# Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Homology and Persistent Homology

#### 2.2. Time-Delay Embedding

#### 2.3. Zigzag Persistent Homology

#### 2.4. Bifurcations Using Zigzag (Buzz)

#### 2.5. Algorithms

`simplex_list`, and a list of lists,

`times_list`, where

`times_list[i]`consists of a list of indices in the zigzag where the simplex,

`simplex_list[i]`, is added and removed. A small example is shown in Figure 3. Looking at that example, the two vertices and one edge in $R\left({X}_{0}\right)$ appear at time 0 and disappear at time 1. There are two edges and a triangle in $R({X}_{0}\cup {X}_{1})$ that appear at time 0.5 (recalling that $R({X}_{i}\cup {X}_{i+1})$ is time $i+0.5$) and disappear at time 1. Lastly, the one vertex in $R\left({X}_{1}\right)$ appears at time 0.5 and never disappears in the zigzag sequence; therefore, by default, we set death time to be 2, which is the next index beyond the end of the zigzag sequence. This is done to avoid persistence points of the form $(i,\infty )$, as our zigzag sequences are always finite and these points have no additional meaning. Note that there are other special cases that can occur. If a simplex is added and removed multiple times, then the corresponding entry in

`times_list`has more than two entries, where the zero and even entries in the list correspond to when it appears and the odd entries correspond to when it disappears. An example with this special case is shown in Figure 4 and will be described in more detail later.

`times_list`is $[0.5,1,1.5,2]$. Thus, the inputs to Dionysus can be computed using the same method as above except the Rips complex needs to be computed for each point cloud, not just the unions, and additional checks need to be done to make sure a simplex being added did not already appear and disappear once before. If it did, the entry in

`times_list`needs to be extended to account for the newest appearance and disappearance.

## 3. Results

#### 3.1. Synthetic Point Cloud Example

#### 3.2. Synthetic Time-Series Example

#### 3.3. Sel’kov Model

`odeint`in python, with initial conditions $(0,0)$. We also removed the first 50 points to remove transients at the beginning of the model (this is sometimes referred to as a “burn-in period”). This data was constructed using full knowledge of the model; however, in practice, typically only one measurement function is obtained through an experiment, and then time-delay embedding is used to reconstruct the underlying system. To mimic this setup, we only used the time series corresponding to the x-coordinates from the model and used the delay embedding. These time series were then embedded using time-delay embedding with dimension $d=2$ for the purpose of easy visualization and delay $\tau =3$.

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Outline of the Bifurcations using Zigzag (BuZZ) method. (

**a**) the input time series; (

**b**) the corresponding point clouds embedded via time-delay embedding; (

**c**) the zigzag of Rips complexes constructed as described in Section 2.3; (

**d**) the corresponding zigzag persistence diagram.

**Figure 4.**Example zigzag using a changing radius with computed inputs for Dionysus: in this example, ${r}_{0}>{r}_{1}$ and ${r}_{2}>{r}_{1}$.

**Figure 5.**

**Top**: Example zigzag of point clouds with unions considered in Section 3.2.

**Middle**: Zigzag filtration applied to point clouds using the Rips complex with specified radii. Note that 2-simplices are not shown in the complexes.

**Bottom**: The resulting zigzag persistence diagram.

**Figure 6.**

**First and second rows**: generated time-series data and corresponding time-delay embeddings.

**Bottom left**: The zigzag filtration using Rips complex with fixed radius of 0.72. Note that 2-simplices are not shown in the complexes.

**Bottom middle**: The corresponding zigzag persistence diagram.

**Bottom right**: Persistence diagram showing how the 1-dimensional off diagonal point varies depending on the Rips complex radius parameter choice.

**Figure 7.**Top: Examples of samplings of the state space of the Sel’kov model for varying parameter value b. Middle: Time series corresponding to only retaining the x-coordinates of the solutions shown in the top figure. Bottom left: zigzag filtration using the Rips complex of the reconstruction with fixed radius of 0.25. Note that 2-simplices are not shown in the complexes. Bottom right: resulting zigzag persistence diagram.

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**MDPI and ACS Style**

Tymochko, S.; Munch, E.; Khasawneh, F.A. Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems. *Algorithms* **2020**, *13*, 278.
https://doi.org/10.3390/a13110278

**AMA Style**

Tymochko S, Munch E, Khasawneh FA. Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems. *Algorithms*. 2020; 13(11):278.
https://doi.org/10.3390/a13110278

**Chicago/Turabian Style**

Tymochko, Sarah, Elizabeth Munch, and Firas A. Khasawneh. 2020. "Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems" *Algorithms* 13, no. 11: 278.
https://doi.org/10.3390/a13110278