Rotating spirals in oscillatory media with nonlocal interactions and their normal form

Gabriela Jaramillo

doi:10.3934/dcdss.2022085

Article Contents

2022, Volume 15, Issue 9: 2513-2551. Doi: 10.3934/dcdss.2022085

This issue Previous Article Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability Next Article Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems

Rotating spirals in oscillatory media with nonlocal interactions and their normal form

Gabriela Jaramillo^,

Department of Mathematics, University of Houston, 3551 Cullen Blvd., Houston, TX 77204-3008, USA

^* Corresponding author: Gabriela Jaramillo
^* Corresponding author: Gabriela Jaramillo

Received: March 2021

Revised: January 2022

Early access: April 6, 2022

Published online: April 6, 2022

This work is supported by NSF grant DMS-1911742.

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Abstract

Biological and physical systems that can be classified as oscillatory media give rise to interesting phenomena like target patterns and spiral waves. The existence of these structures has been proven in the case of systems with local diffusive interactions. In this paper the more general case of oscillatory media with nonlocal coupling is considered. We model these systems using evolution equations where the nonlocal interactions are expressed via a diffusive convolution kernel, and prove the existence of rotating wave solutions for these systems. Since the nonlocal nature of the equations precludes the use of standard techniques from spatial dynamics, the method we use relies instead on a combination of a multiple-scales analysis and a construction similar to Lyapunov-Schmidt. This approach then allows us to derive a normal form, or reduced equation, that captures the leading order behavior of these solutions.

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Mathematics Subject Classification: Primary: 45K05, 45G15, 46N20; Secondary: 35Q56, 35Q92.

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Figure 1. Spirals obtained by numerical simulations of a FitzHugh-Nagumo system (see Section 6) with $\delta = 0.2, \beta = 1, \tau = 0.1$, and $K \ast u = \frac{\sigma}{D} [ - u + K_0(|x|/ \sqrt{D}) \ast u]$. Here $K_0$ is the Modified Bessel function of the second kind. In figures a) and b) the choice of $\sigma = 5 $ and $D = 0.5$, results in a spiral pattern. In contrast, in figures c) and d) the choice of $\sigma = 5$ and $D = 1$ results in a spiral chimera pattern (incoherent core). The figures on the right zoom in into the core of the spirals appearing on the left

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