Analyzing finite-time convergence for variable-order fractional discrete dynamics in Degn–Harrison reaction–diffusion systems

This study presents a novel investigation into finite-time stability (FTS) and synchronization phenomena within a discrete reaction–diffusion system (RDS) governed by variable-order fractional (VOF) operators, inspired by the Degn–Harrison (D–H) model. By employing Caputo-type VOF differences, we model memory effects and time-varying dynamics typical of complex biological and chemical processes. Theoretical contributions include rigorous Lyapunov function (LF)-based criteria for establishing tempered Mittag-Leffler stability (MLS) and global FTS, as well as explicit expressions for the settling time T^∗. A fractional-order (FO) error system is also analyzed, demonstrating that linear coupling ensures finite-time synchronization under variable-order conditions. Extensive numerical simulations confirm the theoretical predictions across various FO profiles δ(t) and parameter regimes. These findings bridge discrete fractional modeling with practical control strategies for systems exhibiting hereditary and anomalous diffusion effects.