Characterizing Two-timescale Nonlinear Dynamics Using Finite-time Lyapunov Exponents and Vectors∗

Finite-time Lyapunov exponents and vectors are used to define and diagnose boundarylayer type, two-timescale behavior and to construct a method for determining an associated slow manifold when one exists. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for the state-space. The slow-fast splitting defined by finite-time Lyapunov exponents and vectors is interpreted in relation to the asymptotic theory of partially hyperbolic sets. The method of determining a slow manifold developed in this paper is potentially more accurate than an existing approach that is based on local eigenvalues and eigenvectors, at the expense of more computation, and is more generally applicable than approaches, such as the singular perturbation method, that require a special coordinate representation. The approach is illustrated via several