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 $T^{\ast }$. 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 $\delta (t)$ and parameter regimes. These findings bridge discrete fractional modeling with practical control strategies for systems exhibiting hereditary and anomalous diffusion effects.