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Planar Radial Spots in a Three-Component FitzHugh–Nagumo System

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Abstract

Localized planar patterns arise in many reaction-diffusion models. Most of the paradigm equations that have been studied so far are two-component models. While stationary localized structures are often found to be stable in such systems, travelling patterns either do not exist or are found to be unstable. In contrast, numerical simulations indicate that localized travelling structures can be stable in three-component systems. As a first step towards explaining this phenomenon, a planar singularly perturbed three-component reaction-diffusion system that arises in the context of gas-discharge systems is analysed in this paper. Using geometric singular perturbation theory, the existence and stability regions of radially symmetric stationary spot solutions are delineated and, in particular, stable spots are shown to exist in appropriate parameter regimes. This result opens up the possibility of identifying and analysing drift and Hopf bifurcations, and their criticality, from the stationary spots described here.

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  1. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA

    Peter van Heijster & Björn Sandstede

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  1. Peter van Heijster
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  2. Björn Sandstede
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Correspondence to Peter van Heijster.

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Communicated by P. Newton.

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van Heijster, P., Sandstede, B. Planar Radial Spots in a Three-Component FitzHugh–Nagumo System. J Nonlinear Sci 21, 705–745 (2011). https://doi.org/10.1007/s00332-011-9098-x

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