Using Homogeneous Linear Difference Equations as a Mathematical Tool for Building and Analyzing Discrete Dynamical Systems

Abstract

In light of the rapid developments in applied and mathematical sciences, mathematical modeling has become an essential tool for understanding and analyzing complex systems that evolve in the fields of environment, economics, or computer science. Among the important mathematical methods in this context, linear homogeneous difference equations stand out for their pivotal role in describing and analyzing discrete dynamical systems that evolve in fixed time steps. These equations provide accurate models that bridge the gap between mathematical theory and practical applications. This paper aims to highlight the role of linear homogeneous difference equations as a mathematical tool for constructing and analyzing discrete dynamical systems. It seeks to answer the following question: To what extent can linear homogeneous difference equations be employed to build accurate mathematical models that describe the behavior of discrete dynamical systems? The study begins by defining the difference equation and its associated operators, such as the difference operator, the shift operator, and the identity operator. It then presents the classification of difference equations and explores methods for solving linear homogeneous difference equations. The paper also focuses on the mathematical modeling of discrete dynamical systems, adopting an analytical modeling approach. This approach involves converting a system into a mathematical model using difference equations and then analyzing it to study its properties and behavior over time. To demonstrate this, the analytical modeling method was applied to a discrete dynamical system representing population growth. The study identified the variables and parameters, constructed the appropriate model, solved the resulting difference equation, and graphically represented the results to examine the effect of the growth parameter. The paper concludes that the linear difference equation is a powerful and effective tool for constructing mathematical models that describe the dynamic behavior of discrete systems, opening broad prospects for its application in various scientific fields.