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Current Research in Statistics & Mathematics(CRSM)

ISSN: 2994-9459 | DOI: 10.33140/CRSM

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Research Article - (2025) Volume 4, Issue 3

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Stability Analysis of a System of Stochastic Difference Equations with Exponential Nonlinearity

Leonid Shaikhet *
 
Department of Mathematics, Ariel University, Ariel 40700, Israel
 
*Corresponding Author: Leonid Shaikhet, Department of Mathematics, Ariel University, Ariel 40700, Israel

Received Date: Oct 20, 2025 / Accepted Date: Nov 14, 2025 / Published Date: Nov 24, 2025

Copyright: ©©2025 Leonid Shaikhet. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Citation: Shaikhet, L. (2025). Stability Analysis of a System of Stochastic Difference Equations with Exponential Nonlinearity. Curr Res Stat Math 4(3), 01-09.

Abstract

A system of two difference equations with exponential nonlinearity in each equation is studied under stochastic perturbations. Conditions of the stability in probability of a positive equilibrium are studied by virtue of the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The obtained results are illustrated via examples and figures with numerical simulation of the solution of the system of stochastic difference equations. The proposed research method can be applied to nonlinear systems of higher dimension with an order of nonlinearity higher than one, both for stochastic difference equations and for stochastic differential equations with delay in various important applications, for example, in quantum physics, in population models and others.

Keywords

Nonlinear Difference Equations, Positive Equilibrium, Stochastic Perturbations, Asymptotic Mean Square Stability, Stability in Probability, Linear Matrix Inequality (LMI), Numerical Simulations, MATLAB MSC: 39A30; 39A50

Introduction

Systems of both difference and differential equations with different forms of exponential nonlinearities are very popular in research and various applications (see, for instance, [1–17] and references therein), in particular, the model from quantum physics [5], the model of Nicholson’s blowflies [2] or Mosquito population equation [12].

Here, similarly to [14], the stability of the positive equilibrium of a system with exponential nonlinearity is investigated under stochastic perturbations via the general method of Lyapunov functionals construction [16,18–20] and the method of linear matrix inequalities (LMIs) [21–29]. However, unlike, for instance, [7,14], where the exponential nonlinearity in each equation depends on only one variable, here each equation exponentially depends on all variables of the system under consideration. The obtained results are illustrated via examples and figures with the equilibrium and numerical simulation of the solution of the considered system of difference equations. Numerical analysis of the considered LMIs is carried out using MATLAB.

Consider the system of two nonlinear difference equations 

Equilibrium


Figure 1: The Graphs of the Functions x2 = f1 (x1) (green) and x2 = f2 (x1) (red)


Stochastic Perturbations and the System Transformation




Stability

Some Necessary Definitions and Statements

Let ′ be the transposition sign. Put now

Remark 3.1 Note that the system (13) has an order of nonlinearity higher than one. It is known [16] that in this case suficient conditions for asymptotic mean square stability of the zero solution of the linear system (14) are also suficient conditions for stability in probability of the zero solution of the nonlinear system (13).

Stability Conditions




Conclusion

Stability of a system of nonlinear difference equations under stochastic perturbations is investigated. The nonlinearity of exponential form in each equation depends on all variables of the system under consideration. The conditions of stability in probability for positive equilibrium of the considered system, obtained via the general method of Lyapunov functionals construction, are formulated in terms of linear matrix inequalities (LMIs) and are illustrated by numerical examples and figures. The method of stability investigation, used in the paper, can be applied to many other types of nonlinear systems with an order of nonlinearity higher than one for both difference and differential equations in various applications.

Data Availability: No data was used for the research described in the article.

Disclosure Statement: No potential conflict of interest was reported by the author(s).

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