Finding periodic orbits of higher-dimensional flows by including tangential components of trajectory motion

Methods to search for periodic orbits are usually implemented with the Newton-Raphson type algorithms that extract the orbits as fixed points. When used to find periodic orbits in flows, however, many such approaches have focused on using mappings defined on the Poincaré surfaces of section, neglecting components perpendicular to the surface of section. We propose a Newton-Raphson based method for Hamiltonian flows that incorporates these perpendicular components by using the full monodromy matrix. We investigated and found that inclusion of these components is crucial to yield an efficient process for converging upon periodic orbits in high dimensional flows. Numerical examples with as many as nine degrees of freedom are provided to demonstrate the effectiveness of our method.