Regulation of spatiotemporal patterns in a discrete Lengyel–Epstein system using PD control

This paper investigates the dynamical behavior of a discretized Lengyel–Epstein system using the coupled map lattices framework. By fixing the time step, we investigate the onset of the Neimark–Sacker bifurcation and Turing instability using a physical parameter as the critical bifurcation variable. The analysis employs center manifold theory and normal form methods. Furthermore, a Proportional–Derivative control is introduced to study its influence on the system’s dynamics. Numerical simulations demonstrate the control’s effectiveness in regulating and stabilizing complex spatiotemporal patterns. This work provides theoretical insights into the control of pattern formation in discrete systems, offering a foundation for engineering and biological applications.