An Improved Numerical Method for Three-dimensional Hyperbolic Lagrangian Coherent Structures using Differential Algebra
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In dynamical systems, it is advantageous to be able to identify separate regions of qualitatively different flow. Lagrangian Coherent Structures have been introduced to obtain separatrices between regions of qualitatively different flow in three-dimensional dynamical systems with arbitrary time-dependence. However, the numerical method to compute them requires obtaining derivatives associated with the system, often performed through the approximation of divided differences, which leads to significant numerical error and numerical noise. In this paper, we introduce a novel method for the numerical calculation of hyperbolic Lagrangian Coherent Structures using Differential Algebra called DA-LCS. As a form of automatic forward differentiation, it allows direct computation of the Taylor expansion of the flow, its derivatives, and its eigenvectors, with all derivatives obtained algebraically and to machine precision. It does so without a priori information about the system, such as variational equations or explicit derivatives. We demonstrate that this provides significant improvements in the accuracy of the Lagrangian Coherent Structures identified compared to finite-differencing methods in a series of test cases drawn from the literature. We also show how DA-LCS uncovers additional dynamical behaviour in a real-world example drawn from astrodynamics.
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