Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents
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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Jun 01, 2017
- Source ID
- 10.1063/1.4984627
Entities
People
- George Haller
- Hessam Babaee
- Mohamad Farazmand
- Themistoklis Sapsis
Organizations
- Air Force Office of Scientific Research
- Army Research Office
- Defense Advanced Research Projects Agency
- ETH Zurich
- Massachusetts Institute of Technology
- Office of Naval Research
- University of Pittsburgh