# Exact coherent structures and chaotic dynamics in a model of cardiac tissue

## Abstract

Unstable nonchaotic solutions embedded in the chaotic attractor can provide significant new insight into chaotic dynamics of both low- and high-dimensional systems. In particular, in turbulent fluid flows, such unstable solutions are referred to as exact coherent structures (ECS) and play an important role in both initiating and sustaining turbulence. The nature of ECS and their role in organizing spatiotemporally chaotic dynamics, however, is reasonably well understood only for systems on relatively small spatial domains lacking continuous Euclidean symmetries. Construction of ECS on large domains and in the presence of continuous translational and/or rotational symmetries remains a challenge. This is especially true for models of excitable media which display spiral turbulence and for which the standard approach to computing ECS completely breaks down. This paper uses the Karma model of cardiac tissue to illustrate a potential approach that could allow computing a new class of ECS on large domains of arbitrary shape by decomposing them into a patchwork of solutions on smaller domains, or tiles, which retain Euclidean symmetries locally.

- Authors:

- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 (United States)

- Publication Date:

- OSTI Identifier:
- 22402537

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 25; Journal Issue: 3; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANIMAL TISSUES; ATTRACTORS; CHAOS THEORY; DYNAMICS; EUCLIDEAN SPACE; FLUID FLOW; MATHEMATICAL SOLUTIONS; ORGANIZING; SYMMETRY; TURBULENCE

### Citation Formats

```
Byrne, Greg, Marcotte, Christopher D., and Grigoriev, Roman O., E-mail: roman.grigoriev@physics.gatech.edu.
```*Exact coherent structures and chaotic dynamics in a model of cardiac tissue*. United States: N. p., 2015.
Web. doi:10.1063/1.4915143.

```
Byrne, Greg, Marcotte, Christopher D., & Grigoriev, Roman O., E-mail: roman.grigoriev@physics.gatech.edu.
```*Exact coherent structures and chaotic dynamics in a model of cardiac tissue*. United States. doi:10.1063/1.4915143.

```
Byrne, Greg, Marcotte, Christopher D., and Grigoriev, Roman O., E-mail: roman.grigoriev@physics.gatech.edu. Sun .
"Exact coherent structures and chaotic dynamics in a model of cardiac tissue". United States.
doi:10.1063/1.4915143.
```

```
@article{osti_22402537,
```

title = {Exact coherent structures and chaotic dynamics in a model of cardiac tissue},

author = {Byrne, Greg and Marcotte, Christopher D. and Grigoriev, Roman O., E-mail: roman.grigoriev@physics.gatech.edu},

abstractNote = {Unstable nonchaotic solutions embedded in the chaotic attractor can provide significant new insight into chaotic dynamics of both low- and high-dimensional systems. In particular, in turbulent fluid flows, such unstable solutions are referred to as exact coherent structures (ECS) and play an important role in both initiating and sustaining turbulence. The nature of ECS and their role in organizing spatiotemporally chaotic dynamics, however, is reasonably well understood only for systems on relatively small spatial domains lacking continuous Euclidean symmetries. Construction of ECS on large domains and in the presence of continuous translational and/or rotational symmetries remains a challenge. This is especially true for models of excitable media which display spiral turbulence and for which the standard approach to computing ECS completely breaks down. This paper uses the Karma model of cardiac tissue to illustrate a potential approach that could allow computing a new class of ECS on large domains of arbitrary shape by decomposing them into a patchwork of solutions on smaller domains, or tiles, which retain Euclidean symmetries locally.},

doi = {10.1063/1.4915143},

journal = {Chaos (Woodbury, N. Y.)},

number = 3,

volume = 25,

place = {United States},

year = {Sun Mar 15 00:00:00 EDT 2015},

month = {Sun Mar 15 00:00:00 EDT 2015}

}