Stability and finite-time synchronization of discrete FitzHugh–Nagumo systems using Lyapunov theory

This study explores finite-time synchronization (FTS) in a discrete FitzHugh–Nagumo (FHN) reaction–diffusion system. Employing Lyapunov-based techniques and numerical simulations, we establish theoretical criteria to achieve synchronization within a finite duration. The proposed methodology involves discretization of the continuous FHN model using finite difference schemes to reformulate it into a computationally feasible framework. A tailored control strategy is introduced, ensuring rapid convergence to synchronization. Numerical results validate the theoretical framework, highlighting the critical roles of diffusion coefficients, system parameters, and control gains in shaping the spatiotemporal dynamics. The findings underscore the effectiveness of the proposed approach in applications such as neuronal network synchronization, chemical kinetics, and biological pattern formation. This study provides a robust theoretical and computational foundation for advancing FTS in reaction–diffusion systems, with practical implications across diverse scientific domains.