheme with respect to perturbations of the coefficients; these estimates agree with the estimates for the differential problem.     

Global attractivity of the almost periodic solution of a delay logistic population model with impulses

In this paper, we study the existence of almost periodic solutions of a delay logistic model with fixed moments of impulsive perturbations. By using a comparison theorem and constructing a suitable Lyapunov functional, a set of sufficient conditions for the existence and global attractivity of a unique positive almost periodic solution is obtained. As applications, some special models are studied; our new results improve and generalize former results.     

Local Constraints on Einstein–Weyl Geometries: The 3-Dimensional Case

Following an earlier study [3], we consider the Einstein–Weyl equations on a fixed (complex) background metric as an equation for a 1-form and its first few derivatives. If the background is flat then we conclude that the only solutions are conformal rescalings of constant curvature metrics. If the background is a homogeneous 3-geometry in Bianchi class A (i.e., with unimodular isometry group), we find necessary and sufficient conditions on the 3-geometry for solutions of the Einstein–Weyl equations to exist. The solutions we find are complexifications of known ones. In particular, we find that the general left-invariant metric on S³ and the metric 'Sol' admit no local solutions of the Einstein–Weyl equations.     

Stability theory and Lyapunov regularity

We establish the stability under perturbations of the dynamics defined by a sequence of linear maps that may exhibit both nonuniform exponential contraction and expansion. This means that the constants determining the exponential behavior may increase exponentially as time approaches infinity. In particular, we establish the stability under perturbations of a nonuniform exponential contraction under appropriate conditions that are much more general than uniform asymptotic stability. The conditions are expressed in terms of the so-called regularity coefficient, which is an essential element of the theory of Lyapunov regularity developed by Lyapunov himself. We also obtain sharp lower and upper bounds for the regularity coefficient, thus allowing the application of our results to many concrete dynamics. It turns out that, using the theory of Lyapunov regularity, we can show that the nonuniform exponential behavior is ubiquitous, contrarily to what happens with the uniform exponential behavior that although robust is much less common. We also consider the case of infinite-dimensional systems.