Controlling Spiral Wave Solutions in the Barkley System Using a Proportional Feedback Control

An important goal in cardiology and other fields is to identify and control dynamic spiral wave patterns in reaction–diffusion partial differential equations. This research focuses on the Barkley model. The spiral wave motion is controlled and suppressed within the Euclidean group rather than through Euclidean symmetry by applying a controller equation. The eigenfunctions associated with the left eigenspace of the adjoint linear equation can be used to characterize the drift or movement of the spiral wave tip trajectory when the system is perturbed. These eigenfunctions provide details regarding how the spiral wave reacts to disruptions. Perturbations to the Barkley system are examined by applying control functions and calculating the principle eigenvalue numerically. The left eigenfunctions of the Barkley equation are determined by solving the left problem associated with the $2\mathrm{D}$ Barkley equation and a $1\mathrm{D}$ dynamical controller. In addition, the control function can be used to suppress the periodic and meandering regimes of the system. In this work, the focus is on the periodic regime.