Strict stability in terms of two measures for impulsive differential equations with ‘supremum’

Strict stability for a nonlinear system of impulsive differential equations with ‘supremum’ is defined and studied. Razhumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations are the bases of the main proofs. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied. An example illustrating the usefulness of the obtained sufficient conditions is also included.     

Stability of linear impulsive Itô differential equations with bounded delays

We use the method of “model” equations to study the exponential p-stability (2 ≤ p < ∞) of the trivial solution with respect to the initial function for a linear impulsive system of Itô differential equations with bounded delays. The specific form of the equation and the method used permit one to analyze the stability of solutions starting from an arbitrary point of the half-line [0,∞) and obtain constructive sufficient conditions in terms of the parameters of the equations to be studied.     

Integral manifolds for uncertain impulsive differential–difference equations with variable impulsive perturbations

In the present paper sufficient conditions for the existence of integral manifolds of uncertain impulsive differential–difference equations with variable impulsive perturbations are obtained. The investigations are carried out by means the concepts of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequalities and piecewise continuous Lyapunov’s functions.     

A canonical system of two differential equations with periodic coefficients and the Poincaré-Denjoy theory of differential equations on a torus

The passage from Cartesian to polar coordinates in a canonical system with periodic coefficients gives rise to a nonlinear differential equation whose right-hand side is periodic in time and the polar angle and thus this equation can be regarded as a differential equation on a torus. In accord with Poincaré-Denjoy theory, the behavior of a solution to a differential equation on a torus is characterized by the rotation number and some homeomorphic mapping of a circle onto itself. We study connections of strong stability (instability) of a canonical system, including the membership in the nth stability (instability) domain, with the rotation number and fixed points of this mapping.     

Stability of solutions to linear impulsive systems of Itô differential equations with aftereffect with respect to initial data in part of variables

The method of model equations is used to study the moment stability of solutions to linear impulsive systems of Itô differential equations with aftereffect with respect to initial data in part of variables. Sufficient conditions of stability are obtained in terms of parameters of these systems.     

Periodic averaging method for impulsive stochastic differential equations with Lévy noise

The purpose of this paper is to present a periodic averaging method for impulsive stochastic differential equations with Lévy noise under non-Lipschitz condition. It is shown that the solutions of impulsive stochastic differential equations with Lévy noise converge to the solutions of the corresponding averaged stochastic differential equations without impulses     

Stability of non-constant steady-state solutions for bipolar non-isentropic Euler–Maxwell equations with damping terms

In this article, we consider the periodic problem for bipolar non-isentropic Euler–Maxwell equations with damping terms in plasmas. By means of an induction argument on the order of the time-space derivatives of solutions in energy estimates, the global smooth solution with small amplitude was established close to a non-constant steady-state solution with asymptotic stability property. Furthermore, we obtain the global stability of solutions with exponential decay in time near the non-constant steady-states for bipolar non-isentropic Euler–Poisson equations. This phenomenon on the charge transport shows the essential relation and difference between the bipolar non-isentropic and the bipolar isentropic Euler–Maxwell/Poisson equations.     

Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation du/dt=λu(t)+μu(t-τ)$\mathrm{d}u/\mathrm{d}t=\lambda u(t)+\mu u(t-\tau )$, where τ$\tau$ is a constant delay and λ,μ$\lambda ,\mu$ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.     

Stability analysis of exponential Runge–Kutta methods for delay differential equations

A sufficient condition of stability of exponential Runge–Kutta methods for delay differential equations is obtained. Furthermore, a relationship between P-stability and GP-stability is established. It is proved that the numerical methods can preserve the analytical stability for a class of test problems.     

Stability of one-leg θ-methods for nonlinear neutral differential equations with proportional delay

The stability properties of one-leg θ-methods for nonlinear neutral differential equations with proportional delay is investigated. In recent years, the stability of one-leg θ-methods for this class of equations on a quasi-geometric mesh is investigated. Instead, in the present paper, the focus is on stability of one-leg θ-methods for the neutral differential equations with constant delay obtained by applying the approach of transformation to the proportional delay equations. Some sufficient conditions for global stability and asymptotic stability are established. Two numerical examples are also included.     

Properties of solutions of two second-order differential equations with the Painlevé property

Theoretical and Mathematical Physics - We consider a Hamiltonian system equivalent to the Painlevé II equation with respect to one component and to the Painlevé XXXIV equation with...     

Bounded solutions of differential equations with “slow” time

The problem of existence of bounded solutions on the whole number line of a nonlinear system of first order with slow time is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1621–1623, November, 1992.     

Hamilton–Jacobi equations in space of measures associated with a system of conservation laws

We introduce a class of action integrals defined over probability measure-valued path space. We show that extremal point of such action exits and satisfies a type of compressible Euler equation in a weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton–Jacobi equations, in the space of probability measures, are well-posed.     

Stability analysis of impulsive Cohen–Grossberg neural networks with distributed delays and reaction–diffusion terms

In this paper, we investigate a class of impulsive Cohen–Grossberg neural networks with distributed delays and reaction–diffusion terms. By establishing an integro-differential inequality with impulsive initial conditions and applying M-matrix theory, we find some sufficient conditions ensuring the existence, uniqueness, global exponential stability and global robust exponential stability of equilibrium point for impulsive Cohen–Grossberg neural networks with distributed delays and reaction–diffusion terms. An example is given to illustrate the results obtained here.     

Comment on “Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients”

The present paper corrects the way of using Jordan canonical forms for studying the symmetry structures of systems of linear second-order ordinary differential equations with constant coefficients applied in [1]. The approach is demonstrated for a system consisting of two equations.     

An analog to Kamenkov’s critical stability case for nonstationary systems of impulsive differential equations

A new approach to the investigation of the stability of nonlinear nonautonomous differential equations with impulse effects in critical cases is proposed. The approach is based on the direct method of Lyapunov with the use of piecewise differentiable functions. The sufficient conditions of the asymptotic stability of the critical position of equilibrium in one case are obtained. The case is analogous to Kamenkov’s critical case.     

A note on a generalization of Diliberto’s Theorem for certain differential equations of higher dimension

In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.This research was supported by Grant No. 201/99/0295 of the Grant Agency of the Czech Republic.This revised version was published online in April 2005 with a corrected missing date string.