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Advances and Applications of Finite-Time Lyapunov Analysis

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Many physical systems can be modeled through nonlinear time-invariant differential equations. When the dynamics of such systems are hyper-sensitive to the initial conditions, such equations are often challenging to solve. However, if the flow of such dynamical systems is also characterized by multiple timescales (e.g., fast-slow behavior), there may be a manifold structure associated with it. Focusing on this manifold structure, that is, adopting a geometric perspective, holds potential for a simpler solution process and a better understanding of the system behavior. We adopt finite-time Lyapunov analysis (FTLA) as the methodology to diagnose the timescale

behavior and to characterize the manifold structure. FTLA is based on finite-time Lyapunov exponents (FTLEs) and vectors (FTLVs) and its main advantage is that it is more widely applicable with respect to other methodologies. In fact, for nonlinear dynamical systems, FTLA is not restricted to equilibria nor periodic solutions, nor does it require require the equations to be in a special form. We will show that the accuracy of FTLA depends solely on how fast certain linear subspaces computed using FTLVs converge to their invariant counterparts. This work is dedicated to continuing the advancement of the methodology and to its applications to two-timescale partially hyperbolic dynamical systems. In the first part of this dissertation, we provide a detailed presentation of the methodology, with particular attention to the diagnosis of a uniform hyperbolic finite-time two-timescale set and to the computation of points on finite-time approximations of the invariant manifolds.

The second and third parts of the dissertation focus entirely on the application of FTLA to two major problems. Firstly, we use FTLA as a means of computing the approximation of the local stable, center and unstable directions at points on orbits around libration points in the circular restricted three-body problem (CR3BP). After assessing the accuracy of the FTLA results for points on periodic orbits, where Floquet theory gives exact results, we move to aperiodic orbits, demonstrating that FTLA results maintain a degree of accuracy that is otherwise lost when applying Floquet theory in an approximate manner. Secondly, we propose a new stationkeeping strategy for spacecraft orbiting about libration points in the Earth-Moon CR3BP. The strategy is based on performing instantaneous velocity corrections that place the spacecraft on the stable manifold of some orbit in the neighborhood of a prescribed reference orbit. We demonstrate via several test cases that the proposed strategy is reliable and therefore can be considered as a good alternative or addition to existent stationkeeping strategies.

In the last part, we focus our attention on developing a solution strategy for partially hyper-sensitive optimal control problems. The strategy exploits the existence of the invariant center-stable and center-unstable manifolds to construct an approximation to the optimal solution by matching trajectories on such invariant manifolds. The approach is specifically applied to the optimal control of a nonlinear spring-mass-damper system. The approximate solution is shown to be accurate by comparison with a solution obtained by a collocation method.

In conclusion, we have contributed in advancing finite-time Lyapunov analysis and applied it to models of physical problems demonstrating its effectiveness.

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