Efficient manifold tracing for planar maps

Invariant manifolds of unstable periodic orbits organize the dynamics of chaotic orbits in phase space. They provide insight into the mechanisms of transport and chaotic advection and have important applications in physical situations involving three-dimensional flows. The numerical determination of invariant manifolds for planar maps is a problem on its own. Efficient and practical techniques to describe these curves must be made available to the scientific community. In this work we introduce an exact calculation method and an approximation technique for tracing the invariant manifolds of unstable periodic orbits of planar maps. The exact method relies in a refinement procedure that prevents redundant calculations occurring in non-refinement approaches and the approximated method is based on an intuitive geometrical decomposition of the manifold in bare and fine details. The resulting approximated manifold is precise when compared to the exact manifold, and its calculation is computationally more efficient, making it ideal for mappings involving intensive calculations like inverse functions or numerical integration of ODEs between crossings in a surface of section.