# Phase dynamics of nearly stationary patterns in activator-inhibitor systems

## Abstract

The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model are studied using a phase dynamics approach. A Cross-Newell phase equation describing slow and weak modulations of periodic stationary solutions is derived. The derivation applies to the bistable, excitable, and Turing unstable regimes. In the bistable case stability thresholds are obtained for the Eckhaus and zigzag instabilities and for the transition to traveling waves. Neutral stability curves demonstrate the destabilization of stationary planar patterns at low wave numbers to zigzag and traveling modes. Numerical solutions of the model system support the theoretical findings. (c) 2000 The American Physical Society.

- Authors:

- Center for Nonlinear Studies and T-7, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
- The Jacob Blaustein Institute for Desert Research and the Physics Department, Ben-Gurion University, Sede Boker Campus 84990, (Israel)
- Observatoire de la Cote d'Azur, Boite Postale 4229, 06304 Nice Cedex 4, (France)
- (United States)

- Publication Date:

- OSTI Identifier:
- 20216772

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

- Additional Journal Information:
- Journal Volume: 61; Journal Issue: 6; Other Information: PBD: Jun 2000; Journal ID: ISSN 1063-651X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; PHASE TRANSFORMATIONS; REACTION KINETICS; INSTABILITY; THEORETICAL DATA

### Citation Formats

```
Hagberg, Aric, Meron, Ehud, Passot, Thierry, and Department of Mathematics, University of Arizona, Tucson, Arizona 85721.
```*Phase dynamics of nearly stationary patterns in activator-inhibitor systems*. United States: N. p., 2000.
Web. doi:10.1103/PhysRevE.61.6471.

```
Hagberg, Aric, Meron, Ehud, Passot, Thierry, & Department of Mathematics, University of Arizona, Tucson, Arizona 85721.
```*Phase dynamics of nearly stationary patterns in activator-inhibitor systems*. United States. doi:10.1103/PhysRevE.61.6471.

```
Hagberg, Aric, Meron, Ehud, Passot, Thierry, and Department of Mathematics, University of Arizona, Tucson, Arizona 85721. Thu .
"Phase dynamics of nearly stationary patterns in activator-inhibitor systems". United States. doi:10.1103/PhysRevE.61.6471.
```

```
@article{osti_20216772,
```

title = {Phase dynamics of nearly stationary patterns in activator-inhibitor systems},

author = {Hagberg, Aric and Meron, Ehud and Passot, Thierry and Department of Mathematics, University of Arizona, Tucson, Arizona 85721},

abstractNote = {The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model are studied using a phase dynamics approach. A Cross-Newell phase equation describing slow and weak modulations of periodic stationary solutions is derived. The derivation applies to the bistable, excitable, and Turing unstable regimes. In the bistable case stability thresholds are obtained for the Eckhaus and zigzag instabilities and for the transition to traveling waves. Neutral stability curves demonstrate the destabilization of stationary planar patterns at low wave numbers to zigzag and traveling modes. Numerical solutions of the model system support the theoretical findings. (c) 2000 The American Physical Society.},

doi = {10.1103/PhysRevE.61.6471},

journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},

issn = {1063-651X},

number = 6,

volume = 61,

place = {United States},

year = {2000},

month = {6}

}

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