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Local Trajectory Variation Exponent (LTVE) for Visualizing Dynamical Systems

Yun Chen Tsai, Shingyu Leung

Abstract

The identification and visualization of Lagrangian structures in flows plays a crucial role in the study of dynamic systems and fluid dynamics. The Finite Time Lyapunov Exponent (FTLE) has been widely used for this purpose; however, it only approximates the flow by considering the positions of particles at the initial and final times, ignoring the actual trajectory of the particle. To overcome this limitation, we propose a novel quantity that extends and generalizes the FTLE by incorporating trajectory metrics as a measure of similarity between trajectories. Our proposed method utilizes trajectory metrics to quantify the distance between trajectories, providing a more robust and accurate measure of the LCS. By incorporating trajectory metrics, we can capture the actual path of the particle and account for its behavior over time, resulting in a more comprehensive analysis of the flow. Our approach extends the traditional FTLE approach to include trajectory metrics as a means of capturing the complexity of the flow.

Local Trajectory Variation Exponent (LTVE) for Visualizing Dynamical Systems

Abstract

The identification and visualization of Lagrangian structures in flows plays a crucial role in the study of dynamic systems and fluid dynamics. The Finite Time Lyapunov Exponent (FTLE) has been widely used for this purpose; however, it only approximates the flow by considering the positions of particles at the initial and final times, ignoring the actual trajectory of the particle. To overcome this limitation, we propose a novel quantity that extends and generalizes the FTLE by incorporating trajectory metrics as a measure of similarity between trajectories. Our proposed method utilizes trajectory metrics to quantify the distance between trajectories, providing a more robust and accurate measure of the LCS. By incorporating trajectory metrics, we can capture the actual path of the particle and account for its behavior over time, resulting in a more comprehensive analysis of the flow. Our approach extends the traditional FTLE approach to include trajectory metrics as a means of capturing the complexity of the flow.
Paper Structure (29 sections, 1 theorem, 30 equations, 10 figures, 1 table)

This paper contains 29 sections, 1 theorem, 30 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Trajectory Metric
  4. Finite-time Lyapunov Exponent (FTLE) and Lagrangian Coherent Structure (LCS)
  5. Local Trajectory Variation Exponent (LTVE)
  6. Theoretical Framework
  7. Relation with FTLE
  8. Numerical Scheme and Error Analysis
  9. Algorithm
  10. Implementation Details and Analysis
  11. Boundary Condition.
  12. Time Complexity.
  13. Storage.
  14. Trajectory Metric
  15. $L^{2}$-norm
  16. ...and 14 more sections

Key Result

Lemma 1

If $f\in C^{1}$ on $\Omega$, then $\Lambda_{f}$ is continuous on $\Omega$.

Figures (10)

  • Figure 1: (Section \ref{['SubSec:Circular']}) The computed LTVE using (a) the normalized 2-norm, (b) the Fréchet distance, and (c) the Hausdorff metric. (d) The corresponding FTLE field.
  • Figure 2: (Section \ref{['SubSec:StandingWave']}) The computed LTVE using (a) the normalized 2-norm, (b) the Fréchet distance, and (c) the Hausdorff metric. (d) The corresponding FTLE field.
  • Figure 3: (Section \ref{['SubSec:3trajEval']}) The computed LTVE using (a) the normalized 2-norm, (b) the Fréchet distance, and (c) the Hausdorff metric. (d) The corresponding FTLE field.
  • Figure 4: (Section \ref{['SubSec:DoubleGyre']}) The computed LTVE using (a) the normalized 2-norm, (b) the Fréchet distance, and (c) the Hausdorff metric. (d) The corresponding FTLE field.
  • Figure 5: (Section \ref{['SubSec:DoubleGyre']}) The computed LTV using (a) the normalized 2-norm and the Fréchet distance. (b) We also plot the trajectory of several particles. (c) The zoom-in to the red boxed region on left and (d) the red boxed region on right in (b).
  • ...and 5 more figures

Theorems & Definitions (15)

  • Definition 2.1: Trajectory of Particle
  • Definition 2.2: Displacement Trajectory of Particle
  • Definition 2.3: Discrete Trajectory
  • Definition 2.4: Trajectory Metric
  • Definition 3.1: Local Trajectory Variation
  • Definition 3.2: Local Trajectory Variation Exponent
  • Lemma 1
  • proof
  • Definition 3.3: First Order Relaxed-LTV
  • Definition 3.4: Second Order Relaxed-LTV
  • ...and 5 more