Dynamics of difference systems: a mathematical study with applications to neural systems

This paper examines the dynamics of a three-dimensional system of difference equations through mathematical transformations and computational analysis. By transforming the original system into a bilinear form, we were able to simplify its structure and gain deeper insights into its behavior. This transformation also allowed us to study an equivalent two-dimensional system. The analysis revealed that the system possesses closed-form solutions under specific conditions, particularly when examining the discriminant of the quadratic polynomial associated with the system. We examined both cases of repeated and distinct characteristic roots, uncovering varying dynamical behaviors such as oscillations, stability, and growth, depending on the parameters involved in the analyzed examples. The model demonstrated its ability to capture various behaviors through extensive simulations, suggesting its potential applicability in real-world systems, including neural networks and other complex dynamic interactions. The findings highlight the model's robustness in various scenarios, making it a valuable tool for further theoretical and practical applications.