Effect of Inhomogeneity on Spiral Wave Dynamics

Abstract

The effect of weak inhomogeneity on spiral wave dynamics is studied within the framework of the two-dimensional complex Ginzburg-Landau equation description. The inhomogeneity gives spatial dependence to the frequency of spiral waves. This provides a mechanism for the formation of a dominant spiral domain that suppresses other spiral domains. The spiral vortices also slowly drift in the inhomogeneity, and results for the velocity are given. [S0031-9007(98)08283-0] PACS numbers: 82.40.Ck, 47.32.Cc, 47.54. + r Spiral waves occur in such diverse situations as cardiac arrythmias [1], reaction-diffusion systems (such as that describing the Belousov-Zhabotinsky reaction) [2,3], and slime mold colonies [4,5]. In this paper we consider the effect of an inhomogeneity of the supporting medium on spiral wave dynamics. For example, in the case of arrythmias, cardiac tissue is inherently inhomogeneous. For slime mold, an excitation inhomogeneity forms due to the sorting of the prestalk and prespore cells and the inhomogeneity results in spiral vortex motion [5]. As a potential example involving chemical reaction-diffusion systems, in Ref.[3] the chemical reaction rate was varied using its sensitivity to red laser light intensity. This could conveniently provide the means to create an inhomogeneity for a test of our theory by having the intensity vary over the entire system (other sources of reaction rate inhomogeneity are temperature inhomogeneity and inhomogeneity of the gel or porous glass medium in which chemical reaction-diffusion experiments are often done, etc.).We find that, for a weak inhomogeneity, the evolution proceeds on distinct time scales. In particular, we find that, after spirals first form, the inhomogeneity causes certain spirals that are favorably located in the inhomogeneous medium to widen their domains, crowding out and sweeping away less favorably located spirals. In addition, the spiral vortices slowly drift with a velocity linearly related to appropriate gradients of the background medium properties.Our studies of these effects are based on the twodimensional complex Ginzburg-Landau equation,where A͑x, y, t͒ is complex. This equation provides a universal description for extended media in which the homogeneous state is oscillatory and near a Hopf bifurcation [6]. The equation is obtained as a balance between weak growth ͓Re͑m͔͒, weak nonlinearity (jAj 2 A term), and weak spatial coupling (= 2 A term). In the case of a homogeneous medium a and b are real constants, and m can be set to unity by an appropriate rescaling of (1). A spiral wave solution to Eq. (1) then has the general form [7] A͑r, u, t͒ F͑r͒ exp͕i͓su 2 v o t 1 c͑r͔͖͒ . (2) The topological charge s 61 results in a 2ps phase increment of A for a counterclockwise circuit around the vortex center (r 0) where A 0. For large r, the solution (2) is locally a plane wave of wave number k o . Using the rescaling of m to unity, the frequency v o satisfies the dispersion relation v o a 1 ͑b 2 a͒k 2 o . The real functions F͑r͒ ϵ jAj and c͑r͒ have the f...

Experimental, numerical, and mechanistic analysis of the nonmonotonic relationship between oscillatory frequency and photointensity for the photosensitive Belousov–Zhabotinsky oscillator

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