Detection and Reconstruction of Bursting Oscillations in Complex Systems Using the HAVOK Analysis Framework
Abstract
:1. Introduction
2. Preliminary Theory and Related Work
2.1. Dynamical Systems
2.2. Bursting Oscillations
2.3. Time-Delay Embedding
2.4. Koopman Theory
2.5. Singular Value Decomposition (SVD)
2.6. Dynamic Mode Decomposition (DMD)
2.7. Constructing a Forced Linear Model
3. Algorithm Summary: HAVOK Analysis Framework for Bursting Oscillations
Algorithm 1: HAVOK Analysis Framework for Detecting Bursting Oscillations in Dynamical Systems. |
Input: Time series data from the dynamical system, Embedding dimension Output: Visualizations of eigen time series Embedded flow fields Reconstructed flow fields Bursting oscillation prediction Steps: 1. Collect Time Series Data: Gather the time series data from the dynamical system. Denote the collected data as “data”. 2. Construct Hankel Matrix: Construct the Hankel matrix H using the collected time series data and embedding dimension . |
3. Perform Singular Value Decomposition (SVD): Apply SVD to the Hankel matrix H. |
Sigs = [] 4. Choose Truncation Rank for Reduced SVD: Truncate the SVD results to the rank r. |
5. Plot Embedded Flow Fields: Plot the two-dimensional and three-dimensional embedded flow field using the truncated SVD results. |
6. Apply Dynamic Mode Decomposition (DMD): Apply DMD on the reduced matrix and construct the forced linear model. Forced linear model: Plot the two-dimensional and three-dimensional reconstructed flow fields from the forced linear model. 7. Plot and Analyze Eigen Time Series: Plot and analyze the visualizations of the dominant eigen time series and the forced time series 8. Quickly Check for Bursting: Observe the visualization of for intermittent periodic bursts to detect bursting. Mark “bursting detected” if intermittent periodic bursts are observed. 9. Return Results: Return the visualizations of the eigen time series, the embedded flow fields, the reconstructed flow fields, and the bursting detection status. Return |
4. Results and Discussion
4.1. The HRVD Oscillator Model with a Periodic External Excitation
4.2. Advantages
4.2.1. Revealing Hidden Dynamical Behaviors
4.2.2. Reducing Overlapping Phenomena through Higher-Dimensional Embedding
4.2.3. Analyzing Quasi-Periodic Behavior and Steady-State Nonlinear Oscillations
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Embedding Dimension m (r = 4) | MSE | Degree of Smoothness (Embedded Flow Field) | Degree of Smoothing (Time Series of Forced Characteristics) |
---|---|---|---|
m = 10 | 4.8309 × 10−10 | low | low |
m = 50 | 2.4387 × 10−11 | medium | low |
m = 100 | 2.661 × 10−11 | high | medium |
m = 200 | 1.978 × 10−10 | high | high |
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Cai, X.; Qian, Y. Detection and Reconstruction of Bursting Oscillations in Complex Systems Using the HAVOK Analysis Framework. Algorithms 2024, 17, 388. https://doi.org/10.3390/a17090388
Cai X, Qian Y. Detection and Reconstruction of Bursting Oscillations in Complex Systems Using the HAVOK Analysis Framework. Algorithms. 2024; 17(9):388. https://doi.org/10.3390/a17090388
Chicago/Turabian StyleCai, Xueyi, and Youhua Qian. 2024. "Detection and Reconstruction of Bursting Oscillations in Complex Systems Using the HAVOK Analysis Framework" Algorithms 17, no. 9: 388. https://doi.org/10.3390/a17090388