Friday 24 Jun 2016: Dynamics seminar: Spiral wave chaos: tiling, local symmetries, and asymptotic freedom
Christopher Marcotte - Georgia Institute of Technology
Understanding the nature of spatiotemporal chaos in excitable systems is not only of fundamental interest, but also of imminent practical importance. This talk will give an overview of recent progress in understanding the dynamical mechanisms that initiate and sustain spiral wave chaos featuring multiple interacting spiral waves that repeatedly break up and merge. Periodic orbit theory aims to describe chaotic dynamics using the properties of unstable periodic solutions embedded in the chaotic attractor and produced a lot of insight into the dynamics of low-dimensional systems, starting with the work of Poincare on celestial mechanics. In excitable systems, however, it fails rather spectacularly due to the extremely short spatial correlations of spiral waves. The relevant length scale is defined by the width of the adjoint eigenfunctions associated with the dominant modes of the linearization. For typical models of excitable dynamics these eigenfunctions are exponentially localized around the spiral core, with the width much smaller than the wavelength. Hence, interaction between two spiral waves falls off exponentially fast, and the dynamics of individual spirals become effectively independent once the separation between the spiral cores exceeds this length scale (spiral waves become asymptotically free). As a result, typical multi-spiral states break the global Euclidean symmetry of the problem, but respect local symmetries (translations and rotations in 2D). Local symmetries imply that time-periodic solutions are extremely rare due to the slow relative drift in the position and orientation of individual spirals. This drift can be understood by partitioning the domain into tiles, each of which supports a single spiral wave. The dynamics of each spiral can then be understood based on the shape of the corresponding tile and the position of the spiral core. This formalism produces a number of specific predictions that are fully supported by numerical simulations and offers a novel way to understand and describe spiral wave chaos.