# Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
## 2 Materials and Methods
### Bifurcations Using Zigzag (Buzz)  
We can now present our method, Bifurcations using ZigZag (BuZZ), for combining the above tools to detect changes in dynamical systems, specifically changes that appear as a change in circular structure in the solution. We will focus on Hopf bifurcations [40], which are seen when a fixed point loses stability and a limit cycle is introduced. These types of bifurcations are particularly topological in nature, as the solution to the dynamical system changes from a small cluster to a circular structure and sometimes reduces back to a cluster. While persistent homology has been used to study Hopf bifurcations [41; 42; 43; 44], none of the existing methods use zigzag persistence.  
The necessary data for our method is a collection of time series for a varying input parameter value, as shown in Figure 2a. This particular example is a collection of time series given by \(\{a\sin(t)\}\) for \(a=0.5,1.0,1.5,2.0\) (going from top to bottom). Each time series is then embedded using time-delay embedding (shown in Figure 2b using \(d=2\) and \(\tau=3\)). While in general the delay could be varied for each time series, the embedding dimension needs to be fixed so each time series is embedded in the same space. For the sake of interpretability and visualization, we will use a dimension of \(d=2\) throughout this paper. Sorting the resulting point clouds based on the input parameter value, the zigzag filtration can be formed from the collection of point clouds, as shown in Figure 2c. Lastly, computing  
Figure 1: A small example of zigzag filtration with a corresponding zigzag persistence diagram.  
zigzag persistence gives a persistence diagram, as shown in Figure 2d, encoding information about the structural changes moving through the zigzag.  
With the right choices of parameters, the 1-dimensional persistence point with the longest lifetime in the zigzag persistence diagram will have birth and death times corresponding to the indices in the zigzag where the Hopf bifurcation appears and disappears. Lastly, mapping the birth and death times back to the parameter values used to create the corresponding point clouds will give the range of parameter values where the Hopf bifurcation occurs.  
Note that there are several parameter choices that need to be selected during the course of the BuZZ method: first, the dimension \(d\) and delay \(\tau\) converting each time series into a point cloud. Fortunately, there is a vast literature from the time-series analysis literature for this, which leads to standard heuristics, some of which are published in [34; 35; 36; 37; 38; 39]. The second and more difficult parameter to choose is the radius (or radii) for the Rips complexes. In this paper, the given examples are simple enough that the choice of radii in the BuZZ method can be tuned by the user. In the tuning process, the user should select a radius that allows points in the circular structure to be connected but not so large that it fills in the circle or makes the Rips complexes too large to be computationally feasible. However, in future work, we would like to create new methods and heuristics for choosing these radii.