Stability and Oscillatory Dynamics of a Delayed Three‐Dimensional Discrete Dynamical System

ABSTRACT This paper investigates the qualitative dynamics of a delayed three‐dimensional discrete system. The analysis focuses on key properties such as local asymptotic stability, boundedness, persistence, and oscillatory behavior of positive solutions. By incorporating a delay parameter and a nonlinear exponent, the proposed model extends previous multidimensional formulations and highlights the combined influence of delay and nonlinearity on system dynamics. Analytical results establish precise conditions determining stability and instability regions, as well as criteria for the emergence of oscillatory and persistent regimes. Numerical simulations are performed to validate the theoretical findings and to illustrate the variety of dynamic behaviors exhibited by the system under different parameter choices. The results provide deeper insights into the complexity of delayed nonlinear discrete systems and contribute to the ongoing development of multidimensional difference models in applied mathematics.