Size-Dependent Transition to High-Dimensional Chaotic Dynamics in a Two-Dimensional Excitable Medium

The spatiotemporal dynamics of an excitable medium with multiple spiral defects is shown to vary smoothly with system size from short-lived transients for small systems to extensive chaos for large systems. A comparison of the Lyapunov dimension density with the average spiral defect density suggests an average dimension per spiral defect varying between 3 and 7. We discuss some implications of these results for experimental studies of excitable media. [S0031-9007(98)05536-7] Much research in the nonequilibrium physics of excitable media has been motivated by the observation of dynamical states containing defects, i.e., spiral waves in two space dimensions or spiral filaments in three dimensions [1]. Experimental studies in surface oxidation experiments [2] and in fibrillating hearts [3] suggest that many such defects may coexist in dynamically complex states. Although similar states have been reproduced in computer simulations [4‐6], there has not yet been careful quantitative analysis of whether the long-time dynamics of such media can be chaotic and, if so, how the properties of this chaos may be related to the statistics of defects, to the size of the medium, and to intrinsic medium parameters. Detailed analysis of mathematical models of excitable media may thus provide new insights in how to analyze spatially extended excitable media, possibly including fibrillating cardiac tissue. In this Letter, we study numerically a two-dimensional model of a homogeneous excitable medium with an emphasis on determining when spatiotemporal chaos occurs, and on quantitatively analyzing basic time and length scales of observed chaotic states. We use a model introduced by Bar et al. [4], as a reduced description of carbon monoxide oxidation on a surface [7], because of its numerical simplicity and because of prior work suggesting the existence of chaos [8]. Extending some recent work by other researchers [4,5,9], we find that the dynamics is strongly dependent on the system size L. For small L, all initial conditions studied rapidly decay to an asymptotic constant or periodic state. As the system size increases, however, the fraction of initial conditions leading to sustained, nonperiodic dynamics increases smoothly, and we discuss the transition from periodic to nonperiodic dynamics with increasing system size. The nonperiodic dynamics sustained in sufficiently large systems are statistically stationary, and we compute Lyapunov exponents and dimensions, defect statistics, and two-point correlation lengths to characterize these states. These different statistics are compared, both to test previous conjectures about the relationship of these correlation lengths [10] and to evaluate the complexity of the defects. Our results indicate that a two-dimensional excitable medium of moderate size with few defects on average can already sustain extensive, highdimensional chaotic dynamics, a fact with important implications for control of excitable media by small parameter perturbations [11]. In the following, we explain the model, summarize our calculations, and discuss the implications of our results. The Bar model describes the interaction of an activator field ust, x, yd with an inhibitor field yst, x, yd via the partial differential equations ›u