  Finite-time Lyapunov exponents and vectors are used to define and diagnose
boundary-layer type, two-timescale behavior in the tangent linear dynamics and
to determine the associated manifold structure in the flow of a
finite-dimensional nonlinear autonomous dynamical system. Two-timescale
behavior is characterized by a slow-fast splitting of the tangent bundle for a
state space region. The slow-fast splitting is defined using finite-time
Lyapunov exponents and vectors, guided by the asymptotic theory of partially
hyperbolic sets, with important modifications for the finite-time case; for
example, finite-time Lyapunov analysis relies more heavily on the Lyapunov
vectors due to their relatively fast convergence compared to that of the
corresponding exponents. The splitting is used to locate points on normally
hyperbolic center manifolds. Determining manifolds from tangent bundle
structure is more generally applicable than approaches, such as the singular
perturbation method, that require special normal forms or other a priori
knowledge. The use, features, and accuracy of the approach are illustrated via
several detailed examples.
