Renormalization group methods for singularly perturbed systems, normal forms and stability of traveling waves in a reaction-diffusion-mechanics system

This thesis concerns two problems arising in the study of singularly perturbed dynamical systems. First, the Renormalization Group (RG) method of Chen, Goldenfeld and Oono is studied. The RG method is a general approach to deriving reduced or amplitude equations that govern the long term dynamics of the system. The method has been shown to be an effective approach for a number of classical problems from perturbation theory that generally require a host of disparate methods to solve. We examine the mathematical basis of this method and determine that the crucial step is a near-identity change of co-ordinates equivalent to those performed in classical normal form theory. In fact, we show that the RG method is equivalent to normal form theory for two large classes of perturbed differential equations. In addition, we study two examples that require logarithmic gauge functions for a proper asymptotic approximation. These gauge functions are shown to arise naturally in the solution of the reduced amplitude equation given by the RG method.;In the second part of this thesis, we study the existence and stability of traveling wave solutions in a mechano-reaction-diffusion model. The particular model we study is one of cardiac tissue where the electrical activity of the tissue is bi-directionally coupled to the mechanical deformation of the underlying medium. We use geometric singular perturbation techniques to show that the model has a traveling pulse solution. The analysis is complicated by the passage of this solution near a non-hyperbolic fold point of a slow manifold. We also establish the spectral stability of this pulse solution. For the stability problem, we construct an analytic Evans Function and show that the only zero of this function is a regular zero at the origin. This is consistent with the picture in the absence of coupling to the mechanical deformation so the inclusion of this coupling does not affect the existence and stability properties of traveling pulses considered on the real line.