The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefited from remarkable technological advances in computing resources, relating throughput as well as storage, and in observational capabilities, regarding both platforms and instruments.

Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of non-autonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing, deterministic as well as stochastic.

This paper is based on the invited talks given by the two authors in an online series on “Perspectives on climate sciences: From historical developments to research frontiers”. The series had twice-monthly talks from July 2020 to July 2021 and its success led to the idea of having a special issue of

Many of the ideas and methods of dynamical systems theory were introduced into the climate sciences by a generation of pioneers in the 1960s.

None of the pioneering papers mentioned above, though, nor any of the thousands of papers since, exhibits all the phenomena – mathematical and physical – of interest in this review paper. As we proceed, the illustrative examples will be taken from atmospheric, oceanographic and climate models that capture best one or a few of these phenomena.

It is important to realize that Poincaré had already seen the analogy between the chaos he found in the so-called reduced three-body problem of celestial mechanics

Our second example will be very analogous to the first and we shall take it from meteorology. Why have the meteorologists such difficulty in predicting the weather with any certainty? Why do the rains, the tempests themselves seem to us to come by chance, so that many persons find it quite natural to pray for rain or shine, when they would think it ridiculous to pray for an eclipse? We see that great perturbations generally happen in regions where the atmosphere is in unstable equilibrium. The meteorologists are aware that this equilibrium is unstable and that a cyclone is arising somewhere; but where they can not tell; one-tenth of a degree more or less at any point, and the cyclone bursts here and not there, and spreads its ravages over countries it would have spared. This we could have foreseen if we had known that tenth of a degree, but the observations were neither sufficiently close nor sufficiently precise, and for this reason all seems due to the agency of chance. Here again we find the same contrast between a very slight cause, unappreciable to the observer, and important effects, which are sometimes tremendous disasters.

The translations are both fromThe work of

Aside from its applications to the climate sciences, the dynamical systems literature is quite extensive, in covering both mathematical fundamentals and applications to other areas.

Following

The equations of continuum mechanics are nonlinear. Surprisingly many phenomena can be explained by linearization about a particular fixed basic state. Many more cannot.

Behavior of solutions to nonlinear equations – subject to some reasonable mathematical assumptions – changes qualitatively only at isolated points in phase-parameter space, called bifurcation points. Behavior along a single branch of solutions, between such points, is modified only quantitatively and can be explored by linearization about the basic state, which changes as the parameters change. That is, nonlinear dynamics are much like linear dynamics, only more so

Bifurcation trees lead from the simplest, most symmetric states to highly complex and realistic ones, with much lower symmetry in either space or time or both. These trees can be explored partially by analytic methods

The truly nonlinear behavior near bifurcation points involves robust transitions, of great generality, between single and multiple fixed points (saddle–node, pitchfork and transcritical bifurcations), fixed points and limit cycles (Hopf bifurcation), and limit cycles and strange attractors

Behavior in the most realistic, chaotic regime can be described by the ergodic theory of dynamical systems. In this regime, statistical information similar to but more detailed than for truly random behavior can be extracted and used for predictive purposes

Chaos and strange attractors are not restricted to low-order systems. They can be shown to exist for the full equations governing continuum mechanics

Single time series

What is the topology of chaos, and why is it important in the theory of dynamical systems and in the time series analysis for nonlinear and chaotic dynamics? We attempt here to provide answers to these questions, with an emphasis on applications to the climate sciences. Essentially, the concepts and tools of algebraic topology can be applied to the evolution of systems in both phase space and physical space as well as to the interesting back-and-forth trip between the two spaces. This complementary view of the way that dynamics and topology interact is a main motivation of the present article.

The emphasis on time dependence and dynamics here should not allow us to forget, though, the huge role that homologies have already been playing in the fields of image processing and visualization

In Sect.

The most recent developments of the merging of the two strands of Poincaré's heritage – algebraic topology and dynamical systems – are covered in Sect.

In the rest of this section, we provide some quick historical background to the current interest in the ways in which algebraic topology can help one infer a system's chaotic dynamics from one or more time series of its observables. The first methods of time series analysis that associated geometric properties with experimental time series appeared in the early 1980s

But is geometry the best lens one can use to classify data according to underlying differences in dynamics? Classifying dynamics is possible thanks to invariants or quasi-invariants in phase space.

metric invariants, such as dimensions of various types, e.g., correlation dimension

dynamic invariants, such as Lyapunov exponents

topological invariants, linking numbers, relative rotation rates, Conway polynomials and branched manifolds

The “recipe” to “knead” the

Sketch of the topological processes that intervene in obtaining the strange attractor of the

The advantage of using topology instead of geometry or fractality to describe chaos lies in the fact that topology provides information about the elementary stretching, folding, tearing or squeezing mechanisms that act in phase space to shape the flow. Geometric features may differ, but if the underlying dynamics obey certain equivalence principles, the topology should be the same. Topological equivalence between branched manifolds is defined by isotopy. In other words, two objects are isotopic if it is possible to mold one into the other without tearing or gluing it. It is in this sense that we speak of dynamical equivalence. Different geometric deformations of the Lorenz attractor that preserve its topology are sketched in Fig.

Point clouds associated with different geometrical representations of the

A good starting point for this quick historical perspective is the pioneering paper of Henri Poincaré

Isopleths of

Joan Birman and Robert F. Williams used branched manifolds to classify chaotic attractors in terms of the way periodic orbits are “knotted” in dynamical systems

In the late 1990s, it became possible to determine whether or not two three-dimensional (3-D) dissipative dynamical systems are equivalent by using knot theory

As we indicated in Sect.

The contributions of “nonlinear dynamics”, as dynamical systems theory tended to be referred to by physicists and other non-mathematicians by training, were presented for the first time in a quadrennial report (1987–1991) of the US geosciences community to the International Union of Geodesy and Geophysics (IUGG) by

Assume that the state of a system of interest can be described by a vector

Points

The bifurcations of a dynamical system that we deal with in this subsection describe the creation and annihilation of fixed points as well as changes in their linear stability. Further types of bifurcations are considered in the next subsection.

Typically, bifurcations lead to abrupt qualitative changes in the dynamics, explaining why they are often invoked as a mathematical model for abrupt regime shifts or state transitions in real-world systems. Until fairly recently, bifurcations were studied mostly in the context of autonomous dynamical systems. The more realistic situations in which the forcing is allowed to depend explicitly on time are addressed in Sect.

There are at least two different interpretations of “tipping” and “tipping points” in the literature. One of these, emanating from

In the latter case, tipping is necessarily related to a tipping point in phase-parameter space as opposed to just a threshold in some parameter value; thus, not every jump or critical transition arises from a such a point. Both points of view – pun intended, of course – have their merits, but confusion should be avoided to the extent possible. Clearly, in this review article, we follow the more unambiguously defined mathematical version.

As an instructive and widely used example, we briefly introduce a prototype model to describe scalar dynamical systems than can occupy either one of two stable fixed points, separated by an unstable one, as plotted in Fig.

Bifurcation diagrams for

The two stable fixed points correspond to the two minima of the potential

Changing

The bifurcation introduced above is called a

Another example of bistability is given by a pitchfork bifurcation (Fig.

The solutions of Eq. (

The bifurcation occurs as the parameter

With the scalar version

Bistability is only the first step up the bifurcation tree that leads from system behavior with the highest degree of symmetry in space and time – possibly as simple as uniform in both – to behavior that has greater and greater complexity

In polar coordinates, this normal form is given by

The version shown in Fig.

A further step on the route to chaos for deterministic systems with no explicit time dependence

This kind of motion is typical in celestial mechanics

Quasi-periodic motion already looks much more irregular than purely periodic motion. Thus, for instance, the intervals between lunar or solar eclipses are highly irregular. Still, the 14th century scholar Nicole Oresme was already aware of the kinematic consequences of quasi-periodicity for celestial motions

From quasi-periodic motion to a deterministically chaotic one there are several routes

Realistically, the natural systems that we want to describe in terms of dynamical systems theory are non-autonomous, meaning that

Moreover, there is typically high-frequency forcing, such as cloud processes or weather variability. In a drastic simplification, this type of forcing is often represented by white noise

Interest in autonomous dynamical systems and their bifurcations started over 2 centuries ago and can be traced back to Leonhard Euler and the Bernoullis, while that in non-autonomous and random dynamical systems (NDSs and RDSs) only goes back a few decades. We describe some key differences between the two cases next and justify the need for considering pullback attractors (PBAs) in the latter case.

For the sake of simplicity, we assume that the physical system under consideration is described by a set of ODEs. In the autonomous case, such a set of ODEs can be formally written as

For the

There are two key distinctions between the autonomous case and the non-autonomous one:

In the autonomous setting, solutions cannot intersect, since there is only one trajectory through a given point

In the autonomous setting, solutions depend only on the time

Given the uniqueness and the continuous dependence of the global solutions to Eq. (

the

the

the initial value property

the two-parameter semigroup property for all

the continuity property that the mapping

This difference matters, in particular, in determining the asymptotic behavior of the solutions. In the autonomous case, a global solution is invariant with respect to translation in time:

To illustrate the effect of this lost invariance, consider the following simple scalar ODE

One is thus led to the following rigorous definition of a PBA for a forced dissipative dynamical system subject to a time-dependent forcing, where we have generalized

For all

for all

The finite-dimensional definition above follows

Here, the time-dependent forcing

The graph of the PBA for the simple NDS example governed by Eq. (

A system defined in polar coordinates by

Since the dynamics of the phase

The PBA with respect to the coordinate

Trajectories and PBA of the system defined by Eqs. (

Figure

Note that the structure of the system's trajectories depends on the ratio

Let us return now to the more general, nonlinear and stochastic case of Eq. (

When

The noise processes may include “weather” and volcanic eruptions when

Schematic diagram of a random attractor

Four snapshots

Heatmaps of the time-dependent invariant measure

The striking effects of the noise on the nonlinear dynamics that are visible in Fig.

For an example of the second class, assume that the control parameter

Finally, the third class of rate-induced transitions arises when there is no strong separation between the system's intrinsic timescales and those at which the control parameter changes. So far, we implicitly assumed that, for each change in

Sketch of a double-fold bifurcation and how it leads to abrupt transitions and hysteresis in the temporal evolution of a system in a double-well potential with slowly changing parameter

We illustrate in Fig.

To simulate the system's trajectory, the control parameter

Note also that in the generalization from autonomous bifurcations to non-autonomous tippings, the phrase “tipping point” – aside from its threatening implication – is somewhat misleading: a bifurcation point is a point in phase-parameter space, like

At the end of Sect.

Clarifying the difference between these two kinds of flow, in physical space and in phase space, is relevant here because, in the community involved in the work been reviewed here, the phrase “topological chaos” is used when studying how fluid–particle trajectories are entangled in physical space during a mixing experiment. A noteworthy example is the motion induced by spatially periodic obstacles in a two-dimensional flow in order to form nontrivial braids

Topological chaos emerges in stirring or mixing experiments. Here we see stylized streamlines
induced by pairs of rods on a periodic lattice
and we see how these streamlines are stretched in physical space. From

On the other hand, “topology of chaos” or “chaos topology”, for short, considers the problem of how multi-dimensional point clouds or trajectories are topologically structured in phase space. Such a study in phase space is not equivalent to the type of study illustrated in Fig.

Mathematically, a knot is an embedding of a circle in 3-D Euclidean space

There are in fact three steps in this knot-theoretical approach, and the aim of each one is achieved in a particular way:

approximate the neighboring unstable periodic orbits (UPOs) around which the flow is evolving with an orbit or closed curve,

find a topological representation of the orbit structure and

obtain an algebraic description of the topological representation.

Knot theory comes in the procedure's second step and computing the identified knot invariants closes the procedure. Another possibility, instead of using knots, is resorting to braids, as discussed by

In “How topology came to chaos”,

Gilmore continues as follows:

Suppose we have a dissipative [chaotic] flow in three dimensions: There is one positive Lyapunov exponent

A branched manifold, in the strict sense of the two words that make up the term, can in fact be defined mathematically without reference either to a flow or to the Birman–Williams projection mentioned above. Following

A branched manifold is, therefore, a manifold that is not required to fulfill the Hausdorff property. We prefer this more general definition, instead of the one related to the Birman–Williams projection, for several reasons, including the possibility of extending the concept of a branched manifold to the structure of instantaneous snapshots of random attractors, as we shall see in Sect.

As the topological structure of a branched manifold is closely related to the stretching and squeezing mechanisms that constitute the fingerprint of a certain chaotic attractor, its properties can be used to distinguish among different attractors. This is how one can justify the two-way correspondence between topology and dynamics. This correspondence remains valid in the case of four-dimensional semi-conservative systems

The terms “branched manifold” and “template” have often been used interchangeably. We do not regard them as synonyms, for technical reasons that will be important in the development of the concept of templex in Sect.

Homologies provide an algebraization of topology by building compressed representations of a certain object through cell complexes and by computing essential signatures of the object's shape through homology groups that do not depend on the particular representation used to compute them. Homology groups enable the analysis of

Other approaches that characterize aspects of dynamical chaos in arbitrary dimensions

To illustrate how homologies work, let us take as an example a point cloud obtained by the integration of the deterministic

approximate the points as lying on a branched manifold,

find a topological approximation of the branched manifold and

obtain an algebraic description of the topological structure.

A branched manifold is a generalization of a differentiable manifold that may have singularities of a very restricted type, which correspond to the branching, and it admits a well-defined tangent space at each point. In other words,
such a manifold has the property that each point has a neighborhood that is homeomorphic to either a full 2-ball or a half 2-ball, and which is locally homeomorphic to Euclidean space or locally metrizable but not globally so because of the branching

As points in our cloud are assumed to lie on a branched manifold, we can classify the points into subsets that constitute a good local approximation of a

Here we use polygons for the cells that pave the attractor's branched manifold. These cells must be correctly glued to each other in order to retain the topological features of the original point cloud. Once the cell complex is constructed, homologies can be computed to yield an algebraic description of the approximating structure. In this review paper, we will not go into the mathematical definitions and theorems required to fully and correctly understand cell complexes and homology theory but only give a taste of the theoretical framework via challenging applications. The reader is referred to

The key point here is that the homology groups represent essential information about the branched manifold, while being independent of the number of cells used to construct the complex

When Michael Ghil visited the University of Buenos Aires in fall 2018 and got acquainted with this methodology, whose first results were published 2 decades ago

In dynamic problems, and especially in chaotic dynamics, the PH approach has to contend with the difficulty of finding robust criteria for the degree to which a cell complex represents a manifold that underlies a point cloud

For this reason, the Buenos Aires group chose to establish special rules for the construction of a complex, namely rules that take into account that the objective of the reconstruction is not just any arbitrary shape but a branched manifold in phase space. Michael Ghil's suggestion led to the use of Branched Manifold Analysis through Homologies (BraMAH) for this method, a name that says it all and simultaneously recalls the Hindu god of creation and knowledge, which seems very auspicious. The precursors of this technique are four researchers of the Nonlinear Systems Laboratory of the Mathematics Institute at the University of Warwick, who extracted Betti numbers from time series

We review here briefly the improvements that

In their follow-up paper,

BraMAH can also detect the presence of a Klein bottle in the data, like the one discovered by

The topological-analysis program has been applied to many fields of science: voice production

validating or refuting models (simulations vs. observations),

comparing models (time series generated by different models),

comparing datasets (e.g., in situ versus satellite data),

characterizing and labeling chaotic behaviors (towards a systematic classification), and

classifying sets of time series according to their main dynamical traits (e.g., in Lagrangian flow analysis).

BraMAH analysis of the

In fluid mechanics, two viewpoints are possible. In the Eulerian viewpoint, fluid motion is observed at specific locations in space, as time passes. In the Lagrangian viewpoint, instead, the observer follows individual fluid particles as they move through the fluid domain. The Eulerian description is more often used for prediction and other purposes. Lagrangian analysis, though, is a powerful way to analyze fluid flows when tracking and understanding the origins and fates of individual particles are important

The oft observed formation of ordered patterns in fluids with complex behavior has led to the search for a theory that could explain Lagrangian coherence in terms of an underlying skeleton responsible for structuring the pathways of sets of fluid particles. These structures may have a finite lifetime, and so one refers to them as finite-time coherent sets

The unsteady or driven double gyre (DDG) system is an analytic model, often used to show how much Lagrangian patterns may differ from patterns in Eulerian fields.

Clearly, this DDG model is non-autonomous for

From the Eulerian perspective, the DDG has a time-periodic and simple behavior, a snapshot of which is shown in Fig.

Eulerian and Lagrangian perspectives for a fluid flow in the case of the

How can BraMAH help us in Lagrangian analysis? The interesting cases, as shown by the DDG example, correspond to dynamical systems that are non-autonomous. But in such systems, some processes involved in the particle dynamics derived from the Eulerian streamfunction are not explicitly described in the two-dimensional space spanned by the particle positions' coordinates. Many authors choose to work in an “extended phase space”, in which time is added as a phase space coordinate.

But such an extended phase space is in fact deceptive, since it assigns a double status to the time variable, which should not play the role of both an independent and a dependent variable. Due to this double status, some tools from autonomous dynamical systems theory do not apply

Working in a space whose dimension is increased by 1 due to introducing the extra ODE

Many of the properties that are valid in a well-defined phase space are altered in an extended phase space, and topology is one of them. In the case of the DDG model discussed by

Such a transformation gets rid of the explicit time dependence with a legitimate procedure that does not run into the previously explained inconsistency. In this four-dimensional phase space, and for certain initial conditions, the topological structure that is obtained is a Klein bottle. A Klein bottle cannot be immersed into a 3-D space without self-intersections: the role of the fourth dimension that is required to rewrite the system in an autonomous form is, therefore, highly relevant here. Thus, to use topological tools self-consistently, one must be prepared to work in a well-defined phase space, and with as many dimensions as required.

In the fluid-flow problem, the four-dimensional phase space complements the Lagrangian variables by an indirect representation of the Eulerian variables. A knotless approach like BraMAH does allow one to work in such a space, which was previously out of reach for a topological analysis. As we shall see, though, in Sect.

When applied to time series describing particle trajectories in fluid flows, BraMAH falls within a family of methods that measure the complexity of individual trajectories to identify coherent regions, i.e., regions with qualitatively different dynamical trajectory behavior.

Returning now to the oil spill in the middle of the DDG system's domain

The analysis of a set of 8528 particles advected by the DDG flow field yields five topological classes. These five classes are obtained by applying BraMAH to four-dimensional point clouds; the plots in the figure are three-dimensional projections of representative cell complexes for each of the five classes. Four of them involve quasi-periodic particle motion, and only one of them, which is represented by the third cell complex, points to a branched manifold that refers to the so-called chaotic sea (colored in blue in Fig.

From left to right, Class I corresponds to a strip, Class II to a torus and Class III to a branched manifold with three 1-holes – i.e., with a Betti number

Coloring of 8528 particles in motion in a DDG field, with colors corresponding to the topological structure of the particle trajectories in phase space. The boundaries between distinct colors are fairly well defined, displaying the existence of transport barriers that separate non-mixing regions, like the green, orange, red or magenta, vs. the chaotic sea (blue). A direct correspondence is found between the regions identified by the topological BraMAH analysis and those observed dynamically using a Poincaré section, as in

The presence of the Klein bottle as Class IV among the five classes in Fig.

The BraMAH applications reviewed in this subsection demonstrate substantial progress in Lagrangian analysis, by providing a method that enables one to identify coherent sets without previous knowledge of the flow field. This particular set of results also shows methodological progress in chaos topology, since it appears that BraMAH can help describe the topological structure of non-dissipative, Hamiltonian systems. Recall, as a stepping stone in this direction, the analogy between the non-divergence of a fluid flow in physical space, like the DDG model, and the Hamiltonian character of a dynamical system's flow conserving volume in phase space, like the equations of celestial mechanics

Structures in phase space are special because they are not just spatial objects: they are associated with a semi-flow on them, which is sometimes represented by arrows. A cell complex can effectively encapsulate the properties of a branched manifold in standard space, but it will not convey the fact that, when the cells in a complex represent a semi-flow on a spatial object, they can be traversed in an arbitrary order only at the expense of forgetting about the semi-flow. In other words, time is absent from the description. Including the arrow of time in the description calls for a more refined mathematical object, in which the topological properties of a flow in phase space come to light through the combined analysis of both the spatial structure of the underlying branched manifold and of the semi-flow upon it.

Solution trajectories for

The

For strongly dissipative systems, like the Rössler attractor, the number of monotone branches of the first-return map provides
the number of strips required to construct the corresponding
template. Strips are cylinders, in topological terms, but one must beware that the meaning of strip in a template is not introduced to refer to a topological class but to discriminate between the different paths followed by the flow along the branched manifold. A strip is typically defined between a splitting chart and a joining chart, in which the strips are
split and joined, respectively. Thus, in the template terminology, the spiral attractor has two strips, while the funnel attractor has three strips, as shown in Fig.

Templates for the

Strips in a template are associated with a tearing of the flow. They are sometimes split in a fictitious manner, introducing false holes into the branched manifold, even if these strips are not necessarily delimited by boundaries or associated with holes in the sense of homologies. Their number can be obtained, for strongly dissipative systems, by computing the number of monotone branches of the first return map. But where are these strips in a cell complex? As mentioned above, they cannot be directly identified with holes in the latter. Can they be identified all the same from some other properties of the cell complex? The short answer is yes but not without the information that is contained in the flow on the cell complex rather than just in the cell complex itself.

The templex thus combines all the essential information that is relevant to the topology of the branched manifold and to the flow on it. The flow on the cell complex is represented by a directed graph (digraph)

Algebraic computations on a templex provide, on the one hand, the already known properties of the cell complex – such as the homology groups, torsion groups and weak boundaries – that describe the branched manifold; on the other hand, they provide the properties of the flow on this structure. The topology of a templex is described in terms of a set of sub-templexes
that will be called stripexes, since they play the same role as strips in a template. This is no longer done at the price of introducing false holes or boundaries to separate the strips. It is achieved through a set of well-defined operations that include flow-orienting the cell complex; minimizing the cell structure at the joining loci, where the tearing of the flow takes place, to obtain a generating templex; calculating the cycles of the digraph; and checking for local twists, since uneven torsions in a strip correspond to a local twist in a stripex. The reader is referred to the steps in

Templexes for

The cell complex of a templex can be seen as a dynamic kirigami or cutout paper model, made of pieces that fit together; in this case, the pieces are polygons. Note that points or segments with the same label must be glued together when constructing the paper model. The digraph can be seen as a map of the flow-compatible connections between the pieces. Combining the cell complex and the digraph, we can define and algebraically compute the stripexes. For details on this procedure, the reader is again referred to

BraMAH and the associated templexes, as presented so far, provide a topological description that holds within an autonomous and deterministic framework. As discussed in Sect.

An example involving not only deterministic time dependence but also random forcing was presented in Eq. (

Such an analysis was performed by

Three LORA snapshots with the noise variance

The stochastic branched manifold, characterized by a single-cell complex for each snapshot, does not contain any information about the future or the past of the invariant measure. The flow in a cell complex representing the invariant measure on a random attractor can no longer be represented within that cell complex, as done when using a deterministic templex, like the one described in Sect.

But how can one track changes between different cell complexes without using specific individual cells? Let us recall that the number of cells and their distribution in a cell complex are arbitrary and that homology groups are conceived so as to cancel out the extraneous information in the cells and to only retain the essential properties of the topological space. Homologies will thus provide the key to connect a cell complex of a random attractor at a given instant to a cell complex corresponding to another instant. For a random attractor, we will endow a set of cell complexes with a digraph that does not connect cells within a single complex, as in Fig.

Tracking holes requires some caveats, though. Homology groups and the associated Betti numbers are independent of the particular set of cells forming a cell complex. Hence, the holes or generators of a homology group can be expressed in terms of one of several representative cycles that need not strictly follow the boundary of the holes, as shown in Fig.

Cell complex of the deterministic

What does the random templex, thus defined, encode? In the life of a random attractor, there may be time intervals within which the branched manifold evolves geometrically but maintains its homological properties. Topology can be said to change when the holes that are being tracked from one snapshot to the next are created or destroyed. Some of them can be found to split or merge. Such changes are associated with what we call hereafter a

To confirm this further,

Figure

A “day in the life” of two mutually symmetric holes of LORA for a fixed noise realization and noise intensity

The indices in Fig.

A particular constellation out of the set that represents the essence of the evolution of LORA's 1-holes within a
time window

The purpose of this paper was to provide an account of the convergence between two strains of Henri Poincaré's heritage – dynamical systems theory

In Sect.

Section

The material in Sect.

We first explained in this section the essential difference between forward and pullback attraction, i.e., between convergence in time of single-parameter and two-parameter semigroups of solutions to the governing equations. Simple examples of pullback attractors (PBAs) were given to familiarize newcomers with the appropriate concepts and methods; see again Figs.

In Sect.

In Sect.

Finally, in Sect.

The fact that the change in the set of minimal holes of a cell complex at

As usual, when stumbling upon some striking findings, there are two kinds of paths that one might wish to pursue: (i) more general or stronger theoretical results and (ii) interesting applications. Clearly we have some rather striking findings, and we will outline some intriguing paths to pursue, of both kinds, as well as connections between the two kinds of paths.

TTPs in the templex of an NDS or RDS are obviously connected with a lot more detailed information in phase space about the system under investigation than one might suspect from the usual kinds of bifurcation-induced, noise-induced and rate-induced transitions – or B tipping, N tipping and R tipping – discussed by

An interesting example, among many, of localized changes in Earth's physical space is that of persistent anomalies

In applying the multiparameter PH method to the classical

The existence of multiple regimes in a dynamical system is certainly associated with its attractor's nontrivial topological structure, as

Schematic overview of atmospheric LFV mechanisms. From

Which type of phenomena dominate atmospheric LFV? There are two apparently contradictory descriptions: oscillatory, wavelike flow features or geographically fixed, particle-like, episodic flow features; e.g., blocking of the westerlies (particle-like) or intraseasonal oscillations (wavelike), with periodicities of 40–50 d

The simplest approach to persistent anomalies in midlatitude atmospheric flows on 10–100 d timescales is to regard them as due to the slowing down of Rossby waves or to their linear interference

A third approach is associated with the idea of oscillatory instabilities of one or more of the multiple fixed points that can play the role of regime centroids. Thus,

Finally, sketch d in the figure refers to the role of stochastic processes in LFV variability and S2S prediction, whether it be noise that is white in time, as in

How might topological data analysis contribute to clarify this thicket of apparently contradictory descriptions of LFV? One hint is found in the work of

This UPO-based approach did confirm certain theoretical results of

As explained here in Sects.

It is thus conceivable, although it remains to be demonstrated, that the additional tools brought to the table by the mathematical object we called templex – namely the digraph and stripexes – could help explore, in a highly simplified setting, issues like the existence and multiplicity of regimes, as well as of the presence of oscillatory features in the dynamics. As explained in the

Finally, as stated at the end of Sect.

The extension of the templex from autonomous and deterministic systems

More broadly, complex networks

The PH framework to obtain families of nested cell complexes from point clouds has only been mentioned in passing in this review article for the sake of brevity; it should be taken into account, though, as an important of branch of computational topology that is continuously providing us with solutions to algorithmic problems being faced in chaos topology and the climate sciences. So far, the complex network community seems to be lacking a dual object such as the templex to deal with nonstationarity. Finding such an object that captures the spatial structure and is, in addition, endowed with another object that captures the flow structure on the spatial object appears to be a worthwhile challenge.

No datasets were used in this article.

The supplement related to this article is available online at:

The authors have contributed equally to the work on this review paper and to its writing. Their names are in alphabetical order.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Interdisciplinary perspectives on climate sciences – highlighting past and current scientific achievements”. It is not associated with a conference.

Section

This research has been supported by the Centre National de la Recherche Scientifique (NOISE (LEFE/MANU) and EU Funding: TiPES (grant no. 820970)).

This paper was edited by Tommaso Alberti and reviewed by two anonymous referees.