| This thesis is in two parts. The first part is an analytical and numerical study of patterns near a codimension two Turing Hopf point of the one dimensional Brusselator model. For the superdiffusive variant, we derive amplitude equations describing slow time evolution of the Turing and Hopf modes. The main qualitative differences from the regular diffusion analog are the presence of a second long spatial scale owing to non-quadratic behavior near the minimum of the Hopf stability curve, and that the evolution of the Hopf mode is governed by an integro-differential equation. In a numerical study farther in the nonlinear regime, we use a modified Fourier spectral method to compute spatiotemporal patterns and compare to those found in the regular diffusion model. In both cases, we find a large number of solutions characterized by the coexistence of stationary stripes and low wavenumber temporally oscillating "cells,'' the shapes of which depend on superdiffusion exponents. For the regular diffusion model, we employ the AUTO package to continue such Turing-Hopf solutions in parameter space. We find that the solutions are organized on snaking branches characterized by a series of saddle-node bifurcations, analogous to those found for stationary pinning solutions. Observations in wavelength variation, location of snaking region, and direction of front depinning, are explained in terms of the amplitude equations. In the second part of this thesis, we study pulse patterns in a singularly perturbed regime of the regular diffusion model with prescribed boundary feed. We find that the boundary feed breaks the symmetric spacing of equilibrium pulse patterns. A differential-algebraic system of equations (DAE) is derived, governing asymptotically slow translations of quasi-equilibrium pulse patterns. Criteria for slow translational instabilities are determined from a stability analysis of the DAE. Fast amplitude instabilities, characterized by pulse collapse events or synchronous and asynchronous oscillations, are studied by analyzing a nonlocal eigenvalue problem. These results are related back to the slow translations, whereby it is found that the latter may dynamically trigger fast instabilities in an initially stable pulse pattern. |
