Parameterization for Center Manifolds: Existence, Rigorous Bounds, and Applications

In Chapters 1 and 2 we provide the theoretical underpinning of the parameterization method for center manifolds. In Chapter 1 -- which is published in Journal of Differential Equations -- we show how the classical parameterization method can be generalized to center manifolds associated with fixed points in discrete dynamical systems. We extend this new method in Chapter 2 to center manifolds for fixed points of ODEs. Besides showing that the parameterization method can be generalized, we also provide a way to obtain approximations of the center manifolds and explicit error bounds on those approximations. Furthermore, we simultaneously obtain similar bounds for the conjugate dynamics on the center manifold. In particular, we obtain qualitative information about the dynamical behaviour on the center manifold In the remainder of the thesis we use the theory from the first two chapters to compute orbits in two applications. In Chapter 3 -- which is published in Celestial Mechanics and Dynamical Astronomy -- we apply the method to numerically find or rule out homoclinic orbits at bifurcations in the Circular Restricted Four Body Problem. Finally, in Chapter 4, we use the parameterization method to prove the existence of spiral waves in the complex Ginzburg-Landau equation. In particular, in the last chapter we rigorously compute an approximation of the center manifold of the spiral wave equation, obtain an explicit error bound on this approximation, and find (part of) the basin of attraction on the center manifold.