Global Manifolds of Saddle Periodic Orbits Parametrised by Isochrons

Global stable and unstable manifolds of a saddle periodic orbit of a vector field carry phase information of how trajectories on the manifold approach the periodic orbit in forward or backward time. This information is encoded on the respective manifold by its foliation by isochrons, which are submanifolds of codimension one comprising all points that are in asymptotic synchrony with a point of a given phase on the periodic orbit. We present a numerical method that finds a two-dimensional stable or unstable manifold of a saddle periodic orbit by computing a representative number of one-dimensional isochrons as arclength-parametrised curves on the manifold. As is demonstrated with examples of both orientable and nonorientable manifolds, this computational approach allows us to determine and visualise the interplay between their topological, geometric as well as synchronisation properties.