Random templex encodes topological tipping points in noise-driven chaotic dynamics Open Access

Branched manifolds underlying chaotic attractors have topological properties that remain invariant in a deterministic framework and that can be characterized using homologies.^1,2 A more complete description is obtained if the cell complex whose homologies are computed is endowed with a directed graph (digraph) that prescribes cell connections in terms of the flow direction. Such a topological description is given by a templex, which carries the information of the structure of the branched manifold, as well as information on the flow.³ This work revisits the templex concept in a stochastic framework. Stochastic attractors in the pullback approach—like the LOrenz Random Attractor (LORA)⁴—include sharp transitions in a Branched Manifold Analysis through Homologies (BraMAH).^5–7 These sharp transitions can be suitably described using what we call here a random templex, computed from a sequence of BraMAH cell complexes and a digraph. The BraMAH cell complexes are such that changes can be followed in terms of how the generators of the homology groups, i.e., the “holes” of these complexes, evolve. The nodes of the digraph are the generators of the homology groups, and its directed edges indicate the correspondence between holes from one snapshot to the next. Topological tipping points can be identified with the creation, destruction, splitting, or merging of holes, through a definition in terms of the nodes in the digraph.

The topological characterization of noise-driven chaos is a challenging issue that is crucial in the understanding of complex systems, where part of the dynamics remains unresolved and is modeled as noise. While additive noise in a system of equations will blur the topological structure, multiplicative noise may radically change it, as shown by Charó et al.⁷ These authors extended the concept of a branched manifold to account for the integer-dimensional set in phase space that robustly supports the system’s invariant measure at each instant.

Such a branched manifold, however, does not contain—as does its deterministic counterpart—any information about the future or the past of the invariant measure. In other words, the evolution of the system is not completely described by the branched manifold, which is now itself time-dependent. The latter requires, therefore, additional information for its complete description.

The templex was introduced in the realm of deterministic attractors in order to provide more topological information than that contained in a cell complex.³ This missing information concerns the flow around the branched manifold. This information can be spelled out using a digraph⁸ that connects the cells of the complex according to the flow.

But what about the flow in a cell complex representing the invariant measure of a random attractor? One could try to pose it in terms of connections between cells of different cell complexes, but algebraic topology definitions are such that the number and distribution of cells in a cell complex are arbitrary. It is not the individual cells, but the topological properties of each cell complex that characterize the changes from one instant to the next. Such topological properties are encoded by the generators of the homology groups of the cell complex and also by the torsion groups. Herein, we concentrate on the homology groups exclusively and leave torsions for future work. The properties of interest are independent of the particular cell decomposition that is adopted to build the cell complex from data. Homologies hence enable us to connect a cell complex of a random attractor at a certain instant, with a cell complex corresponding to a different instant.

Homological properties can be computed at distinct times, and they can also be tracked across sufficiently small time intervals, thus helping us detect sudden changes in the topology. The information on the mapping of the holes at a certain instant to the holes at the next time step can be encoded using a digraph. This naturally leads to the definition of a “random templex” as the mathematical object that condenses the topological information regarding the evolution of the system’s invariant measure in a finite time window. Topological tipping points (TTPs) are contained in a random templex in the form of creation, splitting, merging, and destruction of generators or, equivalently, in certain simple characteristics of the nodes of the digraph associated with a sequence of cell complexes.

This paper provides the theoretical background that leads to the concept of a random templex and shows how to compute it for the Lorenz Random Attractor (LORA). A brief explanation concerning templexes in a deterministic and autonomous framework appears in Sec. II. The construction of a random templex in a nonautonomous and stochastically perturbed framework is discussed in Sec. III and illustrated by application to LORA. Finally, in Sec. IV, using the LORA templex in Sec. III, we define rigorously the TTPs introduced in a merely intuitive fashion by Charó et al.⁷ and compute them for LORA in a time window that contains the various types of TTPs. Section V contains conclusions and perspectives, including possible applications to the effects of global change on climatic subsystems, often referred to recently as tipping elements.⁹ 

Four appendixes clarify the concepts and associated methodologies of pullback attractors, in which we define pullback, as opposed to forward, attraction ( Appendix A); random attractors, in which the time-dependent forcing is random, as opposed to deterministic ( Appendix B); sample measures and their evolution in time ( Appendix C); and computing topological features from a templex, on the algebraic calculations involved in determining the homologies and the flow on a random templex ( Appendix D).

Homology theory allows us to classify manifolds in terms of a small number of topological properties.¹⁰ In order to study the topological structure of solutions of deterministic dynamical systems, the characterization must be done from a set of discretely sampled trajectories, i.e., a set of points with as many coordinates as time-dependent variables. In the case of the Lorenz¹¹ chaotic attractor, for instance, the points have three coordinates; we refer hereafter to this model as L63. The phase space is three dimensional, but the points lie on a butterfly shaped surface. How can the topological properties of this surface be computed from the set of points? This is achieved by building a cell complex.

A cell complex $K$ is a layered structure formed by a set of $k$-cells with $k=0,1,…,d$⁠. Each cell stands for an Euclidean closed set with a certain dimension: points are $0$-cells, segments are $1$-cells, filled polygons are $2$-cells, and so forth. The highest cell dimension defines the dimension of the complex.

There exist methods to build a complex from a point cloud that are quite different. Our choice will be to build a BraMAH complex, i.e., a cell complex whose cells are formed gathering subsets of points which can be locally approximated by a $d$-disk, with $d$ being the local dimension of the underlying manifold. Further details of this procedure can be found in Sciamarella and Mindlin.⁶ A BraMAH complex for the Lorenz attractor is shown in Fig. 1. Its highest dimensional cell is a 2-cell and therefore $d=2$⁠. The polygon figures forming the 2-cells “pave” the surface of the attractor. The number of cells used to pave this surface is not fixed. It depends on the criteria used to define the sets of points. But this number has no special significance, because the homologies finally obtained will not depend on the particular size or distribution of the cells.¹² 

In order to make homologies computable, $k$-cells with $k>1$ must have a “direction” or “orientation.” In a $1$-cell, this direction can be denoted by an arrow. For instance, if $⟨1,2⟩$ stands for instructions to travel along the path from the $0$-cell $⟨1⟩$ to the $0$-cell $⟨2⟩$⁠, the expression $−⟨1,2⟩$ has a natural meaning: it means traveling in the opposite direction, i.e., from $⟨2⟩$ to $⟨1⟩$⁠. A direction or orientation can also be assigned to $2$-cells: clockwise or counterclockwise. A $2$-complex $K$ is said to be directed if each $1$-cell and $2$-cell is assigned a direction.

A directed complex leads to an algebra of chains. A $k$-chain is a formal linear combination of the $k$-cells in a cell complex. The boundary of a directed $2$-cell, for instance, is the chain formed by the $1$-cells on its boundary, with a positive sign if the direction of the edge is consistent with the direction of the $2$-cell, and with a negative sign otherwise. It is, thus, the linear combination of $1$-cells forming its boundary, while using the signs in accordance with the arrows. All $k$-chains in a complex form an abelian group.¹² 

The purpose of introducing these algebraic concepts is to come up with something that will distinguish the important characteristics of a given cell complex $K$⁠. The chains of $1$-cells forming loops that are not the boundaries of the $2$-cells will be important, as well as a chain of $2$-cells enclosing a cavity, as in a torus or a sphere. These features are summarized by the homology groups of $K$⁠, whose generators identify the “holes” at level $k$ of the cell complex, namely, $H k= [ h 1 k , … , h β k k ]$⁠, where $h i k$ is the $i$th $k$-hole and $β k$ is the Betti number, which counts the number of $k$-holes. It was Poincaré who proved the invariance of these numbers for a given set, independently of the details of the construction of a cell complex for the set. Moreover, $β k=0$ for $k≥n$⁠, where $n$ is the dimension of the space in which the set is embedded.^10,13

The significance of such homology groups can be understood level by level. At level $0$⁠, we have $H 0(K)$⁠, which contains the $0$-holes that identify the disconnected pieces of the cell complex $K$⁠. L63 has a single connected component and therefore $β 0=1$⁠.

Stepping up to $k=1$⁠, $H 1(K)$ lists the 1-generators, which are associated with the non-trivial loops or $1$-holes in the complex. When the manifold underlying a complex has “handles,” the $1$-generators encircle these handles forming closed sequences of $1$-cells that can be traveled through sequentially. In L63, there are two such handles, $β 1=2$⁠, one in each wing of the butterfly, as shown in Fig. 2.

These generators need not strictly contour the boundaries of the geometrical hole. A generator may wander around a hole, without tightly encircling the empty space. But the tight holes encircling handles can be retrieved algebraically, from the 2-complex itself, as long as it is uniformly oriented, i.e., the orientation of the 2-cells is propagated so that shared borders (1-cells) of two adjacent 2-cells are canceled out when the borders of all the 2-cells of the complex $K$ are summed.

A graphic example comparing generators and minimal holes for the BraMAH complex of the L63 attractor is given in Figs. 2(a) and 2(b). Comparison of the two panels shows that the 1-generator in the right wing is minimal, while the 1-generator in the left wing is not.

Passing on to the next layer, generators of $H 2$ identify empty volumes enclosed by a surface, here called $2$-holes. This is achieved by looking for chains of $2$-cells forming the set of polygons that enclose the cavity. As there are no cavities enclosed by polygons in the L63 attractor, $β 2=0$⁠, i.e., there are no $2$-holes.

What does a cell complex lack in order to fully characterize an attractor in phase space? The cell complex describes the shape of the attractor, but not the dynamics of the flow on the attractor. This missing information can be brought into the description by indicating the order in which the cells are visited by the flow. This can be done using a directed graph defined so that its nodes represent the highest dimensional cells of the complex—the $2$-cells in the case of a $2$-complex.

Two nodes are linked by a directed edge if the flow connects the cells in a given order. In the directed graph for L63 shown in Fig. 3, there is an edge connecting $γ 7$ to $γ 8$ and $γ 13$ to $γ 8$ because trajectories can flow from the $2$-cells $γ 7$ and $γ 13$ to the $2$-cell $γ 8$⁠.

This leads to the definition of a dual object formed by a cell complex $K$ and its digraph $G$⁠. Charó et al.³ introduced this novel type of object and called it a templex, a contraction of template + complex.

The properties of a templex that characterize both the shape of the branched manifold and the flow upon it can be derived using the combined properties of the cell complex and the digraph. The result is expressed in terms of what we call stripexes. They have the same role and inherit their name from strips in templates.¹⁴ Stripexes identify the nonequivalent ways of circulating around the attractor according to the flow on it. The L63 attractor has four stripexes, which can be computed looking for the cycles in the digraph and retaining only the nonredundant ones. For further details on how to do these computations, the reader is referred to Charó et al.³ 

In a deterministic framework, the topological structure of an attractor can be accurately described by a templex because the branched manifold is invariant. But this is not the case for a random attractor. As shown by Charó et al.,⁷ the concept of branched manifold and cell complex can still be of help in the stochastic context. Section III will discuss how the templex theory can be extended to deal with a time-varying topological structure.

In a stochastic framework, ensembles of trajectories driven by the same noise path $ω t$ can be tracked and provide considerable insight into the overall behavior of the system, as shown by Chekroun et al.^4,15 and by Tél and associates^16,17 in the climate sciences and by a vast literature on random dynamical systems.^18,19 This pullback approach cancels out the well-known smoothing effect of noise and makes the fractal structure of the noise-driven chaotic dynamics emerge.

In the pullback approach, one is interested in the time-dependent sample measures $μ t$ (see  Appendix C) that are driven by the same noise realization $ω$⁠, when starting from any bounded compact set of initial data at a time $s$ in the remote past and watching them converge at times $t$ as $s→−∞$⁠.

The point clouds that are obtained in this way show the time-dependent stretching and folding mechanisms caused by the noise-driven nonlinearities. These point clouds evolve, combining the smoothness of the L63 deterministic convection with sudden deformations of the pullback attractor’s support. In the stochastic setting, these pullback attractors are called random attractors^18,19 or snapshot attractors.^17,20 See  Appendixes A and  B for details.

The stochastic Lorenz system (2) can be modeled using an RDS $(φ, θ t)$⁠; see  Appendix B for additional details on RDSs. The driving system $θ t$ is defined using the Wiener shift, as in Caravallo, Łukaszewicz, and Real,²¹ and modeled using normally distributed random numbers. At time $t=NΔt$⁠, where $Δt=0.005$ is the time step and $N$ is the number of time steps, $θ t$ is modeled using MATLAB’s function $r a n d n(N,3)$⁠. Solutions of this stochastic model generate data sets in the form of three-dimensional point clouds.

The structure of a random attractor is captured by its sample measure $μ ^ t$⁠; see  Appendix C. A numerical estimation $μ ^ t$ of the time-dependent measure $μ t$ at time $t$ can be computed by letting a large set of $N 0$ initial points “flow” in phase space from the remote past until time $t$⁠, for a fixed noise realization $ω$⁠. In practice, the convergence of the $μ t$’s approximation is observed for a set of $N 0= 10 8$ initial points, as in Chekroun et al.⁴ 

Each initial point is forward-integrated following the stochastic Lorenz system with all trajectories starting at the same initial time. Each point in the random attractor at instant $t$ is associated with a value of $μ ^ t$ that is obtained by averaging over a volume encompassing that point: higher or lower $μ ^ t$-values correspond to more or less populated regions of the random attractor.

Topological characterization of a snapshot using Branched Manifold Analysis through Homologies (BraMAH) is not straightforward, as already discussed in Charó et al.⁷ In order to characterize the topology of the evolving structure that is observed in phase space, we assume that the point cloud’s most populated regions at time $t$ can be fitted by a branched manifold associated with a threshold $μ ¯ t$ for the sample measure. Threshold selection is crucial for a judicious fit of the branched manifold to the sequence of point clouds as the attractor evolves. The topological analysis is, therewith, threshold dependent. This threshold is kept fixed for all time, so as to compare the topologies from one time to the next and to trace their changes.

For illustration purposes, Fig. 4 shows a sequence of the projections of point clouds $C t j$ (in black) onto the $(y,z)$-plane and their corresponding sample measures (in color) obtained from LORA simulations for three different consecutive time instants $0< t 1< t 2$⁠, using the noise amplitude $σ=0.3$⁠. Each snapshot is the result of an integration that is illustrated using the three “pullback tunnels”—blue, red, and green. The yellow tunnel represents the evolution of the theoretical random attractor from real time $t=0$ through $t= t 1$ and on to $t= t 2$⁠, as illustrated by the discrete snapshots $A(0,ω)$⁠, $A( t 1,ω)$⁠, and $A( t 2,ω)$⁠. As in Fig. A2 of Chekroun et al.,⁴ reproduced here as Fig. 11 in  Appendix C, the tunnels occur in the pullback time $s$⁠, while the random attractor $A t(ω)$ evolves in the real (forward) time $t$⁠.

Figure 2 of Charó et al.⁷ shows how the threshold $μ ¯ t$⁠, denoted there by $n ¯$⁠, affects the analysis of the topology of the branched manifold: $n ¯$-values that are too high will tend to disassemble the structure while values that are too low will fill up all the holes. Note that the choice of $n ¯$ was done for each snapshot separately, while here it is the same for the entire set of point clouds that were treated.

Finally, we only retain here the points whose projection onto the $(y,z)$-plane is surrounded by more than $n=100$ points within a grid box of size $δy×δz=0.0684×0.0587$⁠.

For the three sieved clouds in Fig. 4— $C ¯ 0$⁠, $C ¯ t 1$⁠, and $C ¯ t 2$—the analysis of the branched manifolds approximated by cell complexes in terms of homologies yields $H 1( C ¯ 0)∼ Z 21$⁠, $H 1( C ¯ t 1)∼ Z 18$⁠, and $H 1( C ¯ t 2)∼ Z 19$⁠, respectively.

The pullback invariance of the random attractor $A(t,ω)$ and the pullback attraction of random sets $B$ is illustrated in Ghil et al.,²², Fig. A2 reproduced here as Fig. 11 in  Appendix B. The difference between this figure and our Fig. 4 is the topological description shown through the colored point clouds $C ¯ t i$ supporting the sample measure that approximates the most populated regions of the random attractor at the times $t=0$⁠, $t 1$⁠, and $t 2$⁠, respectively. For further details on constructing a BraMAH complex for a snapshot from a point cloud, see Charó et al.⁷ 

Defining a templex for a random attractor involves establishing a link between different snapshots, in order to incorporate time into the description, as done in the deterministic case. But how can the topology of the point cloud at a given instant be related to the topology at another instant? We can compute the approximations $μ ^ t$ for several time instants and build cell complexes for each snapshot. Establishing a correspondence between the cells in subsequent cell complexes is not trivial, since, as we have explained in Sec. II A, the number of cells and their distribution is arbitrary.

The cell complex as a whole flows in phase space for a fixed noise realization $ω$⁠, following the trajectories of the system (2). Due to the fact that the evolution of the deterministic convection is smooth, the minimal $1$-holes in the complexes—which are intrinsic to the snapshot—can be followed from one snapshot to the next. This can be used to check if the holes are simply displaced or if they are modified drastically, through splitting, merging, creation, or destruction events. An example of minimal holes of a LORA point cloud is provided in Fig. 5 for $t=40.09$ and $σ=0.3$⁠.

We are now in a position to propose a definition of a random templex of dimension $2$⁠.

Notice that the digraph in a random templex does not connect 2-cells in a single branched manifold, as is the case in the deterministic case of Definition 1, but 1-holes between distinct time steps. Holes may just move or deform, so that the geometry of the evolving branched manifold changes without changing the topology. But a hole may disappear from a snapshot to the next or be created, split, or merge. In such cases, homology groups will change and so will the topology of the random attractor. Topological changes occur at specific times, which are associated with TTPs, as mentioned in Sec. I. A random templex will be shown to encode such TTPs.

We limit our description here to the homology groups through the minimal 1-holes, without discussion of orientability properties, which are associated with the torsion groups of a cell complex. Neglecting torsion properties works here because the LORA complexes do not present distinct torsion groups, which is the case for many other known deterministic attractors, when changing parameters and hence, supposedly, for their corresponding random attractors. But we do not exclude that torsion groups might be relevant for other random attractors or that twists as defined in Charóet al.³ might also play a role.

Figure 6 shows how holes are tracked from one snapshot to the next using two successive snapshots of LORA, at $t=40.07$ and $t=40.075$⁠. Tracking is performed by searching for the minimal distance between the barycenters of the minimal holes at consecutive snapshots. The symmetry of LORA is used in the tracking process. In Fig. 6(a), $b 1$ and $b 2$ are symmetric at time $t=40.07$⁠, and they must also be mapped to symmetric 1-holes $c 1$ and $c 2$ in the subsequent snapshot at $t=40.075$⁠, Fig. 6(b).

Applying this procedure to every hole in all the snapshots, we obtain the digraph of the random templex. The result for ten LORA snapshots in a time window $T w=[40.065,40.11]$ is shown in Fig. 7. The digraph $D$ here is not singly connected: $D$ has 15 connected components, and each of these directed subgraphs tells the story of one or several holes.

Even if there are only ten snapshots within $T w$⁠, a single connected component in $D$ may have more than ten nodes because of the existence of merging and splitting events that enable connections between the storylines of different holes. Important LORA properties that can be extracted from its random templex are given in Sec. IV.

TTPs, introduced by Charó et al.,⁷ can be now identified and classified using the digraph $D$ of LORA’s random templex.

This definition allows one to classify TTPs. The nodes that receive two or more edges are merging TTPs. The nodes that emanate more than one edge are splitting TTPs. Merging can be followed by splitting. The initial nodes of a connected component of the digraph that do not correspond to time $t 1$ are creation events. Destruction events are terminal nodes of a connected component that do not correspond to time $t s$⁠.

Each tipping point is associated with a particular snapshot, that is, a particular time instant in the time window $T w$⁠, and with a particular location in phase space, as defined by the barycenter of the corresponding hole. Figures 8 and 9 show, in detail, an example of a merging node and of a splitting node, respectively. In both figures, the top plot shows the location of the nodes in phase space, as identified by the holes’ barycenters, with an indication of the time instant, and the two bottom plots show the phase portraits of the snapshots at which the holes merge or split being colored.

The digraph of LORA’s random templex $R$ is shown in Fig. 7. This tree corresponds to $s=10$ snapshots within $t 1≤t≤ t s$ with $t 1=40.065$ and $t s=40.11$⁠. Applying the definition of TTP to the nodes and edges in $D$⁠, one can detect them and classify them according to the type of event. We find 18 splitting TTPs (in red), 12 merging TTPs (in blue), 2 mergings followed by splittings (in magenta), 8 creation TTPs (in green), and 12 destruction TTPs (in black).

A better picture of how the holes are evolving in the system’s phase space can be gained by using the coordinates of the barycenters of the holes for an embedding of $D$ into this space. We call the embeddings of $D$’s distinct connected components into phase space constellations.

Figure 10 presents the 15 constellations characterizing the evolution of LORA in $T w$⁠. While Fig. 7 is a purely graph-theoretical representation of the evolution of LORA’s random templex, Fig. 10 is a step in connecting it with a more geometric representation of this evolution. Such a representation could provide a bridge between the random templex—whose simplicity benefits from the invariance of the topology and of its breakdowns—and a more detailed description of the flow’s dynamics in phase space.

In this paper, we have introduced novel tools into algebraic topology and used them to provide insights into the behavior of random attractors, which combine deterministic chaos with stochastic perturbations.^18,19 We summarize our results in Subsection V A and discuss them in Subsection V B.

Section I reviewed the basic tools of this trade: branched manifolds, cell complexes, and homology groups¹² and it laid out the road plan for the paper.

In Sec. II, we presented the by now fairly well-known theory for deterministic strange attractors with time-independent forcing. In Sec. II A, we illustrated the connection between the branched manifold that supports the Lorenz (L63) strange attractor’s¹¹ invariant measure and a cell complex $K$⁠, constructed by the Branched Manifold Analysis through Homologies (BraMAH) of Sciamarella and Mindlin;⁵ see Fig. 1. We emphasized the independence of the homology groups $H k$ and associated Betti numbers $β k$ from the details of the construction of the cell complex.^10,12,13 In particular, we showed that 1-generators, i.e., loops around a 1-hole with $β=1$⁠, need not be minimal to still provide the homological information; see Fig. 2.

In Sec. II B, we recalled the directed graph (digraph) $G$ associated with the cell complex $K$⁠, as introduced by Charó et al.³ for the L63 attractor, as well as for the spiral and funnel Rössler attractors, among others. This digraph $G$⁠, with its nodes $N$ that are the cells and its edges $E$ that are the connections between them, is shown here for L63 in Fig. 3; it provides the direction of the flow on the branched manifold from one cell to another. Together, $K$ and $G$ form the templex $T=(K,G)$ associated with a particular nonlinear and, possibly, chaotic dynamics.

In Sec. III, we turned to our paper’s main focus, namely, extending the recent digraph and templex concepts and tools reviewed in Sec. II to the study of random attractors that evolve in time.^18–20 To do so, we had to somehow incorporate the time into the definition of the digraph, by relating the cell complexes from one snapshot to another. This was implemented by constructing and labeling the minimal holes of one snapshot and tracking them to the next one; see Figs. 5 and 6.

More precisely, the correspondence between snapshots was established by identifying the 1-generators of the homology groups, i.e., the 1-holes of two consecutive complexes. In order to implement hole tracking between snapshots, the minimal holes were defined using the algebraic procedure described in the appendix. These holes are the nodes $N$ used to encode, through the edges $E$ of the digraph $D$⁠, how the random attractor’s invariant measure evolves in time within a certain time window $T w$⁠.

The random templex $R$ is hence defined as the couple $R=( K, D)$⁠. The object $K$ is the indexed family of cell complexes over the time interval $T w$ under consideration, and the object $D$ is the tree illustrated in Fig. 7, with nodes $N$ at each snapshot and edges $E$ connecting one snapshot with another.

Each node of the digraph is a minimal hole at a certain time. Each hole has the possibility of connecting with itself in the subsequent snapshot, but also with other holes, through mergings or splittings. Holes can disappear as well from a snapshot to the next or be created at an intermediate snapshot. Each connected component of the digraph $D$ tells the story of how a hole evolves and how it connects to other holes, as time progresses; see Fig. 7. Examples of merging and splitting of minimal 1-holes were given in Figs. 8 and 9, respectively.

Much of the motivation of the work herein had to do with the visually striking, rapid changes in the evolution of LORA’s invariant measure as described by Chekroun et al.⁴ These visually rapid changes in time were documented by changes in LORA’s algebraic topology; see Fig. 4 in Charó et al.,⁷ in which $H 1$ changed from $Z 2$ to $Z 10$ and back to $Z 4$ in very small time steps of $Δt=0.09$⁠.

We thus addressed in Sec. IV, using the more sophisticated topological apparatus of Sec. III, the existence and nature of topological tipping points (TTPs) conjectured by Charó et al.⁷ A TTP occurring at time $t ∗$ and at position $x ¯ ∗$ is encoded in the digraph $D=(N,E)$ of a random templex $R$ either (i) as a node that receives or emanates two or more edges or else (ii) as an initial or terminal node of a connected component in $D$ that does not correspond to the beginning or end of the time interval $T w$ under consideration.

First, by considering time steps of $Δt=0.005$ that are even smaller compared to the L63 model’s¹¹ characteristic time of order unity, we confirmed, at least numerically, that the holes actually appear, disappear, and merge or split instantaneously. While a truly rigorous mathematical proof that this is indeed so might be difficult, the numerical evidence is quite substantial. Another matter that is left open at this point is whether changes in torsion—such as between a Möbius band, with torsion, and a regular one, without it—might be as sudden or not.

Given the fairly novel and even surprising nature of the concepts and tools introduced herein, a number of other questions are worth mentioning. One concerns the possibility of establishing closer connections between the metric flow of the solutions of a dynamical system in phase space and the topological description provided herein. The constellations shown in Fig. 10 and discussed at the end of Sec. IV suggest such a possibility, since one uses the embedding of the digraph $D$ of the random templex $R$ into the L63 model’s phase space, by using the coordinates of the barycenters of the nodes $N$⁠.

This representation emphasizes how a random templex contains the information of when—i.e., in which snapshot—and where, i.e., in which phase space location—topological tipping is taking place. Charó et al.⁷ emphasized already that BraMAH-based topological data analysis^5,6 is not restricted to low-dimensional dynamical systems. Clearly, some forms of reduced-order modeling^23,24 will be necessary to treat models with substantial geographical resolution. But it is at least conceivable to use the mixed localization approach of our constellations for the study of tipping in subsystems of a larger system, in the spirit of the rather popular “tipping elements” of Lenton et al.⁹ 

It is a pleasure to acknowledge stimulating discussions with M.D. Chekroun on extending the results of Charó et al.⁷ and improving the detection of topological tipping points. Two anonymous reviewers and one anonymous editor have made insightful comments that helped clarify certain items in the original version of this paper. This work has received funding from the ANR project TeMPlex ANR-23-CE56-0002 (D.S.). G.D.C. gratefully acknowledges her postdoctoral scholarship from CONICET. The present paper is TiPES Contribution No. 210; this project has received funding from the EU Horizon 2020 Research and Innovation Program under Grant Agreement No. 820970, and it helps support the work of M.G.

The authors have no conflicts to disclose.

Gisela D. Charó: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Michael Ghil: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Denisse Sciamarella: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article. The code that computes templex properties is available at git.cima.fcen.uba.ar/sciamarella/templex-properties.git.

In the autonomous case in which the right-hand side of Eq. (A1) does not depend explicitly on $t$⁠, $F=F(x),$ and a one-parameter semigroup suffices to determine the evolution of the system. In this case, only the time interval $t ′=t−s$ matters and $Φ(t,s)$ reduces to $Φ( t ′)$⁠, i.e., to a standard flow. Hence, forward-in-time attraction only is of interest, resulting in attractors that are fixed, time-independent objects such as fixed points, limit cycles, tori, and strange attractors. In the case of nonautonomous systems, such as the climate system,^27,28 there is explicit time dependence of forcing or coefficients, $F=F(t,x),$ and we need to define pullback attraction and pullback attractors (PBAs) or snapshot attractors.^{4,19,21,22,29,30}

In simple words, the family ${ A ( t ) } t ∈ R$ is invariant under the system’s dynamics and it attracts at each time $t$ in the pullback sense all compact initial subsets $B$ from the remote past.

In natural systems, such as the climate system, random time-dependent forcing is present²⁸ and it is necessary to use stochastic differential equations (SDEs) to model the system.^28,31 In the theory of random dynamical systems (RDSs), when the time-dependent forcing is random, PBAs are known as random attractors.^19,25,26

In this framework, to have a reasonably self-consistent description of the system’s random aspects, the noise has to be modeled as a time-dependent stochastic process in a probability space $(Ω, F, P)$⁠.⁴ In this probability space, $Ω$ is the sample space; $F$ is a $σ$-algebra of measurable subsets of $Ω$⁠; and $P$ is a probability measure on $F$⁠.¹⁹ To track the noise effect in time, one defines a time-dependent shift $θ t:Ω→Ω$⁠:

The mapping $θ t$ connects each realization $ω$ at time $0$ to its state after a time $t$ has elapsed and the condition (iii) means that the probability measure $P$ is preserved by $θ t$⁠. Then $(Ω, F, P,θ)$ is called a driving dynamical system.

In the case of a stochastic-dynamic system driven by a Wiener process $W$⁠, $θ t$ is defined on $Ω$ according to $W s( θ tω)= W t + s(ω)− W s(ω)$⁠.¹⁹ To define an RDS, one can use $θ t$⁠, to let the noise realization $ω$ evolve in time, and define a cocycle mapping $φ$ to describe the evolution of the state $x$⁠.

The above equalities set (i) the initial condition for the cocycle; and (ii) the cocycle property. This cocycle property is satisfied for a broad class of SDEs like those of interest here; see Arnold.¹⁹ In fact, Fig. 4 and its discussion in Sec. III of the main text herein illustrate very well the difference between (a) the one-parameter semigroups of autonomous dynamics and their purely forward attractors; and (b) the two-parameter semigroups of nonautonomous dynamics and their attractors, which can be either forward or pullback.

The evolution of a stochastic dynamical system can be modeled using an RDS $(φ,θ)$⁠, and its random attractor ${ A ( t ; ω ) } t ∈ R$ provides the natural extension of a PBA to the random framework.

The time-indexed family of random compact sets ${ A ( t ; ω ) } t ∈ R$ provides a complete description of all the system’s possible states that are likely to be observed at any particular time $t$⁠. Figure 11 shows a schematic diagram of a random attractor and the properties of pullback invariance and pullback attraction of random sets. The latter properties are further emphasized in Fig. 4 of Sec. III herein.

Note that, in practice, one does not need to let $s→−∞$ to compute a good numerical approximation of a random attractor. In the case of LORA^4,7 that we treated here, the convergence time of the sample measure estimate $μ ^ t$ seems to be of the order of $s ∗≃40$⁠, in the nondimensional units of the deterministic Lorenz convection model.¹¹ As far as our numerical results show, this convergence is uniform in $t$⁠.

A fundamental object in any RDS $(θ,φ)$ is its random invariant measure $μ$⁠.

The property (ii) means that $μ$ “lifts” $P$ into the extended phase space $Ω×X$⁠.⁴ Any probability measure $μ$ possesses a disintegration or factorization,¹⁹ also known as a sample measure $μ ω$⁠.^4,32 Sample measures enable one to work in the phase space $X$⁠, rather than in the extended phase space $Ω×X$⁠.

The sample measure $μ θ t ω$ supported by the random attractor $A(t;ω)$ at time $t$ and for the noise realization $ω$ provides the spatiotemporal probability distributions of the parts of the phase space $X$ occupied by the RDS. The sample measure $μ ω$ is unique $P$-almost surely,³³ and the invariance property (i) of Definition 9 translates into $φ(t,ω) μ ω= μ θ t ω$⁠.

When an RDS $(θ,φ)$ has a random attractor $A(t;ω)$⁠, then $μ ω( A(ω))=1$ for almost all $ω$ in $Ω$⁠, i.e., it supports every invariant measure. For a given noise realization $ω$⁠, the random attractor determines the parts of the phase space $X$ onto which any bounded random set $B$ is mapped at time $t$⁠, when $B$ is propagated by $φ$ from a sufficiently remote past. Physical measures are sample measures that are “naturally chosen” by the dynamics.^4,28

An example of a class of probability measures that also coincides with that of physical measures for many dynamical systems are the Sinai–Ruelle–Bowen (SRB) measures.^33–35 

Chekroun et al.⁴ have shown in their  Appendix C that, when the Kolmogorov operator associated with their Lorenz stochastic model (2), is hypoelliptic³⁶ and its leading Lyapunov exponent is positive, LORA supports a random SRB measure. They also provided numerical evidence that the measure to which the LORA’s point clouds converge must be physical, too. But they gave no indication of the uniqueness of LORA’s random SRB measure or of its physical measure.

A point cloud of this spiral attractor appears in Fig. 12.

A cell complex $K ′$ for this system is shown in Fig. 13(a) using a planar diagram of the complex, i.e., a diagram in which the cells that appear to be duplicated are in fact two copies of the same cell and must therefore be glued together and considered as one. The advantage of such a planar diagram over the attractor’s three-dimensional representation is that one can see the whole structure, whereas if we plot the complex in three dimensions—juxtaposed on the point cloud as in Fig. 1—some parts of the plot are obscured by the perspective. In the case of $K ′$⁠, the $1$-cells that must be glued together are drawn with heavy lines. If this planar diagram is drawn on paper, one can obtain a model of the attractor by gluing the heavy lines together.

Up to this point, we have calculated homologies as customary in algebraic topology textbooks.¹² But the cell complex contains information relevant to the characterization of a chaotic attractor that is neither included in the homology groups nor in the torsion groups per se. Locating the $1$-cells which are shared by at least three $2$-cells, we can also compute the joining 1-cells. In $K ′$⁠, $⟨0,1⟩$ is such a joining 1-cell. The $2$-cells that are glued to it can also be identified: they are $γ 1$⁠, $γ 5$ and $γ 6$⁠. Together they form what we call the joining $2$-chain: $γ 1− γ 5+ γ 6=⟨1,3,2,0⟩−⟨7,8,1,0⟩+⟨1,8,9,0⟩$⁠.

Let us now endow the complex $K ′$ shown in Fig. 12(a) with a directed graph $G ′$⁠, shown in Fig. 12(b), in order to form the templex $T ′$ of the Rössler attractor in Fig. 12. The set of operations required to combine the complex $K ′$ and the digraph $G ′$ are detailed in Charó et al.³ The joining 1-cell has two ingoing 2-cells, $γ 5$ and $γ 6$⁠, and one outgoing 2-cell, $γ 1$⁠. This information can be viewed in the joining subgraph ${ γ 5→ γ 1, γ 6→ γ 1}$ of Fig. 13(b).

The Rössler attractor is thus homologically equivalent to a cylinder, but a cylinder as such does not have a joining locus. The templex $T ′$ tells us much more about the attractor’s structure: its joining locus has a single component—as expected for an attractor bounded by a genus-1 torus—and there are two stripexes, $S 1$ given by (D2a) and $S 2$ given by (D2b). The stripex $S 1$ is a Möbius band while $S 2$ is a cylinder or normal band (without torsion).

All the computations herein can be handled algorithmically with the Wolfram Mathematica code freely available at git.cima.fcen.uba.ar/sciamarella/templex-properties.git.