Rotating spiral waves in [formula omitted] systems on circular domains

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.