# Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents

## TL;DR

This paper introduces a reduced-order method using OTD modes to efficiently detect and analyze finite-time instabilities in high-dimensional chaotic systems, significantly lowering computational costs.

## Contribution

It develops a new reduced-order approach leveraging OTD modes for fast convergence to instability directions and efficient FTLE computation.

## Key findings

- OTD modes converge exponentially to eigendirections of the Cauchy-Green tensor.
- The reduced-order method drastically reduces computational costs.
- Validated on two numerical examples.

## Abstract

High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have 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.