Stability of a three‐station fluid network

This paper studies the stability of a three‐station fluid network. We show that, unlike the two‐station networks in Dai and Vande Vate [18], the global stability region of our three‐station network is not the intersection of its stability regions under static buffer priority disciplines. Thus, the “worst” or extremal disciplines are not static buffer priority disciplines. We also prove that the global stability region of our three‐station network is not monotone in the service times and so, we may move a service time vector out of the global stability region by reducing the service time for a class. We introduce the monotone global stability region and show that a linear program (LP) related to a piecewise linear Lyapunov function characterizes this largest monotone subset of the global stability region for our three‐station network. We also show that the LP proposed by Bertsimas et al. [1] does not characterize either the global stability region or even the monotone global stability region of our three‐station network. Further, we demonstrate that the LP related to the linear Lyapunov function proposed by Chen and Zhang [11] does not characterize the stability region of our three‐station network under a static buffer priority discipline. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonlinear stability analysis of a regular vortex pentagon outside a circle

A nonlinear stability analysis of the stationary rotation of a system of five identical point vortices lying uniformly on a circle of radius R ₀ outside a circular domain of radius R is performed. The problem is reduced to the problem of stability of an equilibrium position of a Hamiltonian system with a cyclic variable. The stability of stationary motion is interpreted as Routh stability. Conditions for stability, formal stability and instability are obtained depending on the values of the parameter q = R ²/R ₀ ² .     

On partial stability of a dynamic system that is integrally continuous with respect to a small parameter

For a dynamic system described by Carathéodory ordinary differential equations with a small parameter we introduce the definitions of a type of partial stability, attraction, and asymptotic stability. We state theorems giving sufficient conditions for stability in the new definitions. In particular, in terms of perturbed Lyapunov functions we obtain conditions for partial asymptotic stability that generalize known results. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 29–36.     

Stability of some versions of the Prékopa–Leindler inequality

Two consequences of the stability version of the one dimensional Prékopa–Leindler inequality are presented. One is the stability version of the Blaschke–Santaló inequality, and the other is a stability version of the Prékopa– Leindler inequality for even functions in higher dimensions, where a recent stability version of the Brunn–Minkowski inequality is also used in an essential way.     

Some results related to Hurwitz stability of combinatorial polynomials

Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to observe some results and applications for the Hurwitz stability of polynomials in combinatorics and study other related problems.We first present a criterion for the Hurwitz stability of the Turán expressions of recursive polynomials. In particular, it implies the q-log-convexity or q-log-concavity of the original polynomials. We also give a criterion for the Hurwitz stability of recursive polynomials and prove that the Hurwitz stability of any palindromic polynomial implies its semi-γ-positivity, which illustrates that the original polynomial with odd degree is unimodal. In particular, we get that the semi-γ-positivity of polynomials implies their parity-unimodality and the Hurwitz stability of polynomials implies their parity-log-concavity. Those results generalize the connections between real-rootedness, γ-positivity, log-concavity and unimodality to Hurwitz stability, semi-γ-positivity, parity-log-concavity and parity-unimodality (unimodality). As applications of these criteria, we derive some Hurwitz stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating run polynomials of types A and B and the Hurwitz stability for alternating run polynomials defined on a dual set of Stirling permutations.Finally, we study a class of recursive palindromic polynomials and derive many nice properties including Hurwitz stability, semi-γ-positivity, non-γ-positivity, unimodality, strong q-log-convexity, the Jacobi continued fraction expansion and the relation with derivative polynomials. In particular, these properties of the alternating descents polynomials of types A and B can be implied in a unified approach.     

Stability results involving time-averaged Lyapunov function derivatives

The stability results which comprise the Direct Method of Lyapunov involve the existence of auxiliary functions (Lyapunov functions) endowed with certain definiteness properties. Although the Direct Method is very general and powerful, it has some limitations: there are dynamical systems with known stability properties for which there do not exist Lyapunov functions which satisfy the hypotheses of a Lyapunov stability theorem.In the present paper we identify a scalar switched dynamical system whose equilibrium (at the origin) has known stability properties (e.g., uniform asymptotic stability) and we prove that there does not exist a Lyapunov function which satisfies any one of the Lyapunov stability theorems (e.g., the Lyapunov theorem for uniform asymptotic stability). Using this example as motivation, we establish stability results which eliminated some of the limitations of the Direct Method alluded to. These results involve time-averaged Lyapunov function derivatives (TALFD’s). We show that these results are amenable to the analysis of the same dynamical systems for which the Direct Method fails. Furthermore, and more importantly, we prove that the stability results involving TALFD’s are less conservative than the results which comprise the Direct Method (which henceforth, we refer to as the classical Lyapunov stability results).While we confine our presentation to continuous finite-dimensional dynamical systems, the results presented herein can readily be extended to arbitrary continuous dynamical systems defined on metric spaces. Furthermore, with appropriate modifications, stability results involving TALFD’s can be generalized to discontinuous dynamical systems (DDS).     

On the stability of discrete tripole,quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid

A two-layer quasigeostrophic model is considered in the f-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case when the structure consists of a central vortex of arbitrary intensity Γ and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters (R, Γ, α), where α is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered.A limiting case of a homogeneous fluid is also considered.The theory of stability of steady-state motions of dynamic systems with a continuous symmetry group G is applied. The two definitions of stability used in the study are Routh stability and G-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the G-stability is the stability of a three-parameter invariant set O _G , formed by the orbits of a continuous family of steady-state rotations of a multipole. The problem of Routh stability is reduced to the problem of stability of a family of equilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically.The cases of zero total intensity of a tripole and a quadrupole are studied separately. Also, the Routh stability of a Thomson vortex triangle and square was proved at all possible values of problem parameters. The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.     

Backward differentiation formulas with extended regions of absolute stability

This paper presents a class of (p + 2)-step backward differentiation formulas of orderp. The two extra degrees of freedom obtained by limiting the order of a (p + 2)-step formula top are used to extend the region of absolute stability. A new formula of orderp has a region of absolute stability very similar to that of a classical backward differentiation formula of orderp - 1 forp being in the range 4–6. The backward differentiation formulas with extended regions of absolute stability are constructed by appending two exponential-trigonometric terms to the polynomial basis of the classical formulas. Besides the absolute stability, the paper discusses relative stability and contractivity. The principles of an experimental implementation of the new formulas are outlined, and a linear problem integrated with this computer program indicates that the extended regions of absolute stability can actually be exploited in practice.     

Dynamics of a delayed predator–prey model with predator migration

Based on the availability of prey and a simple predator–prey model, we propose a delayed predator–prey model with predator migration to describe biological control. We first study the existence and stability of equilibria. It turns out that backward bifurcation occurs with the migration rate as bifurcation parameter. The stability of the trivial equilibrium and the boundary equilibrium is delay-independent. However, the stability of the positive equilibrium may be delay-dependent. Moreover, delay can switch the stability of the positive equilibrium. When the positive equilibrium loses stability, Hopf bifurcation can occur. The direction and stability of Hopf bifurcation is derived by applying the center manifold method and the normal form theory. The main theoretical results are illustrated with numerical simulations.     

Stability of stationary solutions of a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u) on a compact interval of spatial variable x with Dirichlet boundary conditions. The stability of stationary solutions of this system is studied by the use of Liapunov's second method. We obtain necessary and sufficient conditions for the stability, asymptotic stability, neutral stability, instability, and conditional stability. These conditions are closely connected with the conditions for the existence of the stationary solutions.     

Stability of hybrid stochastic differential systems with switching and time delay

This article introduces a hybrid stochastic differential system with impulsive, switching and time-delay. Some stability criteria of p-moment global asymptotical stability, p-moment global exponential stability and mean square stability of this system are derived by using switching Lyapunov function approach, Itô formula, impulsive differential inequality method, and linear matrix equality techniques. Three examples are presented to demonstrate the efficiency of the obtained results.     

Novel stability criteria for uncertain delayed Cohen–Grossberg neural networks using discretized Lyapunov functional

This paper deals with the stability analysis of delayed uncertain Cohen–Grossberg neural networks (CGNN). The proposed methodology consists in obtaining new robust stability criteria formulated as linear matrix inequalities (LMIs) via the Lyapunov–Krasovskii theory. Particularly one stability criterion is derived from the selection of a parameter-dependent Lyapunov–Krasovskii functional, which allied with the Gu’s discretization technique and a simple strategy that decouples the system matrices from the functional matrices, assures a less conservative stability condition. Two computer simulations are presented to support the improved theoretical results.     

On the stability of some second order numerical methods for weak approximation of Itô SDEs

In this paper, we first investigate the stability of two weak second order methods introduced by Debrabant and Rößler (Appl Numer Math 59:582–594, 2009) and Platen (Math Comput Simulation 38:69–76, 1995). We then propose a new weak second order predictor-corrector method, with an improved stability properties, based on the Rößler’s method as the predictor and the implicit method of Platen as the corrector. The stability functions of these methods, applied to a scalar linear test equation with multiplicative noise, are determined and their regions of stability are then compared with the corresponding stability regions of the test equation. Furthermore, we also investigate mean square stability (MS-stability) of these methods applied to a linear Itô 2-dimensional stochastic differential test equation. Numerical examples will be presented to support the theoretical results.     

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.     

Stability in the almost everywhere sense: A linear transfer operator approach

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.     

Global stability analysis for stochastic coupled reaction–diffusion systems on networks

Coupled systems on networks (CSNs) can be used to model many real systems, such as food webs, ecosystems, metabolic pathways, the Internet, World Wide Web, social networks, and global economic markets. This paper is devoted to investigation of the stability problem for some stochastic coupled reaction–diffusion systems on networks (SCRDSNs). A systematic method for constructing global Lyapunov function for these SCRDSNs is provided by using graph theory. The stochastic stability, asymptotically stochastic stability and globally asymptotically stochastic stability of the systems are investigated. The derived results are less conservative than the results recently presented in Luo and Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493]. In fact, the system discussed in Q. Luo and Y. Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493] is a special case of ours. Moreover, our novel stability principles have a close relation to the topological property of the networks. Our new method which constructs a relation between the stability criteria of a CSN and some topology property of the network, can help analyzing the stability of the complex networks by using the Lyapunov functional method.     

Structure of stability sets and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: I

We consider families of linear differential systems continuously depending on a real parameter. The stability (respectively, asymptotic stability) set of such a family is defined as the set of all values of the parameter for which the corresponding systems in the family are stable (respectively, asymptotically stable). We show that a set on the real axis is the stability (respectively, asymptotic stability) set of some family of this kind if and only if it is an F _σ -set (respectively, an F _σδ -set). For families in which the parameter occurs only as a factor multiplying the matrix of the system, their stability sets are exactly F _σ -sets containing zero on the real line. The asymptotic stability sets of such families will be described in the second part of the present paper.     

Vector-stability of refinable vectors

The stability is an expected property for refinable functions, which is widely considered in the study of refinement equations. Instead of studying the stability of entries of refinable vectors, we study the stability of refinable vectors themselves where they are considered as elements of super Hilbert spaces. We call this kind of stability the vector-stability. We give a necessary and sufficient condition for refinable vectors to be vector-stable. We also give an example to illustrate the difference between two types of stability.     

The Stability of Self-Organized Rule-Following Work Teams

Self-organized rule-following systems are increasingly relevant objects of study in organization theory due to such systems&2018; capacity to maintain control while enabling decentralization of authority. This paper proposes a network model for such systems and examines the stability of the networks&2018; repetitive behavior. The networks examined are Ashby nets, a fundamental class of binary systems: connected aggregates of nodes that individually compute an interaction rule, a binary function of their three inputs. The nodes, which we interpret as workers in a work team, have two network inputs and one self-input. All workers in a given team follow the same interaction rule.We operationalize the notion of stability of the team&2018;s work routine and determine stability under small perturbations for all possible rules these teams can follow. To study the organizational concomitants of stability, we characterize the rules by their memory, fluency, homogeneity, and autonomy. We relate these measures to work routine stability, and find that stability in ten member teams is enhanced by rules that have low memory, high homogeneity, and low autonomy.     

Stability of Functional Equations in Several Variables

We prove a generalization of Hyers＇ theorem on the stability of approximately additive mapping and a generalization of Badora＇s theorem on an approximate ring homomorphism. We also obtain a more general stability theorem, which gives the stability theorems on Jordan and Lie homomorphisms. The proofs of the theorems given in this paper follow essentially the D. H. Hyers-Th. M. Rassias approach to the stability of functional equations connected with S. M. Ulam＇s problem.