Title:

Spatiotemporal chaos analysed through unstable periodic states

Multifractal properties of a chaotic attractor are usefully quantified by its spectrum of singularities, f(α). Here, α is the pointwise dimension of the natural measure at a point on the attractor, and f(α) is the Hausdorff dimension of all points with pointwise dimension α. Within a more general thermodynamic formalism, the singularity spectrum is one of several ways in which the properties of an attractor can be quantified. The technique used to realize the singularity spectrum is orbit theory. This theory tells one how to take properties of finite time solutions and combine them to approximate the infinite time behaviour, thereby allowing qualitative and quantitative predictions to be made. These techniques are first applied to the Lorenz system, where it is also shown that the variation in the pointwise dimension on a surface of section has selfsimilar structure. The general idea of studying the properties of a nonlinear system through the periodic orbits it supports has, to date, been primarily applied to lowdimensional dynamical systems. In the thesis we develop the technique so that it can be applied to the infinitedimensional KuramotoSivashinsky equation. The continuation and bifurcation package Auto is used to investigate stability and bifurcation properties of different types of special solutions to the KuramotoSivashinsky equation, following an expansion in Fourier modes. One such class of solutions is defined by the Michelson equation, to which a very detailed numerical bifurcation analysis is given. Orbit theory is applied to regimes of an asymmetric KuramotoSivashinsky equation where complicated behaviour is observed in a manner similar to that used in lowdimensional systems. Each periodic orbit can be considered as a spatiotemporal pattern, in which both qualitative (the structure and bifurcations of) and quantitative (the dimension and spectrum of Lyapunov exponents) aspects are discussed.
