## 3 Algorithm Summary: HAVOK Analysis Framework for Bursting Oscillations  
In this paper, the HAVOK analysis framework is utilized to detect bursting oscillations in dynamical systems. This algorithm involves constructing a Hankel matrix from time series data, performing SVD, and applying DMD to identify eigen time series. The process concludes by plotting the embedded and reconstructed flow field and visually checking for intermittent periodic bursts to detect bursting oscillations. Next, we outline the complete general process of applying HAVOK analysis to bursting oscillations.  
First, we start by choosing the initial embedding dimension \(m\) based on the specific model combined with experimental experience.  
```
Input: Time series data from the dynamical system, Embedding dimension \(m\) 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 \(\mathbf{H}\) using the collected time series data and embedding dimension \(m\). \(H\leftarrow\text{construct handel matrix}(\text{data},m)\)
3. Perform Singular Value Decomposition (SVD): Apply SVD to the Hankel matrix \(\mathbf{H}\). \(\left(\mathbf{U},\boldsymbol{\Sigma},\mathbf{V}^{T}\right)\leftarrow\text{ svd}(\mathbf{H})\) \(\text{Sigs}=[\sigma_{1},\sigma_{2},\sigma_{3},\cdots]\) 4. Choose Truncation Rank \(r\) for Reduced SVD: Truncate the SVD results to the rank \(r\). \(\mathbf{U}_{r},\boldsymbol{\Sigma}_{r},\mathbf{V}_{r}^{T}\leftarrow\mathbf{U}[: :r],\boldsymbol{\Sigma}[:r],\boldsymbol{V}^{T}[:r,:]\)
5. Plot Embedded Flow Fields: Plot the two-dimensional and three-dimensional embedded flow field using the truncated SVD results. \(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\boldsymbol{v}_{3}\leftarrow\text{ photembeddedflowfields}(\boldsymbol{U}_{r},\boldsymbol{\Sigma}_{r},\mathbf{V}_{r}^{T})\) 6. Apply Dynamic Mode Decomposition (DMD): Apply DMD on the reduced matrix \(\mathbf{V}_{r}^{T}\) and construct the forced linear model. \(\overset{\sim r-1}{\boldsymbol{V}_{1}}=\boldsymbol{A}\mathbf{V}_{1}^{-1}+ \boldsymbol{B}\mathbf{v}_{r}\) 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 \(\boldsymbol{v}_{1}\) and the forced time series \(\boldsymbol{v}_{r}\). 8. Quickly Check for Bursting: Observe the visualization of \(\boldsymbol{v}_{r}\) 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 {\"eigen time series visualizations\", \"embedded flow fields\", \"reconstructed flow fields\", \"bursting detected\"}
```  
**Algorithm 1**HAVOK Analysis Framework for Detecting Bursting Oscillations in Dynamical Systems.  
In the fourth step of the algorithm regarding the selection of \(r\), we will plot the variation of the cumulative energy of the singular values with the increase of the number of singular values, and select the \(r\) whose \(E(k)\) is at least greater than 99%.  
In addition, we will plot the fit of the first column \(\overset{\sim}{v}_{1}\) of the matrix of the output of the forced linear model obtained in the sixth step to the first column \(v_{1}\) of the matrix of the inputs, and compute the mean-square error (MSE) of the reconstruction. With the mean square error, we can continuously adjust the embedding dimension \(m\) and the truncation order \(r\) to keep the MSE of the reconstruction within a certain range. In this paper, we will choose \(m\) and \(r\) by combining the MSE of the reconstruction with the actual effect of the reconstruction.