Qualitative behavior of higher-order rational difference systems: positivity, asymptotic behavior, and periodicity

This paper gives explicit positive solutions for several higher-order and higher-dimensional systems of rational difference equations (RDEs). We study how solutions behave over time for a wide range of initial data in $(0,\infty )$ , and we prove when solutions stay positive and bounded. We also give clear conditions under which solutions approach fixed points or settle into periodic cycles. To support the theory, we include numerical examples that show how the system order and the initial values shape the long-term dynamics. RDEs form one of the most important classes of nonlinear discrete systems because they naturally appear in population dynamics, epidemiology, economic growth models, and control processes. They are also known to generate rich dynamical behavior, including oscillations and chaos, which makes their qualitative study both challenging and useful. The novelty of our work is that we move beyond the mostly studied low-order cases and provide explicit closed-form solutions and stability results for higher-order, multi-dimensional systems, where such results are rare. The significance is that these formulas make analysis and simulation straightforward, and they offer practical tools for models in areas such as mathematical biology, economics, and control. In addition, our results highlight how higher-order interactions can create new stability patterns and periodic behaviors that do not appear in simpler systems. These findings not only strengthen the theoretical framework of nonlinear discrete-time systems but also extend the range of applications where rational difference equations can be effectively used in practice.