Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

• Main Authors: Babaee, Hessameddin (Author), Farazmand, Mohammad M (Author), Haller, George (Author), Sapsis, Themistoklis Panagiotis (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Mechanical Engineering (Contributor)
• Format: Article
• Language: English
• Published: AIP Publishing, 2021-03-04T16:27:31Z.
• Subjects: Article
• Online Access: Get fulltext

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.
ARO (Grant 66710-EG-YIP)
AFOSR (Grant FA9550-16-1-0231)
ONR (Grant N00014-15-1-2381)
DARPA (Grant HR0011-14-1-0060)