ELSEVIER Physica D 105 (1997) 207-225 Spirals in excitable Media. II: Meandering transition in the diffusive free-boundary limit David A. Kessler a, Raz Kupferman b,, a Minerva Center andDepartment of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel b Lawrence Berkeley National Laboratory, 50A-2152, ! Cyclotron Road, Berkeley, CA 94720, USA Received 1 July 1996; revised 16 October 1996; accepted 17 October 1996 Communicated by L. Kramer Abstract In this paper we present a numerical stability calculation for steadily rotating spirals in an excitable medium. While experiments, as well as numerical simulations of two-field reaction~liffusion models have shown the existence of a Hopf bifurcation from steady rotations to a meandering state, all the analytical approaches so far have failed to predict this transition. This mismatch between analysis and simulations raises the question whether meandering critically depends on the finite diffusivity of the interface separating between the excited and the refractory phases. Our calculations show that this is not the case. The meandering transition takes place even in the limit of an infinitely sharp interface. The boundaries of the meandering transition as function of the model parameters are traced. We discuss possible explanations for the failure of previous analytical approaches. I. Introduction The study of spiral patterns in excitable media has been the focus of considerable attention in recent years (for a review, see [1]). Much progress has been achieved in this area through the use of simulations [2-4], the analysis of various limiting cases [5-8], and most recently, the numerical solution of the steady state problem [9-11 ]. Spirals are relevant not only in the context of chemical reactions, such as the famed Belousov-Zhabotinskii reaction [ 12,13], but also in various biological systems, such as electrical conduction in heart tissue [ 14], and aggregation of the slime mold [15]. Spirals exhibit a wide range of interesting dynamics. In particular, simulations reveal a Hopf bifurcation from steady-state rotation to meandering as some of the control parameters are varied. In the steady state, the spiral tip rotates at constant frequency about some fixed point. In contrast, the meandering state is a compound rotation where the distance of the tip to the center of rotation oscillates periodically. By simulating the model near the transition point, Barkley [2] and Karma [3] were able to demonstrate the Hopf nature of the transition and calculate * Corresponding author. E-mail: raz@mekong.lbl.gov. 0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(96)00298-9