The aim of this thesis was to develop a pseudo-spectral continuation scheme for periodic orbits of planar Hamiltonian dynamical systems. Traditionally, numerical continuation of Hamiltonian periodic orbits is performed by way of shooting methods. While we build our analysis on the more traditional approach, we remove any integration required by the shooting method in favor of solving the Hamiltonian dynamical system directly at every point along the continuation branch. This thesis considers the Korteweg-de Vries (KdV) equation to build the pseudo-spectral continuation scheme. We first derive the standard KdV equation and build the corresponding Hamiltonian dynamical system. We then consider an unfolding parameter and a phase condition to ensure that we converge to unique solutions. Finally, we show that using the pseudo-spectral continuation method, we are able to solve for solutions along the continuation branch much faster and with a higher level of accuracy than shooting methods.
