Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents
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ReducedFTLE.pdf
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3.06 MB
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Author(s)
Babaee, Hessameddin
Farazmand, Mohammad M
Haller, George
Sapsis, Themistoklis Panagiotis
Date Issued
June 2017
Journal
Chaos: An Interdisciplinary Journal of Nonlinear Science
Publisher
AIP Publishing
Citation
Babaee, Hessam et al. “Reduced-Order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents.” Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 6 (June 2017): 063103. © 2017 Author(s).
Version
Final published version
Abstract
High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy-Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1063/1.4984627
