  This paper introduces the trajectory divergence rate, a scalar field which
locally gives the instantaneous attraction or repulsion of adjacent
trajectories. This scalar field may be used to find highly attracting or
repelling invariant manifolds, such as slow manifolds, to rapidly approximating
hyperbolic Lagrangian coherent structures, or to provide the local stability of
invariant manifolds. This work presents the derivation of the trajectory
divergence rate and the related trajectory divergence ratio for 2-dimensional
systems, investigates their properties, shows their application to several
example systems, and presents their extension to higher dimensions.
