Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining the ‘global stability’ of a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline ‘resilience’. Unfortunately, there are many different variants and explanations of resilience, and often, the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalised to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain nonautonomous perturbations to demonstrate how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis.

Let $(A_m)_{m \in {\mathop Z}}$ be a sequence of bounded linear maps acting on an arbitrary Banach space X and admitting an exponential trichotomy and let $f_m:X \to X$ be a Lispchitz map for every $m\in {\mathop Z} $. We prove that whenever the Lipschitz constants of $f_m$, $m \in {\mathop Z} $, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in {\mathop Z} $, has various types of shadowing. Moreover, if X is finite dimensional and each $A_m$ is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results, we study the Hyers–Ulam stability for certain difference equations and we obtain a very general version of the Grobman–Hartman's theorem for nonautonomous dynamics.

Let $\mathbf{S}:={{\{S(t)\}}_{t\ge 0}}$ be a ${{\text{C}}_{0}}$-semigroup of quasinilpotent operators (i.e., $\sigma (S(t))=\{0\}$ for each $t>0$). In dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical finite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a ${{\text{C}}_{0}}$-semigroup is preserved under the perturbations of its infinitesimal generator.

Given a sequence of matrices (Am)m∈ℕ whose Lyapunov exponents are limits, we show that this asymptotic behaviour is reproduced by the sequences xm+1 = Amxm + fm(xm) for any sufficiently small perturbations fm. We also consider the general case of exponential rates ecρm for an arbitrary increasing sequence ρm. Our approach is based on Lyapunov's theory of regularity.

The paper deals with periodic systems of ordinary differential equations (ODEs). A new approach to the investigation of variations of multipliers under perturbations is suggested. It enables us to establish explicit conditions for the stability and instability of perturbed systems.

The following system is considered in this paper: The primary goal is to establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) and a forcing term p(t) for all solutions to converge to the origin (0,0) as . Here, the zero solution of the corresponding homogeneous linear system is assumed to be neither uniformly stable nor uniformly attractive. Sufficient conditions are given for asymptotic stability of the zero solution of the nonlinear perturbed system under the assumption that q(t,0,0)=0.

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

On explicite une classe de champ de vecteurs polynomiaux non analytiquement linéarisables à l'aide de la correction introduite par Écalle-Vallet. Notamment, on étend des résultats de Schuman sur la trivialité des hamiltoniens homogàenes isochrones.

Perturbation analysis estimators for expectations of possibly discontinuous functions of the time-stationary workload were derived in [2]. The expressions obtained may, however, not be valid if the customer-stationary distribution of the workload has atoms (at points other than zero). This was pointed out by Brémaud and Lasgouttes in [1]. In this note we clearly state the additional condition required for the validity of the expressions in [2]. We furthermore show how our approximation scheme can also be used to obtain the correct expressions for the right and left derivatives given in [1].

Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.

We study the asymptotic properties of positive solutions to the semilinear equation — Δu = f(x, u). Existence and asymptotic estimates are obtained for solutions in exterior domains, as well as entire solutions, for n ≧ 2. The study uses integral operator equations in Rn, and convergence theorems for solutions of Poisson's equation in bounded domains. A consequence of the method is that more precise estimates can be obtained for the growth of solutions at infinity, than have been obtained by other methods. As a special case the results are applied to the generalized Emden-Fowler equation — Δu = p(x)uγ, for γ > 0