Dynamics of the Gierer–Meinhardt reaction–diffusion system: Insights into finite-time stability and control strategies

Abstract

This study delves into the dynamics of the Gierer–Meinhardt (GM) reaction–diffusion (RD) system, focusing on finite-time stability (FTS) and synchronization (FTSYN) within integer-order spatiotemporal partial differential frameworks. Unlike traditional studies emphasizing asymptotic stability, this work presents novel insights into achieving synchronization and equilibrium within a predefined finite time. By leveraging Lyapunov-based methodologies, sufficient conditions are derived to guarantee FTS and transient behavior control. A synchronization scheme for master–slave systems is developed, ensuring coherent dynamics within a finite time frame. Numerical simulations, employing the finite difference method, validate the theoretical findings and demonstrate the practical applications of the proposed strategies in systems requiring stringent temporal constraints. The results highlight significant advancements in RD modeling, with implications for biological pattern formation, robotics, and distributed networks, paving the way for innovative control solutions in complex dynamical systems.