On stability in finite models

We search for a set-up in which results from the theory of infinite models hold for finite models. As an example we prove results from stability theory. Received: 23 February 1998     

On input-to-state stability for stochastic multi-group models with multi-dispersal

This paper focuses on the input-to-state stability for a general class of stochastic multi-group models with multi-dispersal. By incorporating graph theory with Lyapunov method as well as stochastic analysis techniques, novel sufficient criteria are derived, which are in the form of Lyapunov-type theorem and coefficient-type criterion, respectively. Moreover, to show the applicability of our findings, we employ coefficient-type criterion to analyze the input-to-state stability for stochastic coupled oscillators. Finally, a numerical example and its simulations are offered to demonstrate the validity and feasibility of the theoretic results.     

On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models

We consider the Euler-Voigt equations and the Navier-Stokes-Voigt equations, which are obtained by an inviscid α-regularization from the corresponding equations. The main result we show is the structural stability of the system in terms of the variations of both viscosity and regularization parameters.     

On the stability of periodic trajectories in some gene network models

Under study is the stability of the periodic trajectories of the gene network models with negative feedback. Some sufficient conditions are given for the existence of stable periodic trajectories.     

On the stability and causality of general time-dependent bilinear models

In this Note, sufficient conditions are given for the existence of a causal stable solution for general bilinear time series with time-dependent coefficients. To cite this article: A. Bibi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Structural stability of market models

A market structure in the Arrow-Debreu framework can be represented by a finite number of traded securities. We examine an interesting structure that exists among markets. The partitioning of the set of all markets. M into its rank classes is a stratification of M. Some properties of an equivalence relation among markets are also studied. Some results on the ‘genericity’ of the properties of market structures are obtained.     

On the existence and stability of cycles in five-dimensional models of gene networks

Sufficient conditions for the existence and stability of closed trajectories in five-dimensional nonlinear dynamical systems that model gene networks with negative feedback are obtained.     

Global stability in population models

If g is a continuous map of an interval, the recursion x_{n + 1} = g(x_n), n = 0, 1… converges globally if and only if the equation has only fixed points of g as roots. This leads to simple geometric conditions for global stability in discrete population models. Related conditions involving the Schwarzian derivative and curvature are discussed.     

Orbital stability of spherical galactic models

We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture (Binney, Tremaine in Galactic Dynamics, Princeton University Press, Princeton, 1987) is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov (Sov. Astron. 4:859–867, 1961). In the previous work (Lemou et al. in A new variational approach to the stability of gravitational systems, submitted, 2011), we derived the stability of anisotropic models under spherically symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics literature (Lynden-Bell in Mon. Not. R. Astron. Soc. 144:189–217, 1969; Gardner in Phys. Fluids 6:839–840, 1963; Wiechen et al. in Mon. Not. R. Astron. Soc. 223:623–646, 1988; Aly in Mon. Not. R. Astron. Soc. 241:15, 1989). In this work, we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical models under general perturbations.     

On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate

We study an epidemic model for infections with non permanent acquired immunity (SIRS). The incidence rate is assumed to be a general nonlinear function of the susceptibles and the infectious classes. By using a peculiar Lyapunov function, we obtain necessary and sufficient conditions for the local nonlinear stability of equilibria. Conditions ensuring the global stability are also obtained. Unlike the recent literature on this subject, here no restrictions are required about the monotonicity and concavity of the incidence rate with respect to the infectious class. Among the applications, the noteworthy case of a convex incidence rate is provided.     

Global stability of nonresident computer virus models

In this paper, applying Lyapunov functional techniques to nonresident computer virus models, we establish global dynamics of the model whose threshold parameter is the basic reproduction number R₀ such that the virus‐free equilibrium is globally asymptotically stable when R₀ ≤ 1, and the infected equilibrium is globally asymptotically stable when R₀ > 1 under the same restricted condition on a parameter, which appeared in the literature on delayed susceptible‐infected‐recovered‐susceptible (SIRS) epidemic models. We use new techniques on permanence and global stability of this model for R₀ > 1. Copyright © 2014 John Wiley & Sons, Ltd.     

Monotonicity and stability of periodic polling models

This paper deals with the stability of periodic polling models with a mixture of service policies. Customers arrive according to independent Poisson processes. The service times and the switchover times are independent with general distributions. The necessary and sufficient condition for the stability of such polling systems is established. The proof is based on the stochastic monotonicity of the state process at the polling instants. The stability of only a subset of the queues is also analyzed and, in case of heavy traffic, the order of explosion of the queues is given. The results are valid for a model with set-up times, and also when there is a local priority rule at the queues.This work was supported in part by a Fellowship of the Netherlands Organization for Scientific Research NWO-ECOZOEK.     

Global stability of discrete-time competitive population models

We develop practical tests for the global asymptotic stability of interior fixed points for discrete-time competitive population models. Our method constitutes the extension to maps of the Split Lyapunov method developed for differential equations. We give ecologically-motivated sufficient conditions for global stability of an interior fixed point of a map of Kolmogorov form. We introduce the concept of a principal reproductive mode, which is linked to a normal at the interior fixed point of a hypersurface of vanishing weighted-average growth. Our method is applied to establish new global stability results for 3-species competitive systems of May-Leonard type, where previously only parameter values for local stability was known.     

Some applied problems of stability of stochastic models

We consider several applied problems of probability theory. These problems are related to biophysics, mathematical economics, and automated control theory, and are considered in the framework of the theory of stability of stochastic models. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Global exponential stability of nonresident computer virus models

This paper is concerned with nonresident computer virus models which are defined on the nonnegative real vector space. By using differential inequality technique, we employ a novel argument to show that the virus-free equilibrium is globally exponentially stable, and the exponential convergent rate can be unveiled. Moreover, a numerical simulation is given to demonstrate our theoretical results.     

On Rayleigh-Taylor stability

We study the nonlinear stability of the rest state for the Rayleigh-Taylor problem related to a plane layer of viscous, heavy fluid under periodicity assumptions in the horizotal directions, with nonzero surface tension. Such a problem is studied for both incompressible and isothermal fluids where, in the class of regular flows, it is proved the exponential decay to the rest state for perturbations, in absence of hypotheses on the characteristic coefficients of the flow, and on the initial conditions as well.

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Sunto Si studia la stabilità nonlineare della quiete per il problema di Rayleigh-Taylor relativo ad uno strato piano di fluido viscoso pesante nelle ipotesi di periodicità nelle direzioni orizzontali, ed in presenza di tensione superficiale. Tale problema è studiato sia per fluidi incompressibili che isotermi. In entrambi i casi, nella classe di moti regolari si prova il decadimento di tipo esponenziale allo stato di quiete per le perturbazioni, senza aggiungere alcuna ipotesi sui coefficienti caratteristici del moto o sulle condizioni iniziali.

     

Measure-transmission metric and stability of structured population models

In Gwiazda, et al. (2012) a framework for studying cell differentiation processes based on measure-valued solutions of transport equations was introduced. Under application of the so-called measure-transmission conditions it enabled to describe processes involving both discrete and continuous transitions. This framework, however, admits solutions which lack continuity with respect to initial data. In this paper, we modify the framework from Gwiazda, et al. (2012) by replacing the flat metric, known also as bounded Lipschitz distance, by a new Wasserstein-type metric. We prove, that the new metric provides stability of solutions with respect to perturbations of initial data while preserving their continuity in time. The stability result is important for numerical applications.