Geometric structure of multiple time-scale nonlinear dynamical systems

A new methodology to analyze time-scale structure of smooth finite-dimensional nonlinear dynamical systems is developed. This approach does not assume apriori knowledge of slow and fast variables for special coordinates that simplify the form of the nonlinear dynamics. Conventional approaches to analyze time-scale structure of nonlinear dynamics such as singular perturbation theory proceed from such specialized apriori knowledge which is often not obtainable. Our approach proceeds from spectral analysis of the linear variational dynamics associated with the nonlinear system. The variational dynamics govern the flow on the tangent bundle to the state-space. We decompose the tangent space at each point into spectral subspaces which separate tangent vectors that evolve at different spectral rates. The existence of such measures of spectral rates and corresponding subspaces is established by Sacker and Sell. We have developed a scheme to computationally determine these spectral measures using finite-time Lyapunov exponents and associated direction fields. In the asymptotic limit, the infinite-time Lyapunov direction fields are shown to satisfy useful invariance properties. As a consequence they are shown to uniquely define an invariant spectral filtration, i.e., a collection of nested distributions which are invariant under the nonlinear flow. Using these results, we establish the consistency of these spectral measures with well known results in special cases such as linear time-invariant systems and periodic linear time-varying systems. Differential equations that govern the propagation of Lyapunov directions along orbits of the nonlinear flow are derived using the invariance properties. Methods to apply these spectral analysis tools to construct coordinate transformations that decompose the variational flow are developed. When the Frobenius theorem is applicable, we also show methods to construct a nonlinear transformation of coordinates from the Lyapunov direction fields to decompose the nonlinear dynamics into slow and fast subsystems. In fact, this procedure can be used to transform the two time-scale nonlinear dynamics into a singularly perturbed standard form. Application of these methods for reducing the order of nonlinear dynamics, locating the slow manifold in the state-space and solving boundary value problems arising from hypersensitive optimal control problems is discussed. Several simple examples are used to demonstrate the methods and elucidate the main concepts.