Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
UUsing Zigzag Persistent Homology to Detect Hopf Bifurcations inDynamical Systems
Sarah Tymochko , Elizabeth Munch , , Firas A. Khasawneh Dept. of Computational Mathematics, Science and Engineering Dept. of Mathematics Dept. of Mechanical EngineeringMichigan State University { tymochko,muncheli,khasawn3 } @egr.msu.edu Abstract
Bifurcations in dynamical systems characterize qualitative changes in the system behavior.Therefore, their detection is important because they can signal the transition from normal systemoperation to imminent failure. While standard persistent homology has been used in this setting,it usually requires analyzing a collection of persistence diagrams, which in turn drives up thecomputational cost considerably. Using zigzag persistence, we can capture topological changesin the state space of the dynamical system in only one persistence diagram. Here we presentBifurcations using ZigZag (BuZZ), a one-step method to study and detect bifurcations usingzigzag persistence. The BuZZ method is successfully able to detect this type of behavior in twosynthetic examples as well as an example dynamical system.
Topological data analysis (TDA) is a field consisting of tools aimed at extracting shape in data.Persistent homology, one of the most commonly used tools from TDA, has proven useful in the fieldof time series analysis. Specifically, persistent homology has been shown to quantify features of atime series such as periodic and quasiperiodic behavior [28, 31, 36, 23, 40] or chaotic and periodicbehavior [25, 18]. Existing applications in time series analysis include studying machining dynamics[19, 20, 41, 18, 42, 21, 17], gene expression [28, 4], financial data [13], video data [38, 37], and sleep-wake states [10, 39]. These applications typically involve summarizing the underlying topologicalshape of each time series in a persistence diagram then using additional methods to analyze theresulting collection of persistence diagrams. While these applications have been successful, the taskof analyzing a collection of persistence diagrams can still be difficult. Many methods have beencreated to convert persistence diagrams into a form amenable for machine learning [5, 3, 30, 29].However, so many methods have been developed, it can be difficult to choose one appropriate forthe task. Additionally, the task of computing numerous persistence diagrams is computationallyexpensive.Our method aims to circumvent these issues using zigzag persistence, a generalization of per-sistent homology that is capable of summarizing information from a sequence of point clouds in asingle persistence diagram. While less popular than standard persistent homology, zigzag persis-tence has been used in applications, including studying optical flow in computer generated videos[1, 2], analyzing stacks of neuronal images [24] and comparing different subsamples of the a dataset1 a r X i v : . [ c s . C G ] S e p Here we will present tools needed to build our method, including the time delay embedding, and anoverview of the necessary topological tools. Specifically, we briefly introduce homology, persistenthomology and zigzag persistent homology. However, we will not go into detail and instead directthe interested reader to [15, 11, 6] for more detail on homology, persistent homology, and zigzaghomology, respectively.
Homology is a tool from the field of algebraic topology that encodes information about shape invarious dimensions. Zero-dimensional homology studies connected components, 1-dimensional ho-mology studies loops, and 2-dimensional homology studies voids. Persistent homology is a methodfrom TDA which studies the homology of a parameterized space.For the purposes of this paper, we will focus on persistent homology applied to point clouddata. Here, we need only assume a point cloud is a collection of points with a notion of distance,however in practice, this distance often arises from a point cloud in Euclidean space inheriting theambient metric. Given a collection of points, we will build connections between points based ontheir distance. Specifically, we will build simplicial complexes, which are spaces built from differentdimensional simplices. A 0-simplex is a vertex, a 1-simplex is an edge, a 2-simplex is a triangle,and in general, a p -simplex is the convex hull of p + 1 affinely independent vertices. Abstractly, a p -simplex can be represented by the set of p + 1 vertices it is built from. So a simplicial complex, K , is a family of sets that is closed under taking subsets. That is, given a p -simplex, σ , in K , thenany simplex consisting of a subset of the vertices of size 0 < k ≤ p , called a k -dimensional face of σ , is also in K .To create a simplicial complex from a point cloud, we use the Vietoris-Rips complex (sometimesjust called the Rips complex). Given a point cloud, X , and a distance value, r , the Vietoris-Ripscomplex, K r , consists of all simplices whose vertices have maximum pairwise distance at most r .Taking a range of distance values, r ≤ r ≤ r ≤ · · · r n gives a set of simplicial complexes, {K r i } .Since the distance values are strictly increasing, we have a nested sequence of simplicial complexes K r ⊆ K r ⊆ · · · ⊆ K r n (1)called a filtration. Computing p -dimensional homology, H p ( K ), for each complex in the filtrationgives a sequence of vector spaces and linear maps, H p ( K r ) → H p ( K r ) → · · · → H p ( K r n ) . (2)Persistent homology tracks homological features such as connected components and loops as youmove through the filtration. Specifically, it records at what distance value a feature first appears,and when a feature disappears or connects with another feature. These distance values are calledthe “birth” and “death” times respectively. These birth and death times are represented as apersistence diagram, which is a multiset of the birth death pairs { ( b, d ) } .2 .2 Time delay embedding One way to reconstruct the underlying dynamics given only a time series is through a time delayembedding. Given a time series, [ x , . . . , x n ], a choice of dimension d and lag τ , the delay embeddingis the point cloud X = { x i := ( x i , x i + τ , . . . , x i +( d − τ ) } ⊂ R d . Takens’ theorem [34] shows thatgiven most choices of parameters, the embedding retains the same topological structure as the statespace of the dynamical system and that this is in fact a true embedding in the mathematical sense.In practice, not all parameter choices are optimal, so heuristics for making reasonable parameterchoices have been developed [12, 16, 27, 9, 26, 22].In existing methods combing TDA with time series analysis, most works analyze a collection oftime series by embedding each one into a point cloud using the time delay embedding. However, thiscan be computationally expensive and requires an analysis of the collection of computed persistencediagrams. Instead, we will employ a generalized version of persistent homology to avoid theseadditional steps. Zigzag persistence is a generalization of persistent homology that can study a collection of pointclouds simultaneously. In persistent homology, you have a nested collection of simplicial complexesas in Eqn. 1. However, for zigzag persistence, you can have a collection of simplicial complexeswhere the inclusions go in different directions. Specifically, the input to zigzag persistence is asequence of simplicial complexes with maps, K ↔ K ↔ · · · ↔ K n (3)where ↔ is either an inclusion to the left or to the right. While in general these inclusions can goin any direction in any order, for this paper we will focus on a specific setup for the zigzag basedon a collection of point clouds.Given an ordered collection of point clouds, X , X , . . . , X n , we can define a set of inclusions, X X X · · · X n − X n X ∪ X X ∪ X X n − ∪ X n . (4)However, these are all still point clouds, which have uninteresting homology. Thus, we can computethe Vietoris-Rips complex of each point cloud for a fixed radius, r . This results in the diagram ofinclusions of simplicial complexes R ( X , r ) R ( X , r ) R ( X , r ) · · · R ( X n − , r ) R ( X n , r ) R ( X ∪ X , r ) R ( X ∪ X , r ) R ( X n − ∪ X n , r ) . (5)Computing the 1-dimensional homology of each complex in Eqn. 5 will result in a zigzag diagramof vector spaces and induced linear maps, H ( R ( X , r )) H ( R ( X , r )) · · · H ( R ( X n − , r )) H ( R ( X n , r )) H ( R ( X ∪ X , r )) H ( R ( X n − ∪ X n , r )) . (6)3igure 1: Small example of zigzag filtration with corresponding zigzag persistence diagram.Zigzag persistence tracks features that are homologically equivalent through this zigzag. Thismeans it records the range of the zigzag filtration where the same feature appears. The zigzagpersistence diagram records “birth” and “death” relating to location in the zigzag. If a featureappears in R ( X i , r ) , it is assigned birth time i , and if it appears at R ( X i ∪ X i +1 , r ) , it is assignedbirth time i + 0 .
5. Similarly, if a feature last appears in R ( X j , r ), it is assigned a death time j + 0 . R ( X j ∪ X j +1 , r ), it is assigned a death time of j + 1. A small exampleis shown in Fig. 1. In this example, there is a 1-dimensional feature that appears in R ( X , r )and disappears going into R ( X ∪ X , r ), thus it appears in the zigzag persistence diagram asthe point (0 , . , R ( X ∪ X , r ), corresponding to the0-dimensional persistence point (0 . , . Note that we can easily generalize this idea to use a different radius for each Rips complex, R ( X i , r i ). For the unions we choose the maximum radius between the two individual point clouds, R ( X i ∪ X i +1 , max { r i , r i +1 } ), to ensure the inclusions hold. We can now present our method, Bifurcations using ZigZag (BuZZ) for combining the above toolsto detect changes in circular features in dynamical systems. We will focus on Hopf bifurcations[14], which are seen when a fixed point loses stability and a limit cycle is introduced. These types ofbifurcations are particularly topological in nature, as the state space changes from a small cluster,to a circular structure, and sometimes reduces back to a cluster.The necessary data for our method is a collection of time series for a varying input parametervalue, as shown in Fig. 2(a). This particular example is a collection of time series given by { a sin( t ) } for a = 0 . , . , . , . d = 2 and τ = 3). While in general, the delaycould be varied for each time series, the embedding dimension needs to be fixed so each time seriesis embedded in the same space. For the sake of interpretability and visualization, we will use adimension of d = 2 throughout this paper. Sorting the resulting point clouds based on the inputparameter value, the zigzag filtration can be formed from the collection of point clouds, as shown inFig. 2(c). Lastly, computing zigzag persistence gives a persistence diagram, as shown in Fig. 2(d),encoding information about the structural changes moving through the zigzag.With the right choices of parameters, the 1-dimensional persistence point with the longestlifetime in the zigzag persistence diagram will have birth and death time corresponding to theindices in the zigzag where the Hopf bifurcation appears and disappears. Lastly, mapping the birthand death times back to the parameter values used to create the corresponding point clouds will4igure 2: Outline of BuZZ method. The input time series is converted to an embedded point cloudvia the time delay embedding. The Rips complexes are constructed for either a fixed r or a choiceof r i for each point cloud. Then, the zigzag persistence diagram is computed for the collection.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 theBuZZ method. First, the dimension d and delay τ for converting each time series into a pointcloud. Fortunately, there is a vast literature from the time series analysis literature for this, whichleads to standard heuristics. The second and more difficult parameter is the choice of radius (orradii) for the Rips complexes. In this paper, the given examples are simple enough that the choiceof radii in the BuZZ method can be tuned by the user. However, in future work, we would like tocreate new methods and heuristics for choosing these radii. While zigzag persistence has been in the literature for a decade, it has not often been used inapplication, and thus the software that computes it is not well developed. A C++ package withpython wrappers, Dionysus 2 , has implemented zigzag persistence; however, it requires significantpreprocessing to create the inputs. We have developed a python package that, provided thecollection of point clouds and radii, will perform all the necessary preprocessing to set up thezigzag diagram as shown in Eqn. 5 to pass as inputs to Dionysus.Dionysus requires two inputs, a list of simplices, simplex list , and a list of lists, times list ,where the 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 Fig. 3. Looking at that example, the two ver-tices and one edge in R ( X ) appear at time 0, and disappear at time 1. There are two edges and atriangle in R ( X ∪ X ) that appear there at time 0.5 (recalling that R ( X i ∪ X i +1 ) is time i + 0 . R ( X ) appears at time 0.5, and never disappearsin the zigzag sequence, so by default we set death time to be 2, which is the next index beyond the https://github.com/sarahtymochko/BuZZ r > r and r > r .end of the zigzag sequence. This is done to avoid infinite lifetime points, as our zigzag sequencesare always finite and an infinite point has no additional meaning. Note there are other special casesthat can occur. If a simplex is added and removed multiple times, then the corresponding entryin times list has more than two entries, where the zero and even entries in the list correspondto when it appears, and the odd entries correspond to when it disappears. An example with thisspecial case is shown in Fig. 4 and will be described in more detail later.If we are using a fixed radius across the whole zigzag, these inputs can be computed rathereasily. In this setting, we only need to compute the Rips complex of the unions, R ( X i ∪ X i +1 , r ),which can be done using the Dionysus package, and the list of simplices can be created by combininglists of simplices for all i , removing duplicates. Next, we will outline how to construct the timeslist. Starting with the set of simplices in R ( X i ∪ X i +1 , r ), we can split them into three groups: (a)simplices for which all 0-dimensional faces are in X i , (b) simplices for which all 0-dimensional facesare in X i +1 , or (c) simplices for which some 0-dimensional faces are in X i and some are in X i +1 .Because of the construction of the zigzag, all simplices in group (a) appear at time i − .
5, since R ( X i , r ) also includes backwards into R ( X i − ∪ X i , r ), and disappear at time i + 1, since the union R ( X i ∪ X i +1 , r ) is the last time simplices in X i are included. Similarly, all simplices in group (b)appear at time i + 0 .
5, since this is the first time simplices X i +1 are included, and disappear attime i + 2, since R ( X i +1 , r ) also includes forward into R ( X i +1 ∪ X i +2 , r ). Lastly, all simplices ingroup (c) exist only at R ( X i ∪ X i +1 , r ), so they appear at time i + 0 . i + 1. Notethat the first case needs to be treated separately, since in R ( X ∪ X , r ), all vertices in group (a)will appear at 0.Using a varied radius, as described in Sec. 2.3, complicates the above procedure. Using thesame radius, we are guaranteed all simplices in group (a) are in both R ( X i , r ) and R ( X i ∪ X i +1 , r ),and similarly for group (b), thus we only need to compute R ( X i ∪ X i +1 , r ). However, with a6hanging radius this is no longer true. In the example shown in Fig. 4, the edge e , appears inboth R ( X ∪ X , r ) and R ( X ∪ X , r ) since r > r and r > r , but it is not in R ( X , r ). Thus,its corresponding list in times list is [0 . , , . , times list needsto be extended to account for the newest appearance and disappearance.Because of the additional Rips complex computations, and the checks for the special case, thecase of a changing radius is significantly more computationally expensive than the case of a fixedradius. In both cases, there is the computational cost of the zigzag persistence computation as well.The computational complexity of zigzag persistence is O ( nm ) where n is the number of simplicesin the entire filtration and m is the number of simplices in the largest single complex [7]. Thus, thelargest barrier to computation is the zigzag itself, so choosing a radius that is as small as possiblewithout breaking the topology is the goal. We will test the BuZZ method on three different examples. The first example is not based on timeseries data, but is instead a simple proof-of-concept example to test our methods ability to detectchanging circular behavior. The second example is based on synthetic time series data generatedfrom noisy sine waves of varying amplitude. This lets us fully utilize the BuZZ method, includingthe time delay embedding, as well as test resiliency to noise. The last example is detecting a Hopfbifurcation in the Sel’kov model of glycolysis [32].
To start, we will consider a small, synthetic example generating point cloud circles of varying size asshown in Fig. 5. Note, because we are starting with point clouds, we skip the time delay embeddingstep for this example. While each point cloud is sampled from a circle, the first and last point cloudsconsist of relatively small circles. So the strongest circular structure we can see visually starts with X and ends with X . This is the range we would like to detect using zigzag persistence.For this example, we will use the generalized version of the zigzag filtration in (5) using a chang-ing radii. Computing the zigzag persistence gives the persistence diagram shown in Fig. 5. Recallthat birth and death times are assigned based on the location in the zigzag that a feature appearsand disappears. Thus, the one-dimensional point (1 , .
5) in the persistence diagram correspondsto a feature that first appears at R ( X ) and last appears in R ( X ) Thus, using the persistencediagram we can detect the appearance and disappearance of the circular feature.This is clearly an overly simplified situation as each point cloud is sampled from a perfect circle.Next, we will look at a more realistic example. For the second example, we generate synthetic time series data and apply the full method describedin Sec. 2.4. We start by generating sine waves of varying amplitudes and add noise drawn fromuniformly from [ − . , . d = 2 and delay τ = 4. The time series and corresponding time delay embeddingsare shown in Fig. 6. Looking at the time series, in the first and last time series any signal is mostlyobscured by noise, resulting in a small clustered time delay embedding. However, for the other7igure 5: Top: Example zigzag of point clouds with unions considered in Sec. 3.2. Middle: Zigzagfiltration applied to point clouds using the Rips complex with specified radii. Note that 2-simplicesare 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 embed-dings. Bottom left: The zigzag filtration using Rips complex with fixed radius of 0.72. Note that2-simplices are not shown in the complexes. Bottom right:the corresponding zigzag persistencediagram. 8igure 7: Top: Examples of samplings of the state space of the Sel’kov model for varying parametervalue b . Bottom left: zigzag filtration using Rips complex with fixed radius of 0.25. Note that 2-simplices are not shown in the complexes. Bottom right: resulting zigzag persistence diagram.time series, the time delay embedding is still circular, picking up the periodic behavior even withthe noise.Next we compute zigzag persistence, resulting in the zigzag of rips complexes and zigzag persis-tence diagram shown in Fig. 6. The zigzag persistence diagram has a one-dimensional point withcoordinates (1 , . R ( X ), and disappears going into R ( X ). This is the region we would expect to see a circular feature. Our last experiment is trying to detect a bifurcation in the Sel’kov model [32], a model for glycolysiswhich is a process of breaking down sugar for energy. This model is defined by the system ofdifferential equations, ˙ x = − x + ay + x y ˙ y = b − ay − x y where the overdot denotes a derivative with respect to time. In this system, x and y represent theconcentration of ADP (adenosine diphosphate) and F6P (fructose-6-phosphate), respectively. Thissystem has a Hopf bifurcation for select choices of parameters a and b . This limit cycle behaviorcorresponds to the oscillatory rise and fall of the chemical compounds through the glycolysis process.For our experiments, we will fix a = 0 . b . We generate 500 time pointsof the data ranging between 0 and 500 using odeint in python, with initial conditions (0 , x -coordinates from the model and use the delay embedding.These time series are then embedded using the time delay embedding with dimension d = 2 anddelay τ = 3.The next step would be to compute zigzag persistence as described in Sec. 2.3, however due tothe large number of points in the time delay embeddings, this becomes computationally expensive.In order to reduce the computation time, we subsample these point clouds using the furthest pointsampling method (also called a greedy permutation) [8]. We subsample down to only 20 pointsin each point cloud, compute the Rips complex zigzag for a fixed radius value of 0 .
25, and thencompute the zigzag persistence. Figure 7 shows the zigzag filtration of Rips complexes along withthe resulting zigzag persistence diagram. In the zigzag persistence diagram, the point with thelongest lifetime has coordinates (2 , . R ( X ) and disappearing at R ( X ∪ X ). Looking back at which values of b were used to generate these point clouds, we see thiscorresponds to a feature appearing at b = 0 .
45 and disappearing at b = 0 .
8. For the fixed parametervalue of a = 0 .
1, the Sel’kov model has a limit cycle approximately between the parameter values0 . ≤ b ≤ . x -coordinates of the model, however the same results can be obtainedusing the y -coordinates and a slightly larger radius value. Here we have introduced a method of detecting Hopf bifurcations in dynamical systems using zigzagpersistent homology called BuZZ. This method was shown to work on two synthetic examples aswell as a more realistic example using the Sel’kov model. Our method is able to detect the rangeof the zigzag filtration where circular features appear and disappear. Thus, this method could beapplied to any application with an ordered set of point clouds and a changing topological structure.While this method has shown success, it also has its limitations. The method is computationallyexpensive due to numerous Rips complex computations in addition to the zigzag persistence com-putation itself. This issue can be alleviated using subsampling, as shown with the Sel’kov model,but this may not be feasible depending on the application. Future extensions of this project couldinclude improvements of the algorithms described in Sec. 2.5. Additionally, while the method workswell in practice, it lacks theoretical guarantees. Given the method requires parameter choices forthe radii of the Vietoris-Rips complexes, we would like some heuristics to be used in practice tochoose these radii more easily. Because our examples in this paper are small, selecting parametersby hand is reasonable. However, in the future when applied to larger, experimental data, thesesorts of heuristics will be necessary.
Acknowledgements
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