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1

Asymptotic stability of solutions for a diffusive epidemic model

Bouaziz Khelifa, Douaifia Redouane, Abdelmalek Salem

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

553--573

EN

The aim of this paper is to study the existence and the asymptotic stability of solutions for an epidemiologically emerging reaction-diffusion model. We show that the model has two types of equilibrium points to resolve the proposed system for a fairly broad class of nonlinearity that describes the transmission of an infectious disease between individuals. The model is analyzed by using the basic reproductive number R0 . Finally, we present the numerical examples simulations that clarifies and confirms the results of the study throughout the paper.

2

On a system of rational difference equation

Din Q.

Demonstratio Mathematica

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2014

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Vol. 47, nr 2

324--335

EN

In this paper, we study local asymptotic stability, global character and periodic nature of solutions of the system of rational difference equations given by (…) , where the parameters (…) , and with initial conditions (…). Some numerical examples are given to illustrate our results.

3

Coupled analytical and numerical approach to uncovering new regulatory mechanisms of intracellular processes

Śmieja J.

International Journal of Applied Mathematics and Computer Science

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2010

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Vol. 20, no 4

781-788

EN

The paper deals with the analysis of signaling pathways aimed at uncovering new regulatory processes regulating cell responses. First, general issues of comparing simulation and experimental data are discussed, and various aspects of data normalization are covered. Then, a model of a particular signaling pathway, induced by Interferon-\beta, is briefly introduced. It serves as an example illustrating how mathematical modeling can be used for inferring the structure of a regulatory system governing the dynamics of intracellular processes. In this pathway, experimental results suggest that a hitherto unknown process is responsible for a decrease in the levels of one of the important molecules used in the pathway. Then, equilibrium points of the model are analyzed, allowing the rejection of all but one explanation of the phenomena observed experimentally. Numerical simulations confirm that the model can mimic the dynamics of the processes in the pathway under consideration. Finally, some remarks about the applicability of the method based on an analysis of equilibrium points are made.

4

Stability of positive linear discrete-time systems

James G., Rumchev V.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2005

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Vol. 53, nr 1

1-8

EN

The main focus of the paper is on the asymptotic behaviour of linear discrete-time positive systems. Emphasis is on highlighting the relationship between asymptotic stability and the structure of the system, and to expose the relationship between null-controllability and asymptotic stability. Results are presented for both time-invariant and time-variant systems.

5

A new approach to stability analysis of nonlinear control systems

Švarc I.

Elektronika : konstrukcje, technologie, zastosowania

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2004

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Vol. 45, nr 8-9

18-21

EN

Three methods for stability analysis of nonlinear control systems are introduced in this contribution: method of linearization, Lyapunov direct method and Popov criterion. Since stability analysis of nonlinear control systems is difficult task in engineering practice, these methods are made easier and tabulated. Method of linearization: The table includes the nonlinear equations and their linear approximation. Lyapunov direct method: The table contains Lyapunov functions for usually used equations second order. Popov criterion: The table will allow us to directly determine the stability of the nonlinear circuit with the transfer function G(s) and the nonlinearity that satisfies the slope k.

PL

W artykule przedstawiono trzy metody analizy stabilności nieliniowych systemów sterowania: metodę linearyzacji, bezpośrednią metodę Ljapunowa oraz kryterium Popowa. Ponieważ w praktyce inżynierskiej analiza stabilności nieliniowych systemów sterowania jest trudnym zadaniem, wspomniane metody zostały uproszczone i stablicowane. W przypadku metody linearyzacji tabela zawiera równania nieliniowe oraz ich liniowe aproksymacje. W przypadku bezpośredniej metody Ljapunowa tabela zawiera funkcje Ljapunowa dla wykorzystywanych zazwyczaj równań drugiego rzędu. W przypadku kryterium Popowa tabela pozwala na bezpośrednie wyznaczenie stabilności układu nieliniowego o transmitancji G(s) oraz nieliniowości, która cechuje się nachyleniem k.