Geometric Structure of Multiple Time-scale Nonlinear Systems

The geometric structure of nite-dimensional nonlinear systems with dynamics on multiple time-scales is studied by analyzing the time-scale structure present in the associated linear variational system which evolves on the tangent bundle to the state-space. As a rst step, we restrict our attention to a chosen reference orbit of the nonlinear system and analyze the time scales of the linear time-varying (LTV) system that arises from linearization about that reference trajectory.We use Lyapunov exponents and associated characteristic directions to characterize the time-scales in the LTV system and their geometry. This information is used to construct decoupling coordinate transformations. An example elucidates the methodology described.