On the Dynamics of a Nonlinear Multidimensional Difference Equations System with Order k+1

In this study, we investigate the qualitative behavior of a higher-order nonlinear system of difference equations. Such systems naturally arise in a wide range of applications, including population dynamics, epidemiology, economics, and discrete-time control theory, where time delays and nonlinear interactions significantly influence the long-term evolution of the process under consideration. The presence of higher-order delays increases the mathematical complexity of the model and enriches its dynamical structure.

We begin by establishing the existence and uniqueness of equilibrium points and deriving the necessary conditions for their positivity. The local asymptotic stability of these equilibria is then examined through linearization methods and spectral analysis of the associated Jacobian matrix. Explicit stability criteria are obtained in terms of the system parameters. Furthermore, we provide sufficient conditions that guarantee the boundedness and persistence of positive solutions, ensuring that trajectories remain biologically and physically meaningful over time.

Special attention is devoted to the role of the delay order, a positive constant parameter, and a nonlinear exponent governing the interaction terms. Their combined influence on stability, convergence, and qualitative behavior is carefully analyzed. In addition, we investigate the possible oscillatory nature of solutions and determine parameter regions that may generate oscillations or complex dynamics.

The theoretical results are rigorously proved and supported by several numerical simulations. These simulations confirm the analytical findings and demonstrate the robustness, applicability, and effectiveness of the derived qualitative and stability criteria.