Local orthogonal rectification (LOR) provides a natural and useful geometric frame for analyzing dynamics relative to manifolds embedded in flows. LOR can be applied to any embedded base manifold in a system of ODEs of arbitrary dimension to establish a corresponding system of LOR equations for analyzing dynamics within the LOR frame. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. Additionally, we illustrate the utility of LOR by showing a wide range of application domains.
In the plane, we use the LOR approach to derive a novel definition for rivers, long-recognized but poorly understood trajectories that locally attract other orbits yet need not be related to invariant manifolds or other familiar phase space structures, and to identify rivers within several example systems.
In higher dimensions, we apply LOR to identify periodic orbits and study the transient dynamics nearby. In the LOR method, %in Rn for any n,
the standard approach of finding periodic orbits by solving for fixed points of a Poincar\'{e} return map is replaced by the solution of a boundary value problem with fixed endpoints, and the computation provides information about the stability of the identified orbit. We detail the general method and derive theory to show that once a periodic orbit has been identified with LOR, the LOR coordinate system allows us to characterize the stability of the periodic orbit, to continue the orbit with respect to system parameters, to identify invariant manifolds attendant to the periodic orbit, and to compute the asymptotic phase associated with points in a neighborhood of the periodic orbit in a novel way.
Finally, we generalize the definition of rivers beyond planar systems, and demonstrate a fundamental connection between canard solutions in two-timescale systems and generalized rivers. We will again use a blow-up transformation on the LOR equations, which provides a useful decomposition for studying trajectories' behavior relative to the embedded base curve
