Persistence of the Conley Index in Combinatorial Dynamical Systems

Tamal K. Dey [email protected] Department of Computer Science and Engineering, The Ohio State University, Columbus, USA Marian Mrozek [email protected] Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland Ryan Slechta [email protected] Department of Computer Science and Engineering, The Ohio State University, Columbus, USA

Abstract

A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman [7, 8] and their recent generalization to multivector fields [16] have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to “track” features in a changing multivector field.

1 Introduction

At the end of the 19th century, scientists became aware that the very fruitful theory of differential equations cannot provide a description of the asymptotic behavior of solutions in situations when no analytic formulas for solutions are available. This observation affected Poincaré’s study on the stability of our celestial system [18] and prompted him to use the methods of dynamical systems theory. The fundamental observation of the theory is that solutions limit in invariant sets. Examples of invariant sets include stationary solutions, periodic orbits, connecting orbits, and many more complicated sets such as chaotic invariant sets discovered in the second half of the 20th century [12]. Today, the Conley index [5, 14] is among the most fundamental topological descriptors that are used for analyzing invariant sets. The Conley index is defined for isolated invariant sets which are maximal invariant sets in some neighborhood. It characterizes whether isolated invariant sets are attracting, repelling or saddle-like. It is used to detect stationary points, periodic solutions and connections between them. Moreover, it provides methods to detect and characterize different chaotic invariant sets. In particular, it was used to prove that the system discovered by Lorenz [12] actually contains a chaotic invariant set [13]. The technique of multivalued maps used in this proof may be adapted to dynamical systems known only from finite samples [15]. Unfortunately, unlike the case when an analytic description of the dynamical system is available, the approach proposed in [15] lacks a validation method. This restricts possible applications in today’s data-driven world. In order to use topological persistence as a validation tool, we need an analog of dynamical systems for discrete data. In the case of a dynamical system with continuous time, the idea comes from the fundamental work of R. Forman on discrete Morse theory [8] and combinatorial vector fields [7]. This notion of a combinatorial vector field was recently generalized to that of a combinatorial multivector field [16]. Since then, the Conley index has been constructed in the setting of combinatorial multivector fields [11, 16]. The aim of this research is to incorporate the ideas of topological persistence into the study of the Conley index.

Refer to caption — Figure 1: Three multivector fields. In each field, there is a periodic attractor in blue. Such an attractor is an example of an invariant set. The reader will notice that all flows which enter the periodic attractor can ultimately be traced back to simplices marked with circles. These simplices are individually invariant sets, and they correspond to the notion of fixed points. In each multivector field, the gold triangle corresponds to the notion of a repelling fixed point, while the triangles and edges with magenta circles are spurious. Notice that in the third multivector field there is a spurious periodic attractor. However, despite spurious invariant sets, in all three multivector fields the predominant feature is a repelling triangle, from which most emanating flow terminates in the periodic attractor. We aim to develop a quantitative summary of this behavior.

Given a simplicial complex $K$ , Forman defined a combinatorial vector field $\mathcal{V}$ as a partition of $K$ into three sets $L\sqcup U\sqcup C$ where a bijective map $\mu:L\rightarrow U$ pairs a $p$ -simplex $\sigma\in L$ with a $(p+1)$ -simplex $\tau=\mu(\sigma)$ . This pair can be thought of as a vector originating in $\sigma$ and terminating in $\tau$ . Using these vectors, Forman defined a notion of flow for discrete vector fields called a $V$ -path. These paths correspond to the classical notion of integral lines in smooth vector fields. Multivector fields proposed in [16] generalize this concept by allowing a vector to have multiple simplices (dubbed multivectors) and more complicated dynamics.

An extension to the idea of the $V$ -path from Forman’s theory is called a solution for a multivector field $\mathcal{V}$ . A solution in $\mathcal{V}$ is a possibly infinite sequence of simplices $\{\sigma_{i}\}$ such that $\sigma_{i+1}$ is either a face of $\sigma_{i}$ or in the same multivector as $\sigma_{i}$ . Solutions may be doubly infinite (or bi-infinite), right infinite, left infinite, or finite. Solutions that are not doubly infinite are partial. Bi-infinite solutions correspond to invariant sets in the combinatorial setting.

In Figure 1, one can see a sequence of multivector fields, each of which contains multiple isolated invariant sets. In principle, one would like to choose an isolated invariant set from each of the multivector fields and obtain a description of how the Conley index of these sets changes. Obtaining such a description is highly nontrivial, and it is the main contribution of this paper. Given a sequence of isolated invariant sets, we use the theory of zigzag persistence [4] to extract such a description. In [6], the authors studied the persistence of the Morse decomposition of multivector fields, but this is the first time that the Conley index has been placed in a persistence framework. We also provide schemes to automatically select isolated invariant sets and to limit the effects of noise on the persistence of the Conley index.

2 Preliminaries

Throughout this paper, we will assume that the reader has a basic understanding of both point set and algebraic topology. In particular, we assume that the reader is well-versed in homology. For more information on these topics, we encourage the reader to consult [10, 17].

2.1 Multivectors and Combinatorial Dynamics

In this subsection, we briefly recall the fundamentals of multivector fields as established in [6, 11, 16]. Let $K$ be a finite simplicial complex with face relation $\leq$ , that is, $\sigma\leq\sigma^{\prime}$ if and only if $\sigma$ is a face of $\sigma^{\prime}$ . Equivalently, $\sigma\leq\sigma^{\prime}$ if $V(\sigma)\subseteq V(\sigma^{\prime})$ , where $V(\sigma)$ denotes the vertex set of $\sigma$ . For a simplex $\sigma\in K$ , we let $\operatorname{\sf cl}(\sigma):=\{\tau\in K\,|\,\tau\leq\sigma\}$ and for a set $A\subseteq K$ , we let $\operatorname{\sf cl}(A):=\{\tau\leq\sigma\,|\,\sigma\in A\}$ . We say that $A\subseteq K$ is closed if $\operatorname{\sf cl}(A)=A$ . The reader familiar with the Alexandrov topology [1, Section 1.1] will immediately notice that this notation and terminology is aligned with the topology induced on $K$ by the relation $\leq$ .

Definition 1 (Multivector, Multivector Field).

A subset $A\subseteq K$ is called a multivector if for all $\sigma,\sigma^{\prime}\in A$ , $\tau\in K$ satisfying $\sigma\leq\tau\leq\sigma^{\prime}$ , we have that $\tau\in A$ . A multivector field over $K$ is a partition of $K$ into multivectors.

Every multivector is said to be either regular or critical. To define critical multivectors, we define the mouth of a set as $\operatorname{\sf mo}(A):=\operatorname{\sf cl}(A)\setminus A$ . The multivector $V$ is critical if the relative homology $H_{p}(\operatorname{\sf cl}(V),\operatorname{\sf mo}(V))\neq 0$ in some dimension $p$ . Otherwise, $V$ is regular. Simplices in critical multivectors are marked with circles in Figures 1, 2, and 3. Throughout this paper, all references to homology are references to simplicial homology. Note that $H_{p}\left(\operatorname{\sf cl}(V),\operatorname{\sf mo}(V)\right)$ is thus well defined because $\operatorname{\sf mo}(S)\subseteq\operatorname{\sf cl}(S)\subseteq K$ . Intuitively, a multivector $V$ is regular if $\operatorname{\sf cl}(V)$ can be collapsed onto $\operatorname{\sf mo}(V)$ . In Figure 2, the red triangle with its two edges is a critical multivector $V$ because $H_{1}(\operatorname{\sf cl}(V),\operatorname{\sf mo}(V))$ is nontrivial. Similarly, the gold colored triangles (denoted $\tau$ ) and the green edge (denoted $\sigma$ ) are critical because $H_{2}(\operatorname{\sf cl}(\tau),\partial\tau)$ and $H_{1}(\operatorname{\sf cl}(\sigma),\partial\sigma)$ are nontrivial, where we use $\partial\sigma$ to denote the boundary of a simplex $\sigma$ .

A multivector field over $K$ induces a notion of dynamics. For $\sigma\in K$ , we denote the multivector containing $\sigma$ as $[\sigma]$ . If the multivector field $\mathcal{V}$ is not clear from context, we will use the notation $[\sigma]_{\mathcal{V}}$ . We now use a multivector field $\mathcal{V}$ on $K$ to define a multivalued map $F_{\mathcal{V}}\;:\;K\multimap K$ . In particular, we let $F_{\mathcal{V}}(\sigma):=\operatorname{\sf cl}(\sigma)\cup[\sigma]$ . Such a multivalued map induces a notion of flow on $K$ . In the interest of brevity, for $a,b\in\mathbb{Z}$ , we set $\mathbb{Z}_{[a,b]}=[a,b]\cap\mathbb{Z}$ and define $\mathbb{Z}_{(a,b]}$ , $\mathbb{Z}_{[a,b)}$ , $\mathbb{Z}_{(a,b)}$ as expected. A path from $\sigma$ to $\sigma^{\prime}$ is a map $\rho\;:\;\mathbb{Z}_{[a,b]}\to K$ , where $\rho(a)=\sigma$ , $\rho(b)=\sigma^{\prime}$ , and for all $i\in\mathbb{Z}_{(a,b]}$ , we have that $\rho(i)\in F_{\mathcal{V}}(\rho(i-1))$ . Similarly, a solution to a multivector field over $K$ is a map $\rho\;:\;\mathbb{Z}\to K$ where $\rho(i)\in F_{\mathcal{V}}(\rho(i-1))$ .

Definition 2 (Essential Solution).

A solution $\rho\;:\;\mathbb{Z}\to K$ is an essential solution of multivector field $\mathcal{V}$ on $K$ if for each $i\in\mathbb{Z}$ where $[\rho(i)]$ is regular, there exists an $i^{-},i^{+}\in\mathbb{Z}$ where $i^{-}<i<i^{+}$ and $[\rho(i^{-})]\neq[\rho(i)]\neq[\rho(i^{+})]$ .

For a set $A\subseteq K$ , let $\operatorname{\sf eSol}(A)$ denote the set of essential solutions $\rho$ such that $\rho(\mathbb{Z})\subseteq A$ . If the relevant multivector field is not clear from context, we use the notation $\operatorname{\sf eSol}_{\mathcal{V}}(A)$ . We define the invariant part of $A$ as $\operatorname{\sf Inv}(A)=\{\sigma\in A\;|\;\exists\rho\in\operatorname{\sf eSol}(A),\rho(0)=\sigma\}$ . We say that $A$ is invariant or an invariant set if $\operatorname{\sf Inv}(A)=A$ . If the multivector field is not clear from context, we use the notation $\operatorname{\sf Inv}_{\mathcal{V}}(A)$ .

Solutions and invariant sets are defined in accordance to their counterparts in the classical setting. For more information on the classical counterparts of these concepts, see [3]. As in the classical setting, we have a notion of isolation for combinatorial invariant sets.

Definition 3 (Isolated Invariant Set, Isolating Neighborhood).

An invariant set $A\subseteq N$ , $N$ closed, is isolated by $N$ if all paths $\rho\;:\;\mathbb{Z}_{[a,b]}\to N$ for which $\rho(a),\rho(b)\in A$ satisfy $\rho(\mathbb{Z}_{[a,b]})\subseteq A$ . The closed set $N$ is said to be an isolating neighborhood for $S$ .

2.2 Conley Indices

The Conley index of an isolated invariant set is a topological invariant used to characterize features of dynamical systems [5, 14]. In both the classical and the combinatorial settings, the Conley index is determined by index pairs.

Definition 4.

Let $S$ be an isolated invariant set. The pair of closed sets $(P,E)$ subject to $E\subseteq P\subseteq K$ is an index pair for $S$ if all of the following hold:

1.

$F_{\mathcal{V}}(E)\cap P\subseteq E$
2.

$F_{\mathcal{V}}(P\setminus E)\subseteq P$
3.

$S=\operatorname{\sf Inv}(P\setminus E)$

In addition, an index pair is said to be a saturated index pair if $S=P\setminus E$ . In Figure 3, the gold, critical triangle $\sigma$ is an isolated invariant set. The reader can easily verify that $\left(\operatorname{\sf cl}(\sigma),\operatorname{\sf cl}(\sigma)\setminus\{\sigma\}\right)$ is an index pair for $\sigma$ . In fact, this technique is a canonical way of picking an index pair for an isolated invariant set. This is formalized in the following proposition.

Proposition 5.

[11, Proposition 4.3] Let $S$ be an isolated invariant set. Then $(\operatorname{\sf cl}(S),\operatorname{\sf mo}(S))$ is a saturated index pair for $S$ .

However, there are several other natural ways to find index pairs. Figure 3 shows another index pair for the same gold triangle $\sigma$ . By letting $P:=\operatorname{\sf cl}(\sigma)\cup S_{P}$ and $E:=\operatorname{\sf mo}(\sigma)\cup S_{E}$ , where $S_{P}$ and $S_{E}$ are the set of simplices reachable from paths originating in $\operatorname{\sf cl}(S)$ , $\operatorname{\sf mo}(S)$ respectively, we obtain a much larger index pair. In Figure 3, $P$ is the set of all colored simplices, while $E$ is the set of all colored simplices which are not gold.

In principle, it is important that the Conley index be independent of the choice of index pair. Fortunately, it is also known that the relative homology given by an index pair for an isolated invariant set $S$ is independent of the choice of index pair.

Theorem 6.

[11, Theorem 4.15] Let $(P_{1},E_{1})$ and $(P_{2},E_{2})$ be index pairs for the isolated invariant set $S$ . Then $H_{p}(P_{1},E_{1})\cong H_{p}(P_{2},E_{2})$ for all $p$ .

The Conley Index of an isolated invariant set $S$ in dimension $p$ is then given by the relative homology group $H_{p}(P,E)$ for any index pair of $S$ denoted $(P,E)$ .

3 Conley Index Persistence

We move to establishing the foundations for persistence of the Conley Index. Given a sequence of multivector fields $\mathcal{V}_{1},\mathcal{V}_{2},\ldots,\mathcal{V}_{n}$ on a simplicial complex $K$ , one may want to quantify the changing behavior of the vector fields. One such approach is to compute a sequence of isolated invariant sets $S_{1},S_{2},\ldots,S_{n}$ under each multivector field, and then to compute an index pair for each isolated invariant set. By Proposition 5, a canonical way to do this is to take the closure and mouth of each isolated invariant set to obtain a sequence of index pairs $(\operatorname{\sf cl}(S_{1}),\operatorname{\sf mo}(S_{1})),(\operatorname{\sf cl}(S_{2}),\operatorname{\sf mo}(S_{2})),\ldots,(\operatorname{\sf cl}(S_{n}),\operatorname{\sf mo}(S_{n}))$ . A first idea is to take the element-wise intersection of consecutive index pairs, which results in the zigzag filtration:

(\operatorname{\sf cl}(S_{1}),\operatorname{\sf mo}(S_{1}))\supseteq(\operatorname{\sf cl}(S_{1})\cap\operatorname{\sf cl}(S_{2}),\operatorname{\sf mo}(S_{1})\cap\operatorname{\sf mo}(S_{2}))\subseteq(\operatorname{\sf cl}(S_{2}),\operatorname{\sf mo}(S_{2}))\cdots(\operatorname{\sf cl}(S_{n}),\operatorname{\sf mo}(S_{n}))

Taking the relative homology groups of the pairs in the zigzag sequence, we obtain a zigzag persistence module. We can extract a barcode corresponding to a decomposition of this module:

However, the chance that this approach works in practice is low. In general, two isolated invariant sets $S_{1},S_{2}$ need not overlap, and hence their corresponding index pairs need not intersect. For example, if one were to take the blue periodic solutions in the multivector fields in Figure 1 to be $S_{1}$ , $S_{2}$ , $S_{3}$ , by using Proposition 5 one gets the index pairs $(S_{1},\emptyset)$ , $(S_{2},\emptyset)$ , and $(S_{3},\emptyset)$ (since $\operatorname{\sf cl}(S_{i})=S_{i})$ . Note that in such a case, the intermediate pairs are $(S_{1}\cap S_{2},\emptyset)$ and $(S_{2}\cap S_{3},\emptyset)$ . But $S_{1}\cap S_{2}$ and $S_{2}\cap S_{3}$ intersect only at vertices, so none of the $1$ -cycles persist beyond their multivector field. This is problematic in computing the persistence, because intuitively there should be an $H_{1}$ generator that persists through all three multivector fields. To increase the likelihood that two index pairs intersect, we consider a special type of index pair called an index pair in $N$ .

Definition 7.

Let $S$ be an invariant set isolated by $N$ under $\mathcal{V}$ . The pair of closed sets $(P,E)$ satisfying $E\subseteq P\subseteq N$ is an index pair for $S$ in $N$ if all of the following conditions are met:

1.

$F_{\mathcal{V}}(P)\cap N\subseteq P$
2.

$F_{\mathcal{V}}(E)\cap N\subseteq E$
3.

$F_{\mathcal{V}}(P\setminus E)\subseteq N$ , and
4.

$S=\operatorname{\sf Inv}(P\setminus E)$ .

As is expected, such index pairs in $N$ are index pairs.

Theorem 8.

Let $(P,E)$ be an index pair in $N$ for $S$ . The pair $(P,E)$ is an index pair for $S$ in the sense of Definition 4.

Proof.

Note that by condition three of Definition 7, if $\sigma\in P\setminus E$ , then $F_{\mathcal{V}}(\sigma)\subseteq N$ . Condition one of Definition 7 implies that $F_{\mathcal{V}}(\sigma)\cap N=F_{\mathcal{V}}(\sigma)\subseteq P$ , which is condition two of Definition 4. Likewise, by condition two of Definition 7, if $\sigma\in E$ , then $F_{\mathcal{V}}(\sigma)\cap N\subseteq E$ . Note that $P\subseteq N$ , so it follows that $F_{\mathcal{V}}(\sigma)\cap P\subseteq F_{\mathcal{V}}(\sigma)\cap N\subseteq E$ , which is condition one of Definition 4. Finally, condition four of Definition 7 directly implies condition three of Definition 4. ∎

An additional advantage to considering index pairs in $N$ is that the intersection of index pairs in $N$ is an index pair in $N$ . In general, $(\operatorname{\sf cl}(S_{1})\cap\operatorname{\sf cl}(S_{2}),\operatorname{\sf mo}(S_{1})\cap\operatorname{\sf mo}(S_{2}))$ is not an index pair. However, for index pairs in $N$ , we get the next two results which involve the notion of a new multivector field obtained by intersection. Given two multivector fields $\mathcal{V}_{1}$ , $\mathcal{V}_{2}$ , we define $\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}:=\{V_{1}\cap V_{2}\;|\;V_{1}\in\mathcal{V}_{1},\;V_{2}\in\mathcal{V}_{2}\}$ .

Theorem 9.

Let $(P_{1},E_{1}),(P_{2},E_{2})$ be index pairs in $N$ for $S_{1},S_{2}$ under $\mathcal{V}_{1},\mathcal{V}_{2}$ . The set $\operatorname{\sf Inv}((P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2}))$ is isolated by $N$ under $\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}$ .

Proof.

To contradict, we assume that there exists a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to N$ under $\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}$ where $\rho(a),\rho(b)\in\operatorname{\sf Inv}((P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2}))$ and there exists some $i\in(a,b)\cap\mathbb{Z}$ where $\rho(i)\not\in\operatorname{\sf Inv}((P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2}))$ . Note that by the the definition of an index pair, $F_{\mathcal{V}}(P)\cap N\subseteq P$ . Hence, it follows by an easy induction argument that since $F_{\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}}(\sigma)\subseteq F_{\mathcal{V}_{1}}(\sigma),F_{\mathcal{V}_{2}}(\sigma)$ , we have that $\rho(\mathbb{Z}_{[a,b]})\subseteq P_{1},P_{2}$ . This directly implies that $\rho(\mathbb{Z}_{[a,b]})\subseteq P_{1}\cap P_{2}$ . In addition, it is easy to see that $\rho$ can be extended to an essential solution in $P_{1}\cap P_{2}$ , which we denote $\rho^{\prime}\;:\;\mathbb{Z}\to N$ , by some simple surgery on essential solutions. This is because there must be essential solutions $\rho_{1},\rho_{2}\;:\;\mathbb{Z}\to(P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2})$ where $\rho_{1}(a)=\rho(a)$ and $\rho_{2}(b)=\rho(b)$ , as $\rho(a)$ and $\rho(b)$ are both in essential solutions. Hence, $\rho^{\prime}(x)=\rho_{1}(x)$ if $x\leq a$ , $\rho^{\prime}(x)=\rho(x)$ if $a\leq x\leq b$ , and $\rho^{\prime}(x)=\rho_{2}(x)$ if $b\leq x$ . Since $\rho^{\prime}$ is an essential solution, we have that $\rho(\mathbb{Z}_{[a,b]})\subseteq\operatorname{\sf Inv}(P_{1}\cap P_{2})$ , but also that $\rho(\mathbb{Z}_{[a,b]})\not\subseteq\operatorname{\sf Inv}((P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2}))$ . Therefore, we must have that $\rho(i)\in E_{1}\cap E_{2}$ . But by the same reasoning as before, it follows that $\rho(\mathbb{Z}_{[i,b]})\subseteq E_{1}\cap E_{2}$ . Hence, $b\not\in(P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2})$ , a contradiction. ∎

Theorem 10.

Let $(P_{1},E_{1})$ and $(P_{2},E_{2})$ be index pairs in $N$ under $\mathcal{V}_{1},\mathcal{V}_{2}$ . The tuple $(P_{1}\cap P_{2},E_{1}\cap E_{2})$ is an index pair for $\operatorname{\sf Inv}((P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2}))$ in $N$ under $\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}$ .

Proof.

We proceed by using the conditions in Definition 7 to show that $(P_{1}\cap P_{2},E_{1}\cap E_{2})$ is an index pair in $N$ . Note that $F_{\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}}(P_{1}\cap P_{2})\cap N\subseteq F_{\mathcal{V}_{1}}(P_{1})\cap F_{\mathcal{V}_{2}}(P_{2})\cap N$ , which is immediate by the definition of $F$ and considering $\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}$ . Note that since $(P_{1},E_{1})$ and $(P_{2},E_{2})$ are index pairs in $N$ , we know from Definition 7 that $F_{\mathcal{V}_{1}}(P_{1})\cap N\subseteq P_{1}$ and $F_{\mathcal{V}_{2}}(P_{2})\cap N\subseteq P_{2}.$ Therefore $F_{\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}}(P_{1}\cap P_{2})\cap N\subseteq P_{1}\cap P_{2}$ . This implies the first condition in Definition 7. This argument also implies the second condition by replacing $P$ with $E$ .

Now, we aim to show that $(P_{1}\cap P_{2},E_{1}\cap E_{2})$ satisfies condition three in Definition 7. Consider $\sigma\in(P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2})$ . Without loss of generality, we assume $\sigma\not\in E_{1}$ . Therefore, $\sigma\in P_{1}\setminus E_{1}$ , so $F_{\mathcal{V}_{1}}(\sigma)\subseteq N$ by the definition of an index pair in $N$ . Hence, since $F_{\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}}(\sigma)\subseteq F_{\mathcal{V}_{1}}(\sigma)$ , condition three is satisfied.

Finally, note that $\operatorname{\sf Inv}((P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2}))$ is obviously equal to $\operatorname{\sf Inv}((P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2}))$ , so condition four holds as well. ∎

Hence, if $(P_{i},E_{i})$ are index pairs in $N$ , these theorems gives a meaningful notion of persistence of Conley index through the decomposition of the following zigzag persistence module:

Because of the previous two theorems, when one decomposes the above zigzag module, one is actually capturing a changing Conley index. This contrasts the case where one only considers index pairs of the form $(\operatorname{\sf cl}(S_{i}),\operatorname{\sf mo}(S_{i}))$ , because $(\operatorname{\sf cl}(S_{i})\cap\operatorname{\sf cl}(S_{i+1}),\operatorname{\sf mo}(S_{i})\cap\operatorname{\sf mo}(S_{i+1}))$ need not be an index pair for any invariant set.

As has been established, the pair $(\operatorname{\sf cl}(S),\operatorname{\sf mo}(S))$ is an index pair, but it need not be an index pair in $N$ . We introduce a canonical approach to transform $(\operatorname{\sf cl}(S),\operatorname{\sf mo}(S))$ to an index pair in $N$ by using the push forward.

Definition 11.

The push forward $\operatorname{\sf pf}(S)$ of a set $S$ in $N$ , $N$ closed, is the set of all simplices in $S$ together with those $\sigma\in N$ such that there exists a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to N$ where $\rho(a)\in S$ and $\rho(b)=\sigma$ .

If $N$ is not clear from context, we use the notation $\operatorname{\sf pf}_{N}(S)$ . The next series of results imply that an index pair in $N$ can be obtained by taking the push forward of $(\operatorname{\sf cl}(S),\operatorname{\sf mo}(S))$ .

Proposition 12.

If $S\subseteq K$ is an isolated invariant set with isolating neighborhood $N$ under $\mathcal{V}$ , then $\operatorname{\sf pf}(\operatorname{\sf mo}(S))\cap\operatorname{\sf cl}(S)=\operatorname{\sf mo}(S)$ .

Proof.

Note that by definition, $\operatorname{\sf mo}(S)\subseteq\operatorname{\sf cl}(S)$ and $\operatorname{\sf mo}(S)\subseteq\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ , so it follows that $\operatorname{\sf mo}(S)\subseteq\operatorname{\sf cl}(S)\cap\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ . Hence, it is sufficient to show that $\operatorname{\sf cl}(S)\cap\operatorname{\sf pf}(\operatorname{\sf mo}(S))\subseteq\operatorname{\sf mo}(S)$ . Aiming for a contradiction, assume there exists a $\sigma\in\operatorname{\sf cl}(S)\cap\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ where $\sigma\not\in\operatorname{\sf mo}(S)$ . This directly implies that $\sigma\in\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)$ . But by Proposition 5, $\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)=S$ , so $\sigma\in S$ . But since $\sigma\in\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ , there exists a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to N$ where $\rho(a)\in\operatorname{\sf mo}(S)$ and $\rho(b)=\sigma$ . Because $\rho(a)\in\operatorname{\sf mo}(S)$ , there exists a $\sigma^{\prime}\in S$ such that $\rho(a)\leq\sigma^{\prime}$ . This implies that there exists a path $\rho^{\prime}\;:\;\mathbb{Z}_{[a-1,b]}\to N$ where $\rho(a-1)=\sigma^{\prime}$ and $\rho(b)=\sigma$ , but $\rho(a)\not\in S$ . Hence, $S$ is not isolated by $N$ , a contradiction. ∎

Proposition 13.

If $S\subseteq K$ is an isolated invariant set with isolating neighborhood $N$ under $\mathcal{V}$ , then $\operatorname{\sf pf}(\operatorname{\sf mo}(S))\cup\operatorname{\sf cl}(S)=\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ .

Proof.

Note that since $\operatorname{\sf pf}(\operatorname{\sf mo}(S))\subseteq\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ and $\operatorname{\sf cl}(S)\subseteq\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ , it follows immediately that $\operatorname{\sf pf}(\operatorname{\sf mo}(S))\cup\operatorname{\sf cl}(S)\subseteq\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ . Hence, it is sufficient to show that $\operatorname{\sf pf}(\operatorname{\sf cl}(S))\subseteq\operatorname{\sf pf}(\operatorname{\sf mo}(S))\cup\operatorname{\sf cl}(S)$ . Aiming for a contradiction, we assume there exists a $\sigma\in\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ such that $\sigma\not\in\operatorname{\sf pf}(\operatorname{\sf mo}(S))\cup\operatorname{\sf cl}(S)$ . Note that since $\sigma\not\in\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ , it follows that there must exist a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to N$ where $\rho(i)\not\in\operatorname{\sf mo}(S)$ for all $i$ . Else, $\rho(b)=\sigma$ would be in $\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ , a contradiction. Hence, there must exist a $\rho(i)\in\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)=S$ such that $\rho(i+1)\in F_{\mathcal{V}}(\rho(i))$ and $\rho(i+1)\not\in\operatorname{\sf cl}(S)$ . Note that $\rho(i+1)$ cannot be a face of $\rho(i)$ , else $\rho(i+1)\in\operatorname{\sf cl}(S)$ . Hence, $[\rho(i+1)]=[\rho(i)]$ , which means one can construct a path $\rho^{\prime}\;:\;\mathbb{Z}_{[0,2]}\to N$ where $\rho^{\prime}(0)=\rho^{\prime}(2)=\rho(i)$ and $\rho(1)=\rho(i+1)$ . But this is a path with endpoints in $S$ which is not contained in $S$ , which contradicts $S$ being isolated by $N$ . ∎

Proposition 14.

If $S\subseteq K$ is an isolated invariant set with isolating neighborhood $N$ , then $\operatorname{\sf pf}(\operatorname{\sf cl}(S))\setminus\operatorname{\sf pf}(\operatorname{\sf mo}(S))=\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)=S$ .

Proof.

First, we note that by Proposition 13, we have that $\operatorname{\sf pf}(\operatorname{\sf mo}(S))\cup\operatorname{\sf cl}(S)=\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ . From Proposition 12 we have that $\operatorname{\sf pf}(\operatorname{\sf mo}(S))\cap\operatorname{\sf cl}(S)=\operatorname{\sf mo}(S)$ , which together with the fact that $\operatorname{\sf cl}(S)\subseteq\operatorname{\sf mo}(S)$ implies that $\operatorname{\sf pf}(\operatorname{\sf mo}(S))\cup(\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S))=\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ . Since $\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ and $\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)$ are disjoint, it follows that that $\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)=\operatorname{\sf pf}(\operatorname{\sf cl}(S))\setminus\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ . By Proposition 5, $(\operatorname{\sf cl}(S),\operatorname{\sf mo}(S))$ is a saturated index pair for $S$ , so it follows that $\operatorname{\sf pf}(\operatorname{\sf cl}(S))\setminus\operatorname{\sf pf}(\operatorname{\sf mo}(S))=\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)=S$ . ∎

Crucially, from these propositions we get the following.

Theorem 15.

If $S$ is an isolated invariant set then $(\operatorname{\sf pf}(\operatorname{\sf cl}(S)),\operatorname{\sf pf}(\operatorname{\sf mo}(S)))$ is an index pair in $N$ for $S$ .

Proof.

First, we note that since the index pair $(\operatorname{\sf cl}(S),\operatorname{\sf mo}(S))$ is saturated, it follows that $S=\operatorname{\sf Inv}(\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S))=\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)$ . But since by Proposition 14 $\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S)=\operatorname{\sf pf}(\operatorname{\sf cl}(S))\setminus\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ , it follows that $S=\operatorname{\sf pf}(\operatorname{\sf cl}(S))\setminus\operatorname{\sf pf}(\operatorname{\sf mo}(S))=\operatorname{\sf Inv}(\operatorname{\sf pf}(\operatorname{\sf cl}(S))\setminus\operatorname{\sf pf}(\operatorname{\sf mo}(S)))$ , which satisfies condition four of being an index pair in $N$ .

We show that $F_{\mathcal{V}}(\operatorname{\sf pf}(\operatorname{\sf cl}(S)))\cap N\subseteq\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ . Let $x\in\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ , and assume that $y\in F_{\mathcal{V}}(x)\cap N$ . There must be a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to N$ where $\rho(a)\in\operatorname{\sf cl}(S)$ and $\rho(b)=x$ , by the definition of the push forward. Thus, we can construct an analogous path $\rho^{\prime}\;:\;\mathbb{Z}_{[a,b+1]}\to N$ where $\rho^{\prime}(i)=\rho(i)$ for $i\in\mathbb{Z}_{[a,b]}$ and $\rho^{\prime}(b+1)=y$ . Hence, $y\in\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ by definition. Identical reasoning can be used to show that $F_{\mathcal{V}}(\operatorname{\sf pf}(\operatorname{\sf mo}(S)))\cap N\subseteq\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ , so $(\operatorname{\sf pf}(\operatorname{\sf cl}(S)),\operatorname{\sf pf}(\operatorname{\sf mo}(S)))$ also meets the first two conditions required to be an index pair.

Finally, we show that $F_{\mathcal{V}}(\operatorname{\sf pf}(\operatorname{\sf cl}(S))\setminus\operatorname{\sf pf}(\operatorname{\sf mo}(S)))\subseteq N$ . By Proposition 14, this is equivalent to showing that $F_{\mathcal{V}}(\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S))\subseteq N$ . Since $(\operatorname{\sf cl}(S),\operatorname{\sf mo}(S))$ is an index pair for $S$ , it follows that $F_{\mathcal{V}}(\operatorname{\sf cl}(S)\setminus\operatorname{\sf mo}(S))\subseteq\operatorname{\sf cl}(S)$ . Note that since $N\supseteq S$ is closed, it follows that $\operatorname{\sf cl}(S)\subseteq N$ . Hence, $F_{\mathcal{V}}(\operatorname{\sf pf}(\operatorname{\sf cl}(S))\setminus\operatorname{\sf pf}(\operatorname{\sf mo}(S)))\subseteq N$ , and all conditions for an index pair in $N$ are met. ∎

An example of an index pair induced by the push forward can be seen in Figure 3. Hence, instead of considering a zigzag filtration given by a sequence of index pairs $(\operatorname{\sf cl}(S_{1}),\operatorname{\sf mo}(S_{1}))$ , $(\operatorname{\sf cl}(S_{2}),\operatorname{\sf mo}(S_{2}))$ , …, $(\operatorname{\sf cl}(S_{n}),\operatorname{\sf mo}(S_{n}))$ , a canonical choice is to instead consider the zigzag filtration given by the the sequence of index pairs

(\operatorname{\sf pf}(\operatorname{\sf cl}(S_{1})),\operatorname{\sf pf}(\operatorname{\sf mo}(S_{1}))),(\operatorname{\sf pf}(\operatorname{\sf cl}(S_{2})),\operatorname{\sf pf}(\operatorname{\sf mo}(S_{2}))),\ldots,(\operatorname{\sf pf}(\operatorname{\sf cl}(S_{n})),\operatorname{\sf pf}(\operatorname{\sf mo}(S_{n}))).

Choosing $S_{i}$ is highly application specific, so in our implementation we choose $S_{i}:=\operatorname{\sf Inv}_{\mathcal{V}_{i}}(N)$ . This decision together with the previous theorems gives Algorithm 1 for computing the persistence of the Conley Index.

Input: Sequence of multivector fields

\mathcal{V}_{1},\mathcal{V}_{2},\ldots,\mathcal{V}_{n}

, closed set

N\subseteq K

Output: Barcodes corresponding to persistence of Conley Index

i\leftarrow 1

while $i<=n$ do

S_{i}\leftarrow\operatorname{\sf Inv}_{\mathcal{V}_{i}}(N)

\left(P_{i},E_{i}\right)\leftarrow\left(\operatorname{\sf pf}\left(\operatorname{\sf cl}\left(S_{i}\right)\right),\operatorname{\sf pf}\left(\operatorname{\sf mo}\left(S_{i}\right)\right)\right)

i\leftarrow i+1

end while

return ${\tt zigzagPers}\left(\left(P_{1},E_{1}\right)\supseteq\left(P_{1}\cap P_{2},E_{1}\cap E_{2}\right)\subseteq\left(P_{2},E_{2}\right)\supseteq\ldots\subseteq\left(P_{n},E_{n}\right)\right)$

Algorithm 1 Scheme for computing the persistence of the Conley Index, fixed

N

Index pairs and barcodes computed by Algorithm 1 can be seen in Figure 4.

3.1 Noise-Resilient Index Pairs

The strategy given for producing index pairs in $N$ produces saturated index pairs. Equivalently, the cardinality of $P\setminus E$ is minimized. This is problematic in the presence of noise, where if $\mathcal{V}_{2}$ is a slight perturbation of $\mathcal{V}_{1}$ we frequently have that $\operatorname{\sf Inv}_{\mathcal{V}_{1}}(N)\neq\operatorname{\sf Inv}_{\mathcal{V}_{2}}(N)$ . This gives a perturbation in our generated index pairs and in particular a perturbation in $P\setminus E$ . As the Conley Index is obtained by taking relative homology, taking the intersection of index pairs $(P_{1},E_{2})$ and $(P_{2},E_{2})$ where $P_{i}\setminus E_{i}=\operatorname{\sf Inv}_{\mathcal{V}_{i}}(N)$ can result in a “breaking” of bars in the barcode. An example can be seen in Figure 5, where because of noise, the two $P\setminus E$ do not overlap, and hence a $2$ -dimensional homology class which intuitively should persist throughout the interval does not. In Figure 5, the Conley indices of the invariant sets consisting of the singleton critical triangles in $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ (the left and right multivector fields) have rank $1$ in dimension $2$ because the homology group $H_{2}$ of $P$ (which is the entire complex in both cases) relative to $E$ (which is all pink simplices) has rank $1$ . However, the generators for $H_{2}(P_{1},E_{1})$ and $H_{2}(P_{2},E_{2})$ are both in the intersection field $\mathcal{V}_{1}\overline{\cap}\mathcal{V}_{2}$ . Hence, rather than one generator persisting through all three multivector fields, we get two bars that overlap at the intersection field. The difficulty is rooted in the fact that the sets $W_{1}=P_{1}\setminus E_{1}$ , $W_{2}=P_{2}\setminus E_{2}$ , and $W_{12}=(P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2})$ do not have a common intersection.

To address this problem, we propose an algorithm to expand the size of $P\setminus E$ . It is important to note that a balance is needed to ensure a large $E$ as well as a large $P\setminus E$ . If $E_{1}$ and $E_{2}$ are too small, then it is easy to see that $E_{1}$ and $E_{2}$ may not intersect as expected even though consecutive vector fields are very similar. The following proposition is very useful for computing a balanced index pair.

Proposition 16.

Let $(P,E)$ be an index pair for $S$ in $N$ under $\mathcal{V}$ . If $V\subseteq E$ is a regular multivector where $E^{\prime}:=E\setminus V$ is closed, then $(P,E^{\prime})$ is an index pair for $S$ in $N$ .

Proof.

Note that since $P$ does not change, it is immediate the $P$ satisfies $F_{\mathcal{V}}(P)\cap N\subseteq P$ , and the first condition of an index pair in $N$ is met.

We show that $F_{\mathcal{V}}(E^{\prime})\cap N\subseteq E$ . For a contradiction, we assume that there exists an $x\in E^{\prime}$ such that there is a $y\in F_{\mathcal{V}}(x)\cap N$ , $y\not\in E^{\prime}$ . Note that if $y\leq x$ , then by definition $y$ is in the closure of $x$ . Since $E$ is closed and $x\in E$ , it follows that $y\in E$ . But since $y\not\in E^{\prime}$ , it follows that $y\in V$ . But this is a contradiction since by assumption, $E\setminus V$ is closed. Hence, by definition of $F_{\mathcal{V}}$ , since $y\not\in\operatorname{\sf cl}(x)$ , $y$ and $x$ must be in the same multivector. Note that in such a case $y\not\in V$ , as if it were, $x$ would not be in $E^{\prime}$ since $x$ and $y$ are in the same multivector. This implies that $F_{\mathcal{V}}(E)\cap N\not\subset E$ , as $y\in F_{\mathcal{V}}(x)$ and $y\not\in E$ , a contradiction. Thus, we conclude that $F_{\mathcal{V}}(E^{\prime})\cap N\subseteq E$ .

Now, we show that $F_{\mathcal{V}}(P\setminus E^{\prime})\subseteq N$ . Assume there exists an $x\in P$ satisfying $F_{\mathcal{V}}(x)\setminus N\neq\emptyset$ . Then since $(P,E)$ is an index pair in $N$ , it follows that $x\in E$ . To contradict, we assume $x\not\in E^{\prime}$ . Hence, $x\in V$ . We let $y\in F_{\mathcal{V}}(x)\setminus N$ . By definition of $F_{\mathcal{V}}$ , either $y\in\operatorname{\sf cl}(x)$ or $y$ and $x$ are in the same multivector. In the later case, $y\in N$ as it was assumed that $V\subseteq E$ , a contradiction. Hence, $y\leq x$ . But this implies that $y\in\operatorname{\sf cl}(x)\subseteq E\subseteq N$ , a contradiction. Ergo, $x\in E^{\prime}$ , and $F_{\mathcal{V}}(P\setminus E^{\prime})\subseteq N$ .

Finally, we show that $\operatorname{\sf Inv}(P\setminus E^{\prime})=\operatorname{\sf Inv}(P\setminus E)$ . Trivially, $\operatorname{\sf Inv}(P\setminus E)\subseteq\operatorname{\sf Inv}(P\setminus E^{\prime})$ , so it is sufficient to show that $\operatorname{\sf Inv}(P\setminus E^{\prime})\subseteq\operatorname{\sf Inv}(P\setminus E)$ . For a contradiction, assume that there exists an $x\in\operatorname{\sf Inv}(P\setminus E^{\prime}),x\not\in\operatorname{\sf Inv}(P\setminus E)$ . Thus, there exists an essential solution $\rho\;:\;\mathbb{Z}\to P\setminus E^{\prime}$ where for some $k$ , $y:=\rho(k)\in V$ . Since $V$ is regular, we assume without loss of generality that $z:=\rho(k+1)\not\in V$ . Hence, $z\in\operatorname{\sf cl}(y)$ . In addition, since $z\in\operatorname{\sf cl}(V)$ , we have that $z\in\operatorname{\sf cl}(E)=E$ . Therefore, $z\in E\setminus V=E^{\prime}$ . But $\rho$ is a solution in $P\setminus E^{\prime}$ , a contradiction. ∎

Figure 6 illustrates how enlarging $P\setminus E$ by removing regular vectors as Proposition 16 suggests can help mitigate the effects of noise on computing Conley index persistence. Contrast this example with the example in Figure 5. Denoting $W_{i}=P_{i}\setminus E_{i}$ for $i=1,2$ and $W_{12}=(P_{1}\cap P_{2})\setminus(E_{1}\cap E_{2})$ in both figures, we see that $W_{1}\cap W_{12}\cap W_{2}$ is empty in Figure 5 while in Figure 6 it consists of three critical simplices each marked with a circle. Hence, in Figure 6 a single generator persists throughout the interval, unlike in Figure 5.

3.2 Computing a Noise-Resilient Index Pair

We give a method for computing a noise-resilient index pair by using techniques demonstrated in the previous subsection. Note that by Theorem 15, we have that $(\operatorname{\sf pf}(\operatorname{\sf cl}(S)),\operatorname{\sf pf}(\operatorname{\sf mo}(S)))$ is an index pair for invariant set $S$ in $N$ . Hence, we adopt the strategy of taking $P=\operatorname{\sf pf}(\operatorname{\sf cl}(S))$ and $E=\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ , and we aim to find some collection $R\subseteq E$ so that $(P,E\setminus R)$ remains an index pair in $N$ . Finding an appropriate $R$ is a difficult balancing act: one wants to find an $R$ so that $P\setminus(E\setminus R)$ is sufficiently large, so as to capture perturbations in the isolated invariant set as described in the previous section, but not so large that $E$ is small and perturbations in $E$ are not captured. If $R$ is chosen to be as large as possible, then a small shift in $E$ may results in $(E\setminus R)\cap(E^{\prime}\setminus R^{\prime})$ having a different topology than $E$ or $E^{\prime}$ leading to a “breaking” of barcodes analogous to the case described in the previous section.

Before we give an algorithm for outputting such an $R$ , we first define a $\delta$ -collar.

Definition 17.

We define the $\delta$ -collar of an invariant set $S\subseteq K$ recursively:

1.

The $0$ -collar of $S$ is $\operatorname{\sf cl}(S)$ .
2.

For $\delta>0$ , the $\delta$ -collar of $S$ is the set of simplices $\sigma$ in the $\left(\delta-1\right)$ -collar of $S$ together with those simplices $\tau$ where $\tau$ is a face of some $\sigma$ with a face $\tau^{\prime}$ in the $(\delta-1)$ -collar of $S$ .

For an isolated invariant set $S$ , we will let $C_{\delta}(S)$ denote the $\delta$ -collar of $S$ . Together with Proposition 16, $\delta$ -collars give a natural algorithm for finding an $R$ to enlarge $P\setminus E$ .

Input: Isolated invariant set

S

with respect to

\mathcal{V}

contained in some closed set

N

, Index pair

(P,E)

N

with respect to

\mathcal{V}

\delta\in\mathbb{Z}

Output: List of simplices

R

such that

(P,E\setminus R)

is an index pair for

S

N

R\leftarrow{\tt new\;set()}

vecSet\leftarrow\{\left[\sigma\right]\in\mathcal{V}\;|\;\left[\sigma\right]\subseteq E\cap C_{\delta}(S)\wedge\left[\sigma\right]\cap\partial(E)\neq\emptyset\wedge\left[\sigma\right]\cap\partial(P)=\emptyset\}

vec\leftarrow{\tt new\;queue}()

{\tt appendAll}(vec,vecSet)

while ${\tt size}(vec)>0$ do

\left[\sigma\right]\leftarrow{\tt pop}(vec)

if ${\tt isClosed}(\left(E\setminus R\right)\setminus\left[\sigma\right])\;{\tt and}\;\left[\sigma\right]\subseteq E\setminus R\;{\tt and}\;{\tt isRegular}\left(\left[\sigma\right]\right)$ then

R\leftarrow R\cup\left[\sigma\right]

mouthVecs\leftarrow\{\left[\tau\right]\;|\;\tau\in\operatorname{\sf mo}\left(\left[\sigma\right]\right)\;\wedge\left[\tau\right]\subseteq C_{\delta}(S)\}

{\tt appendAll}(vec,mouthVecs)

end if

end while

return R

Algorithm 2

{\tt findR}(S,P,E,\mathcal{V},\delta)

In particular, we use Algorithm 2 for this purpose.

Theorem 18.

Let $R$ be the output of Algorithm 2 applied to index pair $(P,E)$ in $N$ for isolated invariant set $S$ . The pair $(P,E\setminus R)$ is an index pair for $S$ in $N$ .

Proof.

To contradict, we assume that the $R$ output by Algorithm 2 results in $(P,E\setminus R)$ is not an index pair. We note by inspection of the algorithm that multivectors are removed sequentially, so there exists some first $V$ such that $(P,E\setminus R_{V})$ is an index pair for $S$ in $N$ but $(P,E\setminus\left(R_{V}\cup V\right))$ is not an index pair for $S$ in $N$ , where $R_{V}$ denotes the $R$ variable in Algorithm 2 before $V$ is added to it. By inspection of the algorithm, we observe that since $V$ was added to $R_{V}$ , it must be that $V$ is a regular vector, and $E\setminus\left(R\cup V\right)$ is closed, and $V\subseteq E\setminus R_{V}$ . Proposition 16 directly implies that $\left(P,\left(E\setminus R\right)\setminus V\right)$ is an index pair, which is a contradicton. Hence, there can exist no such $V$ , so it follows that $(P,E\setminus R)$ must be an index pair for $S$ in $N$ . ∎

Hence, Algorithm 2 provides a means by which the user may enlarge $P\setminus E$ . As this algorithm is parameterized, a robust choice of $\delta$ may be application specific. We also include some demonstrations on the effectiveness of using this technique. A real instance of the difficulty can be seen in Figure 7, while the application of Algorithm 2 with $\delta=5$ to solve the problem is found in Figure 8.

4 Tracking Invariant Sets

In the previous section, we established the persistence of the Conley index of invariant sets in consecutive multivector fields which are isolated by a single isolating neighborhood. In this section, we develop an algorithm to “track” an invariant set over a sequence of isolating neighborhoods. A classic example is a hurricane, where if one were to sample wind velocity at times $t_{0},t_{1},\ldots,t_{n}$ , there may be no fixed $N$ which captures the eye of the hurricane at all $t_{i}$ without also capturing additional, undesired invariant sets at some $t_{j}$ .

4.1 Changing the Isolating Neighborhood

Thus far, we have defined a notion of persistence of the Conley Index for some fixed isolating neighborhood $N$ and simplicial complex $K$ . This setting is very inflexible —one may want to incorporate domain knowledge to change $N$ so as to capture changing features of a sequence of sampled dynamics. We now extend the results in Section 3 to a setting where $N$ need not be fixed. Throughout this section, we consider multivector fields $\mathcal{V}_{1},\mathcal{V}_{2},\ldots,\mathcal{V}_{n}$ with corresponding isolated invariant sets $S_{1},S_{2},\ldots,S_{n}$ . In addition, we assume that there exist isolating neighborhoods $N_{1},N_{2},\ldots,N_{n-1}$ where $N_{i}$ isolates both $S_{i}$ and $S_{i+1}$ . We will also require that if $1<i<n$ , the invariant set $S_{i}$ is isolated by $N_{i}\cup N_{i+1}$ . Note that for each $i$ where $1<i<n$ , there exist two index pairs for $S_{i}$ : one index pair $(P_{i}^{(i-1)},E_{i}^{(i-1)})$ in $N_{i-1}$ and another index pair $(P_{i}^{(i)},E_{i}^{(i)})$ in $N_{i}$ . In the case of $i=1$ , there is only one index pair $(P_{1}^{(1)},E_{1}^{(1)})$ for $S_{i}$ . Likewise, in the case of $i=n$ , there is a single index pair $(P_{n}^{(n-1)},E_{n}^{(n-1)})$ .

By applying the techniques of Section 3, we obtain a sequence of persistence modules:

(21)

In the remainder of this subsection, we develop the theory necessary to combine these modules into a single module. Without any loss of generality, we will combine the first modules into a single module, which will imply a method to combine all of the modules into one.

First, we note that by Theorem 6, we have that $H_{p}(P_{2}^{(1)},E_{2}^{(1)})\cong H_{p}(P_{2}^{(2)},E_{2}^{(2)})$ . To combine the persistence modules

(24)

into a single module, it is either necessary to explicitly find a simplicial map which induces an isomorphism $\phi\;:\;H_{p}(P_{2}^{(1)},E_{2}^{(1)})\to H_{p}(P_{2}^{(2)},E_{2}^{(2)})$ , or to construct some other index pair for $S_{2}$ denoted $(P,E)$ such that $P_{2}^{(1)},P_{2}^{(2)}\subset P$ and $E_{2}^{(1)},E_{2}^{(2)}\subset E$ . This would allow substituting both occurrences of $(P_{2}^{(1)},E_{2}^{(1)})$ or $(P_{2}^{(2)},E_{2}^{(2)})$ for $(P,E)$ , and allow the combining of all the modules in Equation 21 into a single module. Since constructing the isomorphism given by Theorem 6 is fairly complicated, we opt for the second approach. First, we define a special type of index pair that is sufficient for our approach.

Definition 19 (Strong Index Pair).

Let $(P,E)$ be an index pair for $S$ under $\mathcal{V}$ . The index pair $(P,E)$ is a strong index pair for $S$ if for each $\tau\in E$ , there exists a $\sigma\in S$ such that there is a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to P$ where $\rho(a)=\sigma$ and $\rho(b)=\tau$ .

Intuitively, a strong index pair $(P,E)$ for $S$ is an index pair for $S$ where each simplex $\tau\in E$ is reachable from a path originating in $S$ . Strong index pairs have the following useful property.

Theorem 20.

Let $S$ denote an invariant set isolated by $N$ , $N^{\prime}$ , and $N\cup N^{\prime}$ under $\mathcal{V}$ . If $(P,E)$ and $(P^{\prime},E^{\prime})$ are strong index pairs for $S$ in $N$ , $N^{\prime}$ under $\mathcal{V}$ , then the pair

(\operatorname{\sf pf}_{N\cup N^{\prime}}\left(P\cup P^{\prime}\right),\operatorname{\sf pf}_{N\cup N^{\prime}}\left(E\cup E^{\prime}\right))

is a strong index pair for $S$ in $N\cup N^{\prime}$ under $\mathcal{V}$ .

Proof.

We proceed by showing that the pair meets the requirements to be a strong index pair in $N\cup N^{\prime}$ . First, note that for all $\sigma\in\operatorname{\sf pf}(P\cup P^{\prime})$ , if $\tau\in F_{\mathcal{V}}(\sigma)\cap\left(N\cup N^{\prime}\right)$ , then it follows by the definition of the push forward that $\tau\in\operatorname{\sf pf}(P\cup P^{\prime})$ . Hence, $F_{\mathcal{V}}\left(\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\right)\cap\left(N\cup N^{\prime}\right)\subset\operatorname{\sf pf}\left(P\cup P^{\prime}\right)$ . Analogous reasoning shows that $F_{\mathcal{V}}\left(\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\right)\cap\left(N\cup N^{\prime}\right)\subset\operatorname{\sf pf}\left(E\cup E^{\prime}\right)$ .

Hence, we proceed to show that $F_{\mathcal{V}}\left(\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\setminus\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\right)\subset N\cup N^{\prime}$ . To contradict, assume that there exists a $\sigma\in\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\setminus\operatorname{\sf pf}\left(E\cup E^{\prime}\right)$ so that there is a $\tau\in F_{\mathcal{V}}\left(\sigma\right)$ where $\tau\not\in N\cup N^{\prime}$ . Since $\sigma\in\operatorname{\sf pf}\left(P\cup P^{\prime}\right)$ , there must exist a $x\in P\cup P^{\prime}$ such that there is a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to N\cup N^{\prime}$ where $\rho(a)=x$ and $\rho(b)=\sigma$ . Without loss of generality, we assume that $x\in P$ . Note that if $\sigma\in P$ , then $\sigma\in P\setminus E$ by our assumption and hence $F_{\mathcal{V}}(\sigma)\subseteq N\subseteq N\cup N^{\prime}$ contradicting our assumption that $F_{\mathcal{V}}(\sigma)\ni\tau\not\in(N\cup N^{\prime})$ . Hence, $\sigma\not\in P$ . Note that since $(P,E)$ is an index pair, we have that $F_{\mathcal{V}}\left(P\setminus E\right)\subset P$ . Hence, there must be some $i$ such that $\rho(i)\in E$ . Hence, $\sigma\in\operatorname{\sf pf}(E\cup E^{\prime})$ , a contradiction. Thus, no such $\sigma$ can exist.

Now, we show that $S=\operatorname{\sf Inv}\left(\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\setminus\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\right)$ . Note that since $S$ is isolated by $N\cup N^{\prime}$ and every $\sigma\in E\cup E^{\prime}$ is reachable by a path originating at $\tau\in S$ , it follows that $\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\cap S=\emptyset$ , else $S$ is not isolated by $N\cup N^{\prime}$ . Hence, this implies that $S\subset\operatorname{\sf Inv}\left(\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\setminus\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\right)$ .

To show that $\operatorname{\sf Inv}\left(\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\setminus\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\right)\subset S$ , we first note that if $x\in\operatorname{\sf pf}\left(P\cup P^{\prime}\right)$ and $x\not\in P\cup P^{\prime}$ , then it follows that $x\in\operatorname{\sf pf}(E\cup E^{\prime})$ . This is because if there exists a $\sigma\in P$ such that there is a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to N\cup N^{\prime}$ which satisfies $\rho(a)=\sigma$ and $\rho(b)=x$ , there must be some $\rho(i)\in E$ . If there is no such $\rho(i)$ , then this contradicts requirement $2$ of Definition 4, which states that $F_{\mathcal{V}}\left(P\setminus E\right)\subset P$ . Hence, by the definition of the push forward, it follows that $x\in\operatorname{\sf pf}\left(E\cup E^{\prime}\right)$ . Thus, it follows that $\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\setminus\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\subset\left(P\cup P^{\prime}\right)\setminus\left(E\cup E^{\prime}\right)$ . Thus, every essential solution in $\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\setminus\operatorname{\sf pf}\left(E\cup E^{\prime}\right)$ is also an essential solution in $\left(P\cup P\right)\setminus\left(E\cup E^{\prime}\right)$ . It remains to be shown that every essential solution in $\left(P\cup P^{\prime}\right)\setminus\left(E\cup E^{\prime}\right)$ is also an essential solution in $P\setminus E$ .

For a contradiction, assume that there exists an essential solution $\rho\;:\;\mathbb{Z}\to\left(P\cup P^{\prime}\right)\setminus\left(E\cup E^{\prime}\right)$ such that there exists an $i$ where $\rho(i)\not\in P\setminus E$ . Note that since $\rho(i)\in\left(P\cup P^{\prime}\right)\setminus\left(E\cup E^{\prime}\right)$ , it follows that $\rho(i)\not\in E$ . Together, these two facts imply that $\rho(i)\not\in P$ . Hence, it follows that $\rho(i)\in P^{\prime}\setminus E^{\prime}$ . We claim in particular that for all $j\in\mathbb{Z}$ , $\rho(j)\in P^{\prime}\setminus E^{\prime}$ . Note that if there exists a $j<i$ such that $\rho(j)\not\in P^{\prime}\setminus E^{\prime}$ , then it follows that $\rho(j)\in P\setminus E$ . Since $(P,E)$ is an index pair, we have that $F_{\mathcal{V}}\left(P\setminus E\right)\subseteq P$ . Thus, there must exist some $k$ where $j<k<i$ where $\rho(k)\in E$ , but this contradicts that $\rho\left(\mathbb{Z}\right)\subseteq\left(P\cup P^{\prime}\right)\setminus\left(E\cup E^{\prime}\right)$ . Hence, no such $j$ exists. Analogous reasoning shows that there is no $j>i$ where $\rho(j)\not\in P^{\prime}\setminus E^{\prime}$ . It follows that $\rho(\mathbb{Z})\subseteq\operatorname{\sf Inv}\left(P^{\prime}\setminus E^{\prime}\right)$ . But this implies that $\rho\left(\mathbb{Z}\right)\subseteq S$ . Note that $(P,E)$ and $(P^{\prime},E^{\prime})$ are both index pairs for $S$ , so it follows that $\rho\left(\mathbb{Z}\right)\subseteq P\setminus E,P^{\prime}\setminus E^{\prime}$ . This contradicts our assumption that $\rho(i)\not\in P\setminus E$ . Thus, there is no such $\rho$ . Hence, we have that $\operatorname{\sf Inv}\left(\left(P\cup P^{\prime}\right)\setminus\left(E\cup E^{\prime}\right)\right)\subseteq\operatorname{\sf Inv}\left(P\setminus E\right),\operatorname{\sf Inv}\left(P^{\prime}\setminus E^{\prime}\right)=S$ , which implies that $\operatorname{\sf Inv}\left(\operatorname{\sf pf}\left(P\cup P^{\prime}\right)\setminus\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\right)\subseteq S$ .

To see that $\left(\operatorname{\sf pf}\left(P\cup P^{\prime}\right),\operatorname{\sf pf}\left(E\cup E^{\prime}\right)\right)$ is a strong index pair, note that there must exist a path from $S$ to every $\sigma\in E\cup E^{\prime}$ , and since $\operatorname{\sf pf}\left(E\cup E^{\prime}\right)$ is the set of simplices $\sigma$ for which there exists a path from $E\cup E^{\prime}$ to $\sigma$ , it follows easily from path surgery that there exists a path from $S$ to $\sigma$ . Hence, $(\operatorname{\sf pf}\left(P\cup P^{\prime}\right),\operatorname{\sf pf}\left(E\cup E^{\prime}\right))$ is a strong index pair for $S$ in $N$ . ∎

Crucially, this theorem gives a persistence module

(26)

where the arrows are given by the inclusion. Note that since these are all index pairs for the same $S$ , it follows that we have $H_{p}\left(P_{2}^{(1)},E_{2}^{(1)}\right)\cong H_{p}\left(\operatorname{\sf pf}\left(P_{2}^{(1)}\cup P_{2}^{(2)}\right),\operatorname{\sf pf}\left(E_{2}^{(1)}\cup E_{2}^{(2)}\right)\right)\cong H_{p}\left(P_{2}^{(2)},E_{2}^{(2)}\right)$ . Hence, we will substitute $H_{p}\left(\operatorname{\sf pf}\left(P_{2}^{(1)}\cup P_{2}^{(2)}\right),\operatorname{\sf pf}\left(E_{2}^{(1)}\cup E_{2}^{(2)}\right)\right)$ into the persistence module. By using the modules in Equation 21, we get a new sequnece of persistence modules

(32)

which can immediately be combined into a single persistence module.

This approach is not without it’s disadvantages, however. Namely, if $(P,E)$ and $(P^{\prime},E^{\prime})$ are index pairs for $S$ in $N$ and $N^{\prime}$ , it requires that $(P,E)$ and $(P^{\prime},E^{\prime})$ are strong index pairs and that $S$ is isolated by $N\cup N^{\prime}$ . Fortunately, the push forward approach to computing an index pair in $N$ gives a strong index pair.

Theorem 21.

Let $S$ be an isolated invariant set where $N$ is an isolating neighborhood for $S$ . The pair $\left(\operatorname{\sf pf}\left(\operatorname{\sf cl}\left(S\right)\right),\operatorname{\sf pf}\left(\operatorname{\sf mo}\left(S\right)\right)\right)$ is a strong index pair in $N$ for $S$ .

Proof.

We note that by Theorem 20, the pair $\left(\operatorname{\sf pf}\left(\operatorname{\sf cl}\left(S\right)\right),\operatorname{\sf pf}\left(\operatorname{\sf mo}\left(S\right)\right)\right)$ is an index pair for $S$ in $N$ . Hence, it is sufficient to show that the index pair is strong. Note that by definition, for all $\sigma\in\operatorname{\sf mo}(S)$ , there exists a $\tau\in S$ such that $\sigma$ is a face of $\tau$ . Hence, $\sigma\in F_{\mathcal{V}}(\tau)$ . Note that $\operatorname{\sf pf}(\operatorname{\sf mo}(S))$ is precisely the set of simplices $\sigma^{\prime}$ for which there exists a path originating in $\operatorname{\sf mo}(S)$ and terminating at $\sigma^{\prime}$ , so it is immediate that there is a path originating in $S$ and terminating at $\sigma$ . Hence, the pair $\left(\operatorname{\sf pf}\left(\operatorname{\sf cl}\left(S\right)\right),\operatorname{\sf pf}\left(\operatorname{\sf mo}\left(S\right)\right)\right)$ is a strong index pair. ∎

Our enlarging scheme given in Algorithm 2 does not affect the strongness of an index pair.

Theorem 22.

Let $R$ be the output of applying Algorithm 2 to the strong index pair $\left(P,E\right)$ in $N$ for $S$ with some parameter $\delta$ . The pair $\left(P,E\setminus R\right)$ is a strong index pair for $S$ in $N$ .

Proof.

Theorem 18 gives that $\left(P,E\setminus R\right)$ is an index pair for $S$ in $N$ , so it is sufficient to show that such an index pair is strong. Note that $P$ does not change, but the strongness of index pairs only requires paths to be in $P$ . Since all paths in $\left(P,E\right)$ are also paths in $\left(P,E\setminus R\right)$ , it follows that $\left(P,E\setminus R\right)$ is a strong index pair in $N$ . ∎

These theorems give us a canonical scheme for choosing invariant sets from a sequence of multivector fields and then computing the barcode of persistence module given in Equation (32). We give our exact scheme in Algorithm 3.

Input: Sequence of multivector fields

\mathcal{V}_{1},\mathcal{V}_{2},\ldots,\mathcal{V}_{n}

, closed set

N_{0}\subset K

\delta\in\mathbb{Z}

Output: Barcodes corresponding to persistence of Conley Index

i\leftarrow 1

while $i<=n$ do

S_{i}\leftarrow\operatorname{\sf Inv}_{\mathcal{V}_{i}}(N_{i-1})

\left(P^{\prime}_{i,1},E^{\prime}_{i,1}\right)\leftarrow\left(\operatorname{\sf pf}_{N_{i-1}}\left(\operatorname{\sf cl}\left(S_{i}\right)\right),\operatorname{\sf pf}_{N_{i-1}}\left(\operatorname{\sf mo}\left(S_{i}\right)\right)\right)

R_{i,1}\leftarrow{\tt findR}(S_{i},P^{\prime}_{i,1},E^{\prime}_{i,1},\mathcal{V},\delta)

\left(P^{(1)}_{i},E^{(1)}_{i}\right)\leftarrow\left(P^{\prime}_{i,1},E^{\prime}_{i,1}\setminus R_{i,1}\right)

N_{i}\leftarrow{\tt find}(S_{i},N_{i-1},\mathcal{V},\delta)

\left(P^{\prime}_{i,2},E^{\prime}_{i,2}\right)\leftarrow\left(\operatorname{\sf pf}_{N_{i}}\left(\operatorname{\sf cl}\left(S_{i}\right)\right),\operatorname{\sf pf}_{N_{i}}\left(\operatorname{\sf mo}\left(S_{i}\right)\right)\right)

R_{i,2}\leftarrow{\tt findR}(S_{i},P^{\prime}_{i,2},E^{\prime}_{i,2},\mathcal{V},\delta)

\left(P^{(2)}_{i},E^{(2)}_{i}\right)\leftarrow\left(P^{\prime}_{i,2},E^{\prime}_{i,2}\setminus R_{i,2}\right)

if $i=1$ then

\left(P_{i},E_{i}\right)\leftarrow\left(P^{(2)}_{i},E^{(2)}_{i}\right)

else if $i=n$ then

\left(P_{i},E_{i}\right)\leftarrow\left(P^{(1)}_{i},E^{(1)}_{i}\right)

else

\left(P_{i},E_{i}\right)\leftarrow\left(\operatorname{\sf pf}_{N_{i-1}\cup N_{i}}\left(P^{(1)}_{i}\cup P^{(2)}_{i}\right),\operatorname{\sf pf}_{N_{i-1}\cup N_{i}}\left(E^{(1)}_{i}\cup E^{(2)}_{i}\right)\right)

end if

i\leftarrow i+1

end while

return ${\tt zigzagPers}\left(\left(P_{1},E_{1}\right)\supseteq\left(P_{1}^{(2)}\cap P_{2}^{(1)},E_{1}^{(2)}\cap E_{2}^{(1)}\right)\subseteq\left(P_{2},E_{2}\right)\supseteq\ldots\subseteq\left(P_{n},E_{n}\right)\right)$

Algorithm 3 Scheme for computing the persistence of the Conley Index, variable

N

The astute reader will notice an important detail about Algorithm 3. Namely, the find function is parameterized by a nonnegative integer $\delta$ , and the function has not yet been defined. In particular, said function must output a closed $N_{i}\supseteq S_{i}$ such that $S_{i}$ is isolated by $N_{i-1}\cup N_{i}$ . An obvious choice is to let $N_{i}:=N_{i-1}$ , but such an approach does not allow one to capture essential solutions that “move” outisde of $N_{i-1}=N_{i}$ as the multivector fields change. We give a nontrivial find function in the next subsection that can be used to capture such changes in an essential solution.

4.2 Finding Isolating Neighborhoods

Given an invariant set $S$ isolated by $N$ with respect to $\mathcal{V}$ , we now propose a method to find a closed, nontrivial $N^{\prime}\subseteq K$ such that $N\cup N^{\prime}$ isolates $S$ . Our method relies heavily on the concept of $\delta$ -collar introduced in Section 3. In fact, we will let $N^{\prime}=C_{\delta}(S)\setminus R$ such that $N\cup N^{\prime}$ isolates $S$ . Hence, it is sufficient to devise an algorithm to find $C_{\delta}(S)\setminus R$ . Before we give and prove the correctness of the algorithm, we briefly introduce the notion of the push backward of some set $S$ in $N$ , denoted $\operatorname{\sf pb}_{N}(S)$ . We let $\operatorname{\sf pb}_{N}(S)=\{x\in N\;|\;\exists\;\rho\;:\;\mathbb{Z}_{[a,b]}\to N,\;\rho(a)=x,\rho(b)\in S\}$ . Essentially, the push backward of $S$ in $N$ is the set of simplices $\sigma\in N$ for which there exists a path in $N$ from $\sigma$ to $S$ .

Input: Invariant set

S

isolated by

N

under

\mathcal{V}

\delta\in\mathbb{Z}

Output: Closed set

N^{\prime}\supseteq S

such that

N\cup N^{\prime}

isolates

S

under

\mathcal{V}

V\leftarrow{\tt new\;stack()}

R\leftarrow{\tt new\;set()}

pb\leftarrow\operatorname{\sf pb}_{N}(S)

foreach $\sigma\in C_{\delta}(S)\cup N$ do

{\tt setUnvisited}(\sigma)

end foreach

foreach $\sigma\in S$ do

adj\leftarrow\operatorname{\sf cl}(\sigma)\cup\left[v\right]_{\mathcal{V}}

foreach $\tau\in adj$ do

if $\tau\not\in S\;{\tt and}\;\tau\in C_{\delta}(S)\cup N$ then

{\tt push}(V,\tau)

end if

end foreach

while ${\tt size}(V)>0$ do

v\leftarrow{\tt pop}(V)

if $!{\tt hasBeenVisited}(v)$ then

{\tt setVisited}(v)

if $\left(\operatorname{\sf cl}\left(v\right)\cup\left[v\right]_{\mathcal{V}}\right)\cap pb\neq\emptyset$ then

{\tt add}(R,v)

cf\leftarrow{\tt cofaces}(v)

{\tt addAll}(R,cf)

else

foreach $\sigma\in\left(\operatorname{\sf cl}\left(v\right)\cup\left[\sigma\right]_{\mathcal{V}}\right)\cap\left(C_{\delta}(S)\cup N\right)$ do

{\tt push}(V,\sigma)

end foreach

end if

end while

return $C_{\delta}(S)\setminus R$

Algorithm 4

{\tt find}(S,N,\mathcal{V},\delta)

We now prove that $N\cup\left(C_{\delta}(S)\right)\setminus R$ isolates $S$ . Note that since $S\subseteq C_{\delta}(S)\setminus R$ , this also implies that $C_{\delta}(S)\setminus R$ isolates $S$ .

Theorem 23.

Let $S$ denote an invariant set isolated by $N\subseteq K$ under $\mathcal{V}$ . If $C_{\delta}(S)\setminus R$ is the output of Algorithm 4 on inputs $S,N,\mathcal{V},\delta$ , then the closed set $N\cup\left(C_{\delta}(S)\setminus R\right)$ isolates $S$ .

Proof.

For a contradiction, assume that there exists a path $\rho\;:\;\mathbb{Z}_{[a,b]}\to N\cup\left(C_{\delta}(S)\setminus R\right)$ so that $\rho(a),\rho(b)\in S$ where there is an $i$ satisfying $a<i<b$ with $\rho(i)\not\in S$ . Note that since $N$ isolates $S$ , if $N\cup C_{\delta}(S)\setminus R$ does not isolate $S$ , then there must exist a first $k\in\mathbb{Z}_{[a,b]}$ such that $\rho(k)\in C_{\delta}(S)\setminus N$ and $F_{\mathcal{V}}(\rho(k))\cap\operatorname{\sf pb}_{N}(S)\neq\emptyset$ . If this were not the case, then $N$ would not isolate $S$ . Without loss of generality, we assume that for all $a<j<k$ , we have that $\rho(j)\not\in S$ . Note that for all $j\in\mathbb{Z}_{[a+1,k-1]}$ , when $\rho(j)$ is removed from the stack $V$ , if $\rho(j+1)$ has not been visited, then $\rho(j+1)$ is added to the stack. Hence, this implies that if any $\rho(j)$ is visited, then $\rho(k)$ will be added to $R$ . If this were not the case, there would exist some $\rho(j)$ such that when $\rho(j)$ was removed from the stack, $\rho(j+1)$ was not visited and was not added to the stack. This implies that $F_{\mathcal{V}}(\rho(j))\cap\operatorname{\sf pb}_{N}(S)\neq\emptyset$ , which contradicts $\rho(k)$ being the first such simplex in the path.

Hence, since $\rho(a+1)$ is added to the stack, it follows that $\rho(k)$ is added to $R$ , which implies that $\rho\left(\mathbb{Z}_{[a,b]}\right)\not\subset N\cup\left(C_{\delta}(S)\setminus R\right)$ . Note too that $N\cup C_{\delta}(S)\setminus R$ must be closed, as if there is a $\sigma\in N$ such that $\rho(k)\leq\sigma$ , then $\rho(k)\in N$ because $N$ is closed, a contradiction. But when $\rho(k)$ is removed from $C_{\delta}(S)$ , any of its cofaces which are in $C_{\delta}(S)$ are also removed. Hence, $N$ is closed, $C_{\delta}(S)\setminus R$ is closed, so their union must be closed. ∎

Hence, we use Algorithm 4 as the find function in our scheme given in Algorithm 3. We give an example of our implementation of Algorithm 3 using the ${\tt find}$ function in Figure 9.

5 Conclusion

In this paper, we focused on computing the persistence of Conley indices of isolated invariant sets. Our preliminary experiments show that the algorithm can effectively compute this persistence in the presence of noise. It will be interesting to derive a stability theory for this persistence. Toward that direction, the stability result for the isolated invariant sets presented in Appendix A seems promising. In designing the tracking algorithm, we have made certain choices about the isolated neighborhoods and the invariant sets. Are there better choices? Which ones work better in practice? A thorough investigation with data sets in practice is perhaps necessary to settle this issue.

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Appendix A Stability of invariant sets

We now establish a result on the stability of invariant sets under perturbations to the underlying multivector field. Note that since multivector fields are discrete, metrics on multivector fields over a given simplicial complex must likewise be discrete. Hence, rather than defining a metric on the space of multivector fields for some simplicial complex $K$ , we consider a topology on the space of multivalued maps induced by multivector fields on $K$ . To define the topology on this space, we first define a relation on the set of multivector fields. In particular, if $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ are multivector fields on simplicial complex $K$ with the property that for all $V_{1}\in\mathcal{V}_{1}$ there exists a $V_{2}\in\mathcal{V}_{2}$ where $V_{1}\subset V_{2}$ . We say that $V_{1}$ is inscribed in $V_{2}$ and $\mathcal{V}_{1}$ is a refinement of $\mathcal{V}_{2}$ . We follow the notation in [6] and denote this relation as $\mathcal{V}_{1}\sqsubset\mathcal{V}_{2}$ .

Unfortunately, there is no clear notion of stability between invariant sets under multivector field refinement. This is because if $\mathcal{V}_{1}\sqsubset\mathcal{V}_{2}$ , there can exist $\sigma\in K$ such that $\left[\sigma\right]_{\mathcal{V}_{1}}$ is critical while $\left[\sigma\right]_{\mathcal{V}_{2}}$ is not, or vice versa. Hence, we consider the notion of a strong refinement. The multivector field $\mathcal{V}_{1}$ is a strong refinement of $\mathcal{V}_{2}$ if $\mathcal{V}_{1}$ is a refinement of $\mathcal{V}_{2}$ and for each regular $V_{2}\in\mathcal{V}_{2}$ , we have that $\operatorname{\sf eSol}_{\mathcal{V}_{1}}(V_{2})=\emptyset$ . We use the symbol $\mathcal{V}_{1}\overline{\sqsubset}\mathcal{V}_{2}$ to denote the strong refinement relation. In addition, if $\mathcal{V}_{1}\overline{\sqsubset}\mathcal{V}_{2}$ , then we write that $F_{\mathcal{V}_{1}}\overline{\subset}F_{\mathcal{V}_{2}}$ . This notation is motivated by the fact that the definition of $F_{\mathcal{V}}$ established in Section 2 implies that $F_{\mathcal{V}_{1}}(\sigma)\subset F_{\mathcal{V}_{2}}(\sigma)$ for all $\sigma\in K$ .

We use the relation $\overline{\subset}$ to define a partial order on the space of dynamics

F_{K}=\{F_{\mathcal{V}}\,|\,{\mathcal{V}}\mbox{ is on }K\}.

In particular, if $F_{\mathcal{V}_{1}}\overline{\subset}F_{\mathcal{V}_{2}}$ , we write $F_{\mathcal{V}_{2}}\leq_{F}F_{\mathcal{V}_{1}}$ .

Proposition 24.

The relation $\leq_{F}$ is a partial order.

Proof.

Note that for every multivector field $\mathcal{V}\overline{\sqsubset}\mathcal{V}$ , so it follows that for all $F_{\mathcal{V}}\in F_{K}$ , we have that $F_{\mathcal{V}}\leq_{F}F_{\mathcal{V}}$ . Hence, $\leq_{F}$ is reflexive. Similarly, if $\mathcal{V}_{1}\overline{\sqsubset}\mathcal{V}_{2}$ and $\mathcal{V}_{2}\overline{\sqsubset}\mathcal{V}_{1}$ , then it follows that for $V_{1}\in\mathcal{V}_{1}$ there exists a $V_{2}\in\mathcal{V}_{2}$ such that $V_{1}\subset V_{2}$ . But likewise, there must exist a $V^{\prime}_{1}\in\mathcal{V}_{1}$ such that $V_{2}\subset V^{\prime}_{1}$ . But a multivector is a partition, and $V_{1}\subset V_{2}\subset V^{\prime}_{1}$ . Hence, $V^{\prime}_{1}\cap V_{1}\neq\emptyset$ , so $V^{\prime}_{1}=V_{1}$ . Thus, $V_{1}\subset V_{2}$ and $V_{2}\subset V_{1}$ , so $\mathcal{V}_{1}=\mathcal{V}_{2}$ which implies that $F_{\mathcal{V}_{1}}=F_{\mathcal{V}_{2}}$ . Thus, $\leq_{F}$ is antisymmetric.

Finally, we show that $\leq_{F}$ is transitive. Note that if $F_{\mathcal{V}_{1}}\leq F_{\mathcal{V}_{2}}$ and $F_{\mathcal{V}_{2}}\leq F_{\mathcal{V}_{3}}$ , we have that $F_{\mathcal{V}_{3}}\overline{\subset}F_{\mathcal{V}_{2}}$ and that $F_{\mathcal{V}_{2}}\overline{\subset}F_{\mathcal{V}_{1}}$ . Hence, since $\mathcal{V}_{1},\mathcal{V}_{2},\mathcal{V}_{3}$ are all defined on the same $K$ , we have that for $V_{1}\in\mathcal{V}_{1}$ , there exists a $V_{2}\in\mathcal{V}_{2}$ such that $V_{1}\subset V_{2}$ . Similarly, there exists a $V_{3}\in\mathcal{V}_{3}$ such that $V_{2}\subset V_{3}$ . Hence, $\mathcal{V}_{1}$ is a refinement of $\mathcal{V}_{3}$ , but it is not obvious that this is a strong refinement. Note that $\mathcal{V}_{2}$ is a strong refinement of $\mathcal{V}_{3}$ , so for every $V_{3}\in\mathcal{V}_{3}$ , we have that $\operatorname{\sf eSol}_{\mathcal{V}_{2}}(V_{3})=\emptyset$ . We aim to show that $\operatorname{\sf eSol}_{\mathcal{V}_{1}}(V_{3})=\emptyset$ . Aiming for a contradiction, assume that $\operatorname{\sf eSol}_{\mathcal{V}_{1}}(V_{3})\neq\emptyset$ . In particular, let $\rho\in\operatorname{\sf eSol}_{\mathcal{V}_{1}}(V_{3})$ denote such an invariant set. Note that $\rho(\mathbb{Z})\not\subset V_{2}$ , where $V_{2}\subset V_{3}$ . If the image of $\rho$ were in some $V_{2}$ , then this contradicts $\mathcal{V}_{1}$ being a strong refinement of $\mathcal{V}_{2}$ . In addition, there cannot exist an $a$ such that $\rho\left(\mathbb{Z}_{[-\infty,a]}\right)\subset V_{2}$ , as this also contradicts $\mathcal{V}_{1}$ being a strong refinement. There likewise can’t be an $a$ so that $\rho\left(\mathbb{Z}_{[a,\infty]}\right)\subset V_{2}$ . Hence, for all $i$ , there exists $j,k$ satisfying $j<i<k$ and $\left[\rho(j)\right]_{\mathcal{V}_{2}}\neq\left[\rho(i)\right]_{\mathcal{V}_{2}}\neq\left[\rho(k)\right]_{\mathcal{V}_{2}}$ . This implies that $\rho\in\operatorname{\sf eSol}_{\mathcal{V}_{2}}(V_{3})$ , a contradiction. ∎

Given a poset $(X,\leq_{X})$ , a set $U\subset X$ is said to be upper if for all $x\in X,u\in U$ where $u\leq x$ , we have that $x\in U$ . Lower sets are defined as is expected. It is well known that upper sets induce a topology.

Definition 25.

The Alexenadrov topology on a poset $(X,\leq_{X})$ is the topology on $X$ in which all upper sets with respect to $\leq_{X}$ are open.

In addition, it is well known that lower sets correspond to closed sets. Alexenandrov topologies have several convenient properties not found in arbitrary topological spaces. Notably, in the Alexandrov topology on $X$ , every point in $X$ has a minimal neighborhood.

Throughout the remainder of this section, we consider the Alexandrov topology on $(F_{K},\leq_{F})$ . In particular, we show that invariant sets are strongly upper semicontinuous with respect to this topology. Since we are only dealing with finite spaces, we use a definition from [2].

Definition 26 (Strongly Upper Semicontinuous).

Let $F\;:\;X\multimap Y$ denote a multivalued map from a $T_{0}$ topological space $X$ to some set $Y$ . The map $F$ is strongly upper semicontinuous if for all $x_{1},x_{2}\in X$ where $x_{1}\leq x_{2}$ , $F(x_{1})\subseteq F(x_{2})$ .

Let $\operatorname{\sf Inv}=\{\operatorname{\sf Inv}(F_{\mathcal{V}},N)\;|\;F_{\mathcal{V}}\in F_{K}\}$ denote the set of invariant sets induced by multivector fields over $K$ in some fixed $N$ . As each $F_{\mathcal{V}}$ defines an invariant set, we consider the map $p\;:\;F_{K}\to\operatorname{\sf Inv}$ which maps each $F_{\mathcal{V}}$ to its invariant part in $N$ . We conclude this subsection by showing that $p$ is strongly upper semicontinous.

Proposition 27.

Let $F_{\mathcal{V}_{1}}\leq_{F}F_{\mathcal{V}_{2}}$ denote multivalued maps induced by multivector fields $\mathcal{V}_{2}\overline{\sqsubset}\mathcal{V}_{1}$ on some simplicial complex $K$ . For any closed $N\subset K$ , $\operatorname{\sf Inv}(F_{\mathcal{V}_{2}},N)\subset\operatorname{\sf Inv}(F_{\mathcal{V}_{1}},N)$ .

Proof.

For contradiction, assume that there exists a solution $\rho\;:\;\mathbb{Z}\to N$ where $\rho\in\operatorname{\sf Inv}(F_{\mathcal{V}_{2}},N)$ but $\rho\not\in\operatorname{\sf Inv}(F_{\mathcal{V}_{1}},N)$ . This implies that $\rho$ is not essential with respect to $\mathcal{V}_{1}$ , so there must exist some $a\in\mathbb{Z},V\in\mathcal{V}_{1}$ , $V$ regular such that $\rho\left(\left[a,\infty\right)\right)\subset V$ or $\rho\left(\left(-\infty,-a\right]\right)\subset V$ . But this implies that $\operatorname{\sf eSol}_{\mathcal{V}_{1}}(V)\neq 0$ , which contradicts $\mathcal{V}_{2}$ being a strong refinement of $\mathcal{V}_{1}$ . ∎

Corollary 28.

The map $p$ is strongly upper semicontinuous.