We study numerically the dynamics of a scroll wave in a three-dimensional ͑3D͒ excitable medium in the presence of substantial meandering of the corresponding 2D spiral wave in the Aliev-Panfilov model. We identify three types of dynamics of the scroll wave filament-quasi-2D, periodic, and aperiodic meanderingand we study their dependence on parameter settings and thickness of the medium. DOI: 10.1103/PhysRevE.72.022902 PACS number͑s͒: 87.19.Hh, 87.19.Nn, 87.18.Pj, 82.40.Np Scroll waves are three-dimensional ͑3D͒ vortices which are extensions of the well-known spiral waves that occur in a variety of excitable media. Scroll waves have been observed in the Belousov-Zhabotinski ͑BZ͒ chemical reaction ͓1,2͔, in the slug phase of the life cycle of slime molds ͓3͔ and in the ventricles of the heart during cardiac arrhythmias ͓4,5͔. Numerous modeling studies of scroll waves have been performed using analytical and numerical methods. The importance of scroll waves for our understanding of the behavior of excitable media has been consistently emphasized in the literature ͓6͔.A scroll wave is usually characterized by its filament ͓7͔, which is an extension into three dimensions of the notion of the core of the spiral wave. In general, the dynamics of a filament in 3D is governed by its curvature, twist ͓8,9͔, and by the anisotropy of the excitable medium ͓6,10,11͔. Mathematically, drifts occur as a result of a convective term in the Laplacian in a coordinate system with an axis directed along the filament and can be effectively studied using singular perturbation theory ͓9͔. The curvature-induced drift of the filament changes the filament length. This property of the filament is often regarded as the filament tension ͓12͔, which can be positive or negative ͓8͔. If the tension is negative, the filament increases in length, leading to instability, which can cause the multiplication of scrolls ͓12,13͔. If the tension of the filament is positive, the filament tends to become shorter, which results either in the collapse of scroll waves with a closed ͑circular͒ filament or in the stabilization of a straight filament between two opposite boundaries in a uniform medium ͓14͔. Filament behavior is highly important for the system in which scroll waves occur. For example, it is a widely accepted hypothesis that different orientations of filaments in the heart and different types of their dynamics determine the type of cardiac arrhythmia and its possible deterioration into fibrillation in the ventricles of the heart ͓15,16͔.Until recently, the dynamics of filaments was mainly studied for those parameters of an excitable medium that show stable rotation of a 2D spiral wave, i.e., for so-called circular cores. However, the assumption of a circular core does not hold for several important practical cases: spiral waves in a BZ reaction ͓17͔ and spiral waves in models of cardiac tissue ͓18͔ both show pronounced meandering ͓19͔. In this article, we study filament dynamics in the presence of scroll wave meandering in a model for cardiac...
