Feedback-controlled dynamics of spiral waves in the complex Ginzburg–Landau equation

Spiral dynamics in the complex Ginzburg–Landau equation (CGLE) with a feedback control are studied. It is shown that the spiral tip follows a circular pathway centered at the measuring point after some transients. The attractor radius is a piecewise constant function of the initial distance between the spiral tip and the measuring point, and a piecewise linear increasing function of the time delay in the feedback loop. Many bumps or lobes can be formed from the circular attractor if the time delay becomes relatively long, and the tip path becomes complex. With an increase in the positive feedback gain, the attractor is not changed, but the drift velocity of the spiral tip along such an attractor is linearly increased. When the gain is negative, the tip is attracted to the measuring point from its initial position if the initial distance is relatively small. The variation of the scaling parameter μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} in CGLE initiates also transitions between attractors of different size, and the attractor radius is a piecewise function of μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}, where it is inversely proportional to a square root of the parameter on each piece. It is also demonstrated that the tip drift velocity has a power-law relation with μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and the wave speed. Some results can be understood by analyzing the drift direction of the spiral tip in the transient process.