Guaranteed finite-time convergence in discrete generalized Gray–Scott reaction–diffusion system

This paper investigates the finite-time stability (FTS) characteristics of a discrete generalized Gray–Scott reaction–diffusion system (GS-RDs). In contrast to most existing studies, which establish only asymptotic or exponential stability, we derive verifiable conditions that guarantee the global stabilization of the system’s unique equilibrium point (EP) within a finite number of discrete time steps. By constructing suitable Lyapunov functions (LFs) and combining summation-by-parts formulas with eigenvalue estimates, an explicit closed-form upper bound on the settling time (ST) is obtained. The sharpness of this analytical bound is validated via numerical simulations on a master–slave configuration, for which the system converges precisely within 29 steps, demonstrating exact agreement with the theoretical prediction. The novelty of this work resides in providing a rigorous framework for FTS in discrete RDs, thereby bridging the gap between continuous-time theory and discrete models. Moreover, the proposed methodology extends naturally to synchronization problems. These contributions advance the state of the art in stability analysis of RDs and broaden the applicability of FTS techniques to chemical, biological, and complex networked systems.