The backward phase flow method for the Eulerian finite time Lyapunov exponent computations

Abstract

We propose a simple Eulerian approach to compute the moderate to long time flow map for approximating the Lyapunov exponent of a (periodic or aperiodic) dynamical system. The idea is to generalize a recently proposed backward phase flow method which is specially designed for long time level set propagation. Unlike the original phase flow method or the backward phase flow method, which is applicable only to autonomous systems, the current approach can also be applied to any time-dependent (periodic or aperiodic) flow. We will discuss the stability of the proposed method. Numerical examples will be given to demonstrate the effectiveness of the algorithm.

Eulerian Methods for Visualizing Continuous Dynamical Systems using Lyapunov Exponents

“…Based on the first algorithm, we are now able to determine the forward flow map on the fly so that the PDE is solved forward in time. Unlike the original Eulerian algorithm as developed in [23,24], there is no need to store the whole velocity field at all time steps at the beginning of the computations. This makes the forward flow map computations practical.…”

“…One disadvantage about the previous Eulerian approach for forward flow map computation as discussed in Section 2.2 and [23,24,38] is that the level set equation…”

“…We will first improve the numerical algorithm for forward flow map construction. The papers [23,24] have proposed to construct the forward flow map defined on a fixed Cartesian mesh by solving the level set equations or the Liouville equations in the backward direction. In particular, to compute the forward flow map from the initial time t = 0 to the final time t = T , one needs to solve the Liouville equations backward in time from t = T to t = 0 with the initial condition given at t = T .…”

“…We first introduce the definition of various closely related Lyapunov exponents including the finite time Lyapunov exponent (FTLE) [16,12,13,34,21], the finite size Lyapunov exponent (FSLE) [2,3,22] and also the infinitesimal size Lyapunov exponent (ISLE) [18]. Then we discuss typical Lagrangian approaches and summarize the original Eulerian approaches as in [23,24,38,39]. The discussions here, however, will definitely not be a complete survey.…”

“…

We propose a new Eulerian numerical approach for constructing the forward flow maps in continuous dynamical systems. The new algorithm improves the original formulation developed in [23,24] so that the associated partial differential equations (PDEs) are solved forward in time and, therefore, the forward flow map can now be determined on the fly. Due to the simplicity in the implementations, we are now able to efficiently compute the unstable coherent structures in the flow based on quantities like the finite time Lyapunov exponent (FTLE), the finite size Lyapunov exponent (FSLE) and also a related infinitesimal size Lyapunov exponent (ISLE).

…”