^{1,a)}

### 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.

Finite-time Lyapunov exponent (FTLE) is a widely used quantity for visualizing the Lagrangian coherent structure (LCS) in a complex dynamical system. Unfortunately, typical numerical approaches to FTLE are computationally very expansive. In this work, we further develop an Eulerian approach to this Lagrangian quantity and propose a partial differential equation (PDE) based phase flow approach to speed up the overall computations, even for aperiodic flows. This numerical method can efficiently compute the moderate to long time flow map for approximating the Lyapunov exponent of a dynamical system.

The author would like to thank the anonymous reviewers for their extremely helpful and constructive comments and suggestions that greatly improve the quality of the manuscript. The work was supported in part by the Hong Kong RGC under Grant Nos. 602210 and 605612.

I. INTRODUCTION

II. THE PROPOSED METHOD

A. An Eulerian method for short-time flow maps

B. A backward phase flow method for long-time flow maps

C. FTLE

III. SOME PROPERTIES AND FURTHER EXTENSIONS

A. Boundary condition and the interpolation scheme

B. Computational complexity

C. Time evolution of the FTLE

D. Extension to aperiodic flows

IV. EXAMPLES

A. Double-gyre flow

B. Point vortex flow on a sphere

V. ADVANTAGES, LIMITATIONS, AND FUTURE WORK

### Key Topics

- Flow simulations
- 32.0
- Interpolation
- 27.0
- Lagrangian mechanics
- 24.0
- Boundary value problems
- 9.0
- Computational complexity
- 7.0

## Figures

Lagrangian and Eulerian interpretations of the function . (a) Lagrangian ray tracing from a given grid location x at t = 0. Note that y might be a non-grid point. (b) Eulerian values of at a given grid location y at t = T gives the corresponding take-off location at t = 0. Note the take-off location might not be a mesh point.

Lagrangian and Eulerian interpretations of the function . (a) Lagrangian ray tracing from a given grid location x at t = 0. Note that y might be a non-grid point. (b) Eulerian values of at a given grid location y at t = T gives the corresponding take-off location at t = 0. Note the take-off location might not be a mesh point.

(Section IV A with ϵ = 0.0) The scaled forward FTLE using the proposed backward phase flow method, i.e., T ^{ T }: x = 1/256, T* = 0.1. We apply the backward phase flow method to obtain the scaled FTLE at (a)–(c) t = T* (2^{15} – 2^{5} ^{ k }) for k = 1, 2, and 3 with T = T*2^{5} ^{k}. (d) Once we have obtained 3276.8 ^{3276.8}(x, 0), we propagate the solution to obtain 3276.8 ^{3276.8}(x, 3276.8).

(Section IV A with ϵ = 0.0) The scaled forward FTLE using the proposed backward phase flow method, i.e., T ^{ T }: x = 1/256, T* = 0.1. We apply the backward phase flow method to obtain the scaled FTLE at (a)–(c) t = T* (2^{15} – 2^{5} ^{ k }) for k = 1, 2, and 3 with T = T*2^{5} ^{k}. (d) Once we have obtained 3276.8 ^{3276.8}(x, 0), we propagate the solution to obtain 3276.8 ^{3276.8}(x, 3276.8).

(Section IV A with ϵ = 0.0) Trajectories of several particles corresponding to Figure 2(a) . Initial locations are plotted in (red) square, and final locations are plotted in (blue) circle.

(Section IV A with ϵ = 0.1) The scaled forward FTLE at t = 0 using the Lagrangian approach, i.e., T ^{ T }: T = 320, x = 1/64, and 1/128, respectively.

(Section IV A with ϵ = 0.1) The scaled forward FTLE at t = 0 using the Lagrangian approach, i.e., T ^{ T }: T = 320, x = 1/64, and 1/128, respectively.

(Section IV A with ϵ = 0.1) The scaled forward FTLE using the proposed backward phase flow method, i.e., T ^{ T }: x = 1/256, T ^{*} = 10. (a)–(c) We apply the backward phase flow method to obtain the scaled FTLE's t k ^{tk } (x, 0) for t k = T* • 2^{5} • k/5 and k = 1, 2, and 5. (d) Once we have obtained 320 ^{320} (x, 0), we propagate it to obtain 320 ^{320}(x, 320).

(Section IV A with ϵ = 0.1) The scaled forward FTLE using the proposed backward phase flow method, i.e., T ^{ T }: x = 1/256, T ^{*} = 10. (a)–(c) We apply the backward phase flow method to obtain the scaled FTLE's t k ^{tk } (x, 0) for t k = T* • 2^{5} • k/5 and k = 1, 2, and 5. (d) Once we have obtained 320 ^{320} (x, 0), we propagate it to obtain 320 ^{320}(x, 320).

(Section IV A with an aperiodic perturbation (8) ) The scale backward FTLE for an aperiodic flow using the proposed backward phase flow method, i.e., T ^{−T }: x = y = 1/256, z = 10/128, and T* = 10/2^{12}. We iterate the obtained flow map for (a) 3, (b) 6, (c) 9, and (d) 12 times to obtain the scaled backward FTLE (a) 2^{3} (x, 10), (b) 2^{6} T* (x, 10), (c) 2^{9} T* (x, 10), and (d) 2^{12} t* (x, 10).

(Section IV A with an aperiodic perturbation (8) ) The scale backward FTLE for an aperiodic flow using the proposed backward phase flow method, i.e., T ^{−T }: x = y = 1/256, z = 10/128, and T* = 10/2^{12}. We iterate the obtained flow map for (a) 3, (b) 6, (c) 9, and (d) 12 times to obtain the scaled backward FTLE (a) 2^{3} (x, 10), (b) 2^{6} T* (x, 10), (c) 2^{9} T* (x, 10), and (d) 2^{12} t* (x, 10).

(Section IV B ) The scaled backward FTLE using the proposed backward phase flow method, i.e., T ^{−T}: x = 3/128, T ^{*} = 0.0625. We apply the backward phase flow method to obtain the scaled FTLE's 64 ^{−64}(x, 64).

(Section IV B ) The scaled backward FTLE using the proposed backward phase flow method, i.e., T ^{−T}: x = 3/128, T ^{*} = 0.0625. We apply the backward phase flow method to obtain the scaled FTLE's 64 ^{−64}(x, 64).

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