Spiral dynamics under feedback derived from a square-shaped domain in the complex Ginzburg–Landau equation

The complex Ginzburg–Landau equation (CGLE) is used to investigate the effect of a nonlocal feedback control on spiral dynamics in oscillatory media, where the feedback is derived from a square-shaped domain, and three kinds of controlled tip trajectories can be found. The tip trajectory usually tends to a square limit cycle concentric with the measuring domain when the initial tip location is near the periphery of the domain. If the spiral tip is initially located in the measuring domain, it will finally be fixed at a point in the domain. For some larger delay time, the spiral tip may have a motion pathway of small limit cycles. For the case of a square-shaped pathway, the modulus function of the feedback signal has a maximum value when the spiral tip approaches the midpoint of each side, and a slower change when the tip is close to each turning point. As the feedback gain is increased, the spiral tip moves along the same pathway with a greater speed, and multiple tips can be generated when the gain is very large. The final trajectory is also affected by the distance between the measuring center and the initial spiral tip, the size of the measuring domain, the delay time, the choice of positive and negative gains and the scaling parameter.