Multiple time-scales in nonlinear flight mechanics: diagnosis and modeling

There are often disparate time-scales in the dynamics of flight, creating the potential for reduced-order modeling to simplify simulation, analysis and design. There have been notable successes in developing reduced-order models; however, in the case of nonlinear dynamics, which one must typically deal with in guidance problems, there has not been a systematic, reliable means of diagnosing disparate time-scales and developing reduced-order models. Focusing on two time-scale behavior in nonlinear dynamical systems, we recall Fenichel's characterization of the geometric structure of the flow and his theorem establishing the existence and properties of coordinates adapted to this structure. Adapted coordinates are difficult to construct directly, without an appropriate singularly perturbed model of the dynamics. We discuss the use of Lyapunov exponents and vectors to diagnose two time-scale behavior and to determine the corresponding tangent space structure for the linearized dynamics. The structure of the linearized flow can then be translated into the manifold structure of the nonlinear flow. We briefly mention the use of Lyapunov vectors to locate a slow manifold and contrast this approach with two existing approaches. The minimum time to climb problem provides an example of two time-scale behavior and motivates the discussion.