The finite-time averaged Lyapunov exponent and its Eulerian algorithms

In this paper, we introduce the finite-time averaged Lyapunov exponent (FTALE) to characterize chaotic mixing in dynamical systems from a fresh perspective. Different from the traditional finite-time Lyapunov exponent (FTLE), which adopts a single-perturbation-evaluation paradigm and only measures the amplification of an initial disturbance at the final instant, FTALE employs an infinite-perturbation-evaluation protocol. It continuously tracks the average sensitivity of particle trajectories to impulsive perturbations introduced at any instant throughout the entire time window, thereby capturing the full sequence of instabilities experienced along the way. To make the framework practical, we develop two efficient Eulerian algorithms tailored for distinct scenarios and provide rigorous complexity and error estimates. Numerical experiments demonstrate that FTALE, together with its Eulerian algorithms, unveils the intricate chaotic skeleton of the underlying flow robustly and efficiently.