### Abstract

λ - ω systems are reaction-diffusion systems whose reaction kinetics admit a stable limit cycle. It is known that λ - ω systems can possess various types of solutions. Among them, spiral waves are the most fascinating pattern. However, the effects of the diffusivity, the sizes of the domains, and the reaction kinetics on spiral waves are largely unknown. In this paper, we investigate how these quantities affect the properties of m-armed spiral waves in a generalized class of λ - ω system on a circular disk with no-flux boundary condition. First we derive a criterion for the existence of m-armed spiral waves. Specifically, we show that m-armed spiral waves do not exist for d ≥ λ_{0} R^{2} / j_{m}^{2}, while for d ∈ (0, λ_{0} R^{2} / j_{m}^{2}), there exists an m-armed spiral wave if the twist parameter q is small. Here d is the diffusivity for the λ - ω system, R is the radius of the circular disk, λ_{0} is the value of the function λ (A) at A = 0, and j_{m} is the first positive zero of the first derivative of the Bessel function of the first kind of order m. We also show that the critical diffusivity d = λ_{0} R^{2} / j_{m}^{2} is a bifurcation point. Next we use the numerical simulation to show that, for small twist parameter, the rotational frequency increases with increasing domain size, while for large twist parameter, the dependence of the rotational frequency on the domain size is not monotonic. Moreover, small circular domains may change the properties of spiral waves drastically. These numerical results are in contrast to those in excitable media. Finally, the stability of spiral waves is investigated numerically.

Original language | English |
---|---|

Pages (from-to) | 1007-1025 |

Number of pages | 19 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 239 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2010 Jun 15 |

### Fingerprint

### Keywords

- Reaction-diffusion systems
- m-armed spiral waves
- λ - ω systems

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

### Cite this

**Rotating spiral waves in λ - ω systems on circular domains.** / Tsai, Je-Chiang.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 239, no. 12, pp. 1007-1025. https://doi.org/10.1016/j.physd.2010.02.009

}

TY - JOUR

T1 - Rotating spiral waves in λ - ω systems on circular domains

AU - Tsai, Je-Chiang

PY - 2010/6/15

Y1 - 2010/6/15

N2 - λ - ω systems are reaction-diffusion systems whose reaction kinetics admit a stable limit cycle. It is known that λ - ω systems can possess various types of solutions. Among them, spiral waves are the most fascinating pattern. However, the effects of the diffusivity, the sizes of the domains, and the reaction kinetics on spiral waves are largely unknown. In this paper, we investigate how these quantities affect the properties of m-armed spiral waves in a generalized class of λ - ω system on a circular disk with no-flux boundary condition. First we derive a criterion for the existence of m-armed spiral waves. Specifically, we show that m-armed spiral waves do not exist for d ≥ λ0 R2 / jm2, while for d ∈ (0, λ0 R2 / jm2), there exists an m-armed spiral wave if the twist parameter q is small. Here d is the diffusivity for the λ - ω system, R is the radius of the circular disk, λ0 is the value of the function λ (A) at A = 0, and jm is the first positive zero of the first derivative of the Bessel function of the first kind of order m. We also show that the critical diffusivity d = λ0 R2 / jm2 is a bifurcation point. Next we use the numerical simulation to show that, for small twist parameter, the rotational frequency increases with increasing domain size, while for large twist parameter, the dependence of the rotational frequency on the domain size is not monotonic. Moreover, small circular domains may change the properties of spiral waves drastically. These numerical results are in contrast to those in excitable media. Finally, the stability of spiral waves is investigated numerically.

AB - λ - ω systems are reaction-diffusion systems whose reaction kinetics admit a stable limit cycle. It is known that λ - ω systems can possess various types of solutions. Among them, spiral waves are the most fascinating pattern. However, the effects of the diffusivity, the sizes of the domains, and the reaction kinetics on spiral waves are largely unknown. In this paper, we investigate how these quantities affect the properties of m-armed spiral waves in a generalized class of λ - ω system on a circular disk with no-flux boundary condition. First we derive a criterion for the existence of m-armed spiral waves. Specifically, we show that m-armed spiral waves do not exist for d ≥ λ0 R2 / jm2, while for d ∈ (0, λ0 R2 / jm2), there exists an m-armed spiral wave if the twist parameter q is small. Here d is the diffusivity for the λ - ω system, R is the radius of the circular disk, λ0 is the value of the function λ (A) at A = 0, and jm is the first positive zero of the first derivative of the Bessel function of the first kind of order m. We also show that the critical diffusivity d = λ0 R2 / jm2 is a bifurcation point. Next we use the numerical simulation to show that, for small twist parameter, the rotational frequency increases with increasing domain size, while for large twist parameter, the dependence of the rotational frequency on the domain size is not monotonic. Moreover, small circular domains may change the properties of spiral waves drastically. These numerical results are in contrast to those in excitable media. Finally, the stability of spiral waves is investigated numerically.

KW - Reaction-diffusion systems

KW - m-armed spiral waves

KW - λ - ω systems

UR - http://www.scopus.com/inward/record.url?scp=77950949246&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950949246&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2010.02.009

DO - 10.1016/j.physd.2010.02.009

M3 - Article

AN - SCOPUS:77950949246

VL - 239

SP - 1007

EP - 1025

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 12

ER -