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Feedback-mediated control of spiral waves. (English) Zbl 1067.92003
Summary: A theoretical approach is developed which allows one to describe the drift of a spiral wave mediated by a feedback signal taken as an integral of a system variable over a certain domain of an excitable medium. The underlying reaction-diffusion equations are reduced to an autonomous system of ordinary differential equations for the coordinates of the core center of the spiral wave. A new method to determine the drift velocity field of the spiral core is proposed based on a superposition principle. This approach is applied to analyze the drift velocity field for a feedback signal generated from a one-point detector and from integration domains of different size and shape, including global feedback.
It is shown that a variation of the shape of the integration domain can induce bifurcations that change the attractor structure of the drift velocity field. These theoretical predictions are in good quantitative agreement with numerical simulations performed for the Oregonator model and the complex Ginzburg-Landau equation. The results are also confirmed by experiments with the light-sensitive Belousov-Zhabotinsky reaction.

MSC:
92B05 General biology and biomathematics
35Q92 PDEs in connection with biology, chemistry and other natural sciences
37N25 Dynamical systems in biology
35K57 Reaction-diffusion equations
92E20 Classical flows, reactions, etc. in chemistry
92C15 Developmental biology, pattern formation
35Q53 KdV equations (Korteweg-de Vries equations)
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