Many of the ideas and methods of dynamical systems theory were introduced into the climate sciences by a generation of pioneers in the 1960s. formulated a two-box and two-pipe model for the oceans' overturning circulation that had two stable, steady-state solutions with counter-rotating flows. used another form of reduced-order model, by projecting a one-layer, single-gyre wind-driven circulation model onto a small number of Fourier modes and likewise found multiple solutions, both steady and periodic. Most intriguingly, applied the same type of low-truncation Galerkin method to a Boussinesq model of flow between two horizontal plates, heated from below and cooled from above. In this setting, transitions from a quiescent fluid to two mutually symmetric flow patterns and to chaotic solutions occurred. The latter, particularly simple, convection model governed by only three ordinary differential equations (ODEs) provided inspiration for hundreds of papers on deterministically chaotic phenomena in the climate sciences and way beyond.

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 and the “sensitive dependence on initial conditions” that he realized occurred in the evolution of the weather. In fact, in Book I, Chap. IV of , called “Le Hasard”, he states that “it may happen that small differences in the initial conditions produce very great ones in the final phenomena”. And his second example of sensitive dependence is weather:

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 work of , while actually referring to a highly simplified model of thermal convection, illustrates perfectly Poincaré's insights about the role of what we now call deterministic chaos rather than pure chance. presented the applications of dynamical systems theory to large-scale atmospheric and climate dynamics as well as to dynamo theory and geomagnetism, in a systematic book form; they included a gradual introduction to the basic mathematical concepts and tools involved, and this book was reissued by Springer in 2012 as an e-book. provided a considerably expanded version of , in terms of both the mathematical content and the areas of climatic applications, which include oceanographic and coupled ocean–atmosphere phenomena.

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. provides a fine historical overview of the field from 1885 to 1965, along with a fairly complete bibliography. Two important books are and . Further references – including some that treat the subject for infinite-dimensional function spaces, like those that describe the solutions of the partial differential equations of fluid dynamics – are given in the subsequent list of noteworthy insights that dynamical systems theory has provided for the climate sciences.

Following and , we summarize herewith some key insights in the climate sciences from the theory of autonomous dynamical systems, in which time-dependent forcing or coefficients are absent.

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

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 e.g.,and references therein. Significant advances in computational topology have helped substantially in these more static area of applications and will clearly do so in the more dynamic ones contemplated herein.

In Sect. , we present the rather novel approach of Branched Manifold Analysis through Homologies (BraMAH) for approximating the branched manifolds of dynamical systems by a cell complex that allows one to characterize the manifold by its homology groups in phase space . The detection and description of localized coherent sets (LCSs) in two-dimensional flows in physical space by BraMAH-based methods is reviewed 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.  and . In Sect. , we introduce the templex, a novel concept in algebraic topology , which complements the previously mentioned cell complexes of BraMAH by a directed graph (digraph), whose nodes are the cells and which approximates the flow on the branched manifold. The extension of this concept to the noise-perturbed chaotic attractors of RDS theory follows 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 e.g.,. These geometric methods continue to be used, for instance, to analyze datasets of Lagrangian trajectories and understand the geometry of transport .

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. classified invariants as belonging to three distinct categories:

metric invariants, such as dimensions of various types, e.g., correlation dimension or multifractal scaling functions ;

The “recipe” to “knead” the strange attractor is illustrated in Fig.  as a sequence of steps that are topological in nature. Quoting , “sets of initial conditions (cubes) are sliced, by running into an axis with a stable and unstable direction (the z axis for Lorenz-like systems), for example. The different parts flow off in different directions in the phase space, where they may encounter other sliced parts from different regions of the phase space. These are squeezed together and eventually return to regions they originated from (recursion).”

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. . There is a two-way correspondence between topology and dynamics, in a sense that will be clarified in Sect. .

A good starting point for this quick historical perspective is the pioneering paper of Henri Poincaré , who first described the way in which a dynamical system's properties depend upon its topology; see also Chap. 8. The concept of a branched manifold, introduced by , was anticipated in Edward N. Lorenz's famous convection paper: on p. 138, he remarks that “the [computed] trajectory is confined to a pair of surfaces which appear to merge in the lower portion of Fig. 3.” The paper's Fig. 3 is reproduced here, coincidentally, as Fig.  as well. Lorenz plots the isopleths of X as a function of Y and Z of the strange attractor, to approximate surfaces formed by all points on limiting trajectories. The etymology of “isopleth” combines “iso” with the ancient Greek word plêthos, “a great number”, as in the modern English word “plethora”. It is generically used to refer to a curve of points sharing the same value of some quantity. We will return to a stochastically perturbed version of the model in Sects.  and .

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 . These authors discovered that systems whose branched manifolds have the same topology, are dynamically equivalent. From this discovery, a topologist's dream blooms: can one classify types of dynamics as one classifies the elements in Mendeleev's table?

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 . In Sect. , we address the question of how these authors and many more worked with knots, the difficulties that arose with knot theory, and how the latter were solved, at least in part, using the homology groups of algebraic topology. Section – describe the applications of BraMAH to the Lagrangian analysis of fluid flows in physical space, the introduction of digraphs to complement cell complexes in describing the flow on a branched manifold in phase space and the extension of the templexes that describe the latter to noise-driven chaotic systems.

Assume that the state of a system of interest can be described by a vector x∈Rd and that the time evolution of x(t) is governed by the following equation of motion, namely a first-order autonomous ODE: 1x˙=f(x;p). Here (⋅)⋅=d(⋅)/dt, f denotes a generally nonlinear, smooth – i.e., continuously differentiable, up to some order – vector field and p a scalar parameter or, in more general cases, a small set of parameters. For clarity, one separates the variables x from the parameter p by a semicolon. The term “autonomous” refers to the fact that in Eq. () both the coefficients and the forcing are constant in time. This means that changes in p are assumed to be infinitely slow or, at least, very slow compared to the characteristic internal-variability times of the system being modeled.

Points x* for which f(x*;p)=0 are called fixed points. Linearizing the equation of motion around a given fixed point x* yields, for a small perturbation x̃=x-x*, 2x̃˙=f′(x*;p)x̃; here f′(x*;p) is the Jacobian matrix comprised of the elements ∂fi/∂xj. For an initial condition x̃0, the solution to this linearized equation is given by 3x̃(t)=ef′tx̃0. We call a fixed point x* linearly stable if all eigenvalues of f′ have negative real part and linearly unstable otherwise. A scalar example will be given in Sect. .

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. . In this broader context, bifurcations have been called “tippings” in the climate sciences and elsewhere.

There are at least two different interpretations of “tipping” and “tipping points” in the literature. One of these, emanating from and , interprets tipping merely as a sudden change, whether due to a well-defined bifurcation or not. In this interpretation, a tipping point is merely a threshold. The other interpretation sees a tipping point as a generalization to non-autonomous systems of a bifurcation point .

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.

Realistically, the natural systems that we want to describe in terms of dynamical systems theory are non-autonomous, meaning that f in Eq. () above has an explicit time dependence: ∂f/∂t≢0. The Earth system as a whole, as well as all its components, is clearly non-autonomous, being affected by time-dependent forcing, such as quasi-periodic variations in solar insolation due to gravitational perturbations in Earth's orbit , along with anthropogenic forcing due to rising greenhouse gas concentrations .

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 . Including both deterministic and stochastic time dependence requires a description of the dynamics in terms of stochastic differential equations of the form 8dX=F(X,t;p)dt+σ(X)dη, where dη denotes the infinitesimal increments of a Wiener process, which are stationary and independently distributed according to a normal distribution with mean μ=0 and variance E(|dη|2)=dt. Often, a further simplification is made in assuming that the noise is additive or state-independent, and thus σ=const. above. The possibly time-dependent but still deterministic term F(X,t;p) is called the drift.

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.

A straight-line PBA. An even simpler example of a PBA than the one of Eqs. () and () above is given by 16x˙=-αx+σt, with both α and σ being positive. The example was provided by Mickaël D. Chekroun (personal communication, 2011). The autonomous part of this ODE, x˙=-αx, is dissipative, and all solutions x(t;x0)=x(t;x(0)=x0) converge to 0 as t→+∞. What about the non-autonomous, forced ODE?

Here, the time-dependent forcing σt and the state-dependent dissipation -αx will tend to balance. But, again, as in the example of Sect. , there is no forward limit as t→+∞, and one has to use the pullback limit, i.e., replace x(t;x0) by x(s,t;x0)=x(t;x(s)=x0) and let s→-∞. Doing so yields the snapshots 17A(t)=σαt-1α. These snapshots are, in the extremely simple case at hand, just the points along the straight line illustrated in Fig. , which is the graph of the PBA (A(t))t∈ℜ.

A PBA with periodic forcing. To further improve the reader's intuition for PBAs, we provide a second illustrative example here. It was worked out in detail by .

A system defined in polar coordinates by 18ρ˙=α(μ-ρ),ϕ˙=υ,withρ,μ∈R+andϕ∈R/2π, can easily be seen to exhibit a limit cycle in the (x,y) plane with (x=ρcos⁡ϕ,y=ρsin⁡ϕ). An initial deviation of ρ from μ will decay exponentially, and the system converges to an oscillation of radius μ with the angular velocity υ. Here, we transform this autonomous dynamical system into a non-autonomous one by modulating the target radius μ with a sinusoidal forcing 19μ→μ(t)=μ+βsin⁡(νt), where the modulation is moderate, so as to guarantee that μ+βsin⁡(νt)>0 for all t.

Since the dynamics of the phase ϕ and of the radius ρ are decoupled, the corresponding equations can be solved and analyzed separately. While the temporal development of the phase is trivial, the pullback invariant attracting set of the radius for the initial condition ρ(s)=ρ0 is given by 20aA(ρ)(t;ρ0)=lim⁡s→-∞ρ(t,s;ρ0)=αβsin⁡(νt+ϑ)+μ, with 20bϑ=arctan⁡(-ν/α), as shown in Appendix B. Note that, in the limit s→-∞, the dependence on the initial value ρ0 vanishes and the attracting set At(ρ) performs an oscillation of the same frequency as the forcing. It lags the phase of the time-dependent fixed point by the constant ϑ, while its amplitude is amplified by the factor α. Since ρ is restricted to positive values, this solution requires αβ<μ.

The PBA with respect to the coordinate ρ is comprised of the family of all the sets At(ρ) as defined in Eqs. () and () and thus reads 21A(ρ)={αβsin⁡(νt+ϑ)+μ}t∈R. Since the pullback limit for the phase ϕ does not exist, no constraints on it other than ϕ∈[0,2π) are imposed by the dynamics. Hence, for the system () comprised of radius and phase, we find that 22lim⁡t0→-∞dH((ρ(t;t0,ρ0),ϕ(t;t0,ϕ0)),{(αβsin⁡(νt+ϑ)+μ,ϕ):ϕ∈[0,2π)}︸At)=0, where dH denotes the Hausdorff semi-distance. The pullback-attracting sets At at time t are circles in the (x,y) plane with oscillating radius, and the system's PBA is given by the family of these circles: 23A={(αβsin⁡(νt+ϑ)+μ,ϕ):ϕ∈[0,2π)}t∈R.

Figure  shows trajectories of the system starting from different points in the past. In panel a the trajectories are depicted in the 3-D space spanned by the two Cartesian coordinates (x,y) and the time t, where the usual transformation from polar to Cartesian coordinates was applied. The shaded surface in this panel represents the PBA of the system. Panel b shows a heat map that approximates a portion of the PBA's invariant measure projected onto the (x,y) plane. For a clean definition of such a measure in NDSs and RDSs, there are several references e.g.,. Essentially, the heat map here counts the number of times that the trajectories in panel a cross small pixels in the (x,y) plane.

Note that the structure of the system's trajectories depends on the ratio υ/ν, and three different cases must be distinguished. If the radius is modulated with the same frequency as the oscillation itself, i.e.,  υ=ν, after one period the system practically repeats its orbit. More precisely, the radius of the oscillation does differ from one “round trip” to the next, but this difference tends to zero as ρ(t) asymptotically approaches At(ρ). If υ and ν are rationally related, mυ=nν with n,m∈N, then the same quasi-repetition of the orbit occurs after n periods of the radial modulation and m periods of the system's oscillation. Such a trajectory will appear as an n-fold quasi-closed loop. Finally, if υ/ν∉Z, then the trajectory does not repeat itself but instead densely covers the annular disk D={(ρ,ϕ):ρ∈[μ-αβ,μ+αβ] and ϕ∈[0,2π)}. The trivial evolution of the phase is depicted in panel c, while the trajectories of ρ(t) and their convergence to At(ρ) are shown in panel d.

In Sect. , we provided a bird's eye view of the evolution of these two strains of research since the mid-20th century and how they started to be applied to issues related to fluid flows at both engineering and planetary scales. Sections  and developed next in greater detail (a) the concepts and methods associated with dynamical systems and their applications to the climate sciences and (b) those associated with algebraic topology and their applications first to engineering fluid dynamics and then to the climate sciences. Note that the pioneering references mentioned in Sects.  and date back to the early 1960s, while those of Sects.  and start in the early 1980s. It is clear that, on the whole, algebraic topology started playing a noticeable role in the climate sciences about 2 decades later than dynamical systems theory.

Section  covered autonomous dynamical systems, in which neither the forcing nor the coefficients depend explicitly on time. A very extensive and thorough mathematical theory exists and certain aspects of it are well known to a substantial fraction of climate scientists; see, for instance, and . The contents of this section emphasized elementary bifurcations – saddle–node and fold, pitchfork, and Hopf, summarized in Fig.  – ending with bifurcation trees and routes to deterministic chaos.

The material in Sect.  refers to systems with explicit time dependence in the forcing or the coefficients, and it is much newer. The theory of NDSs and RDSs only started in the 1960s – with George Sell, followed by Ed Ott and colleagues and by Ludwig Arnold, Hans Crauel and Franco Einaudi – and its applications to the climate sciences started merely 15 years ago .

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.  and . The sequence of examples was concluded with the striking random attractor of the stochastically perturbed Lorenz model, as introduced and studied by ; see Figs.  and . Finally, tipping points were introduced as the proper generalization to NDSs and RDSs of the elementary bifurcations for autonomous systems described in Sect.  (Fig. ).

In Sect. , we presented topological methods in a dynamical systems perspective. We reviewed the advantages of working with homology theory in order to overcome the limitation of a three-dimensional space imposed by using knot theory, since knots simply disentangle in higher dimensions. In Sect. , we showed that homologies provide a knotless method and that a BraMAH cell complex can be used to describe the spatial structure of a flow in phase space by using homology group generators, weak boundaries and torsion groups (Fig. ). We described an application of these concepts and methods to Lagrangian analysis in Sect. , by showing how to define and detect localized coherent sets (LCSs) for fluid flows in physical space; see again Figs. –.

In Sect. , we dealt with the fact that BraMAH alone does not provide a robust skeleton of the flow in phase space on its branched manifold, even for an autonomous, deterministic system. To obtain such a robust and parsimonious flow description in phase space, we introduced a directed graph (digraph), whose nodes are the cells, while the edges point from one cell to another, in a way that is consistent with the flow on the branched manifold. The mathematical object that combines such a digraph with the underlying cell complex is called a templex. Homologically equivalent attractors – such as the spiral and funnel versions of the attractor – can be distinguished using a templex, given its digraph's properties; see Figs. –.

Finally, in Sect. , we discussed how a digraph, and hence a templex, can be generalized from the autonomous and deterministic version of Sect.  to non-autonomous and random dynamical systems; see Figs. –. To define a random templex, one needs to shift the perspective from defining a digraph on the single-cell complex of an autonomous system to an indexed family of cell complexes at successive instants in time and the vertices pointing from one cell at time t=tj to the corresponding one at time t=tj+1.

The fact that the change in the set of minimal holes of a cell complex at t=tj to the next one is sudden allowed us to rigorously define topological tipping points (TTPs) as happening at an instant at which such a sudden change occurs. It is these TTPs that are a matter of particular interest for future work in the overlap of the two fields that we considered in this review, namely dynamical systems and algebraic topology.

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 and mentioned here toward the end of Sect. . But what does that say about changes in the flow in physical space? Could this localization in phase space say something about the association with localized sudden changes in the flow in physical space, i.e., with the “tipping elements” of ? The use of systematically derived reduced-order models , for which both TTPs and the better understood dynamical tipping points can be computed fairly easily, could help clarify such relationships and the associated precursors of critical transitions.

An interesting example, among many, of localized changes in Earth's physical space is that of persistent anomalies or flow regimes Chap. 6 or weather regimes . have recently applied a multiparameter PH method to decide more objectively the much debated existence of distinct regimes in the large-scale atmosphere's phase space . Their findings certainly strengthen the affirmative reply to the quandary. Before proceeding to the next quandary, though, let us consider briefly the issues that are still open in applying this approach to regime identification.

In applying the multiparameter PH method to the classical convection model, essentially equate the existence of distinct regimes to the existence of two holes in its branched manifold. Doing so, however, is not quite enough. To explain why, we revisit the example of the Rössler attractor discussed in our Sect. . This attractor's spiral form, shown in Fig. a, has a single hole, but it has two strips: one strip related to the system's slow branch and the other strip to its fast branch. These two branches, however, are not separated by a hole in the sense of homology groups, and this is why homologies alone cannot distinguish between them. Instead, the templex introduced in the same subsection captures these two ways of circulating around the attractor in terms of two stripexes, despite the fact that the branched manifold is single-holed.

The existence of multiple regimes in a dynamical system is certainly associated with its attractor's nontrivial topological structure, as state, but this nontrivial topology is not necessarily captured by homologies alone. As explained throughout this work, the templex – with its stripexes and digraph – contributes additional tools to accurately describe the phase-space topology of a flow and of its single or multiple regimes.

, however, asked a more subtle question: is the so-called low-frequency variability (LFV) of the atmosphere – which is closely related to the rapidly growing interest in subseasonal-to-seasonal (S2S) predictability e.g., – oscillatory, i.e., wavelike, or episodic and intermittent, i.e., particle-like. These authors, with an obvious nod to the classical problem of quantum mechanics, formulated the question as “waves” vs. “particles”. Two decades later, this question is still far from settled, as discussed quite recently by and by .

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 . In fact, these two are by now accompanied by several more key dynamical mechanisms of midlatitude LFV variability, summarized in Fig. .

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 . This approach is illustrated in the sketch labeled c within the figure: zonal flow Z and blocked flow B are simply slow phases of a harmonic oscillation, like the neighborhood of t=π/2 or t=3π/2 for a sine wave sin⁡(t); or else they are due to an interference like that occurring for a sum Asin⁡(t)+Bsin⁡(3t) near t=(2k+1)π/2. A more versatile, quasi-linear version of this approach is to study long-lived resonant wave triads between a topographic Rossby wave and two free Rossby waves Sect. 6.2. Neither version of this approach, though, explains the anomalies' organizing into distinct flow regimes.

initiated a different, genuinely nonlinear approach by raising the possibility of multiple equilibria as an explanation of preferred atmospheric flow patterns. These authors drew an analogy between such equilibria and hydraulic jumps and formulated simple models in which similar transitions between faster and slower atmospheric flows could occur. This multiple-equilibrium approach was then pursued quite aggressively in the 1980s Sect. 6.3–6.6, and it is illustrated in Fig.  by the sketch labeled a: one version of the sketch illustrates models that concentrated on the B–Z dichotomy and the other on models e.g., that allowed for the presence of additional clusters, like those found by or , viz. opposite phases of the North Atlantic Oscillation (NAO) and the Pacific North American (PNA) anomalies – dubbed RNA for Reverse PNA and BNAO for Blocked NAO in sketch a of Fig. . The LFV dynamics in this approach are given by the preferred transition paths of a Markov chain between two or more regimes.

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, found a 40 d oscillation arising by Hopf bifurcation off their blocked regime B, as illustrated in sketch b of the figure. An ambiguity arises, though, between this point of view and a complementary possibility, namely that the regimes are just slow phases of such an oscillation, caused itself by the interaction of the midlatitude jet with topography. Thus, found, in their observational data, closed paths within a Markov chain whose states resemble well-known phases of an intraseasonal oscillation. confirmed the likelihood of such a scenario in the intermediate-complexity model of . Furthermore, multiple regimes and intraseasonal oscillations can coexist in a two-layer model on the sphere within the scenario of “chaotic itinerancy” .

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 or in linear inverse models , or red in time, as in empirical model reduction and multilayer stochastic models , or even non-Gaussian . Stochastic processes may enter into models situated on various rungs of the modeling hierarchy, from the simplest conceptual models to high-resolution global climate models. In the latter, they may enter via stochastic parametrizations of subgrid-scale processes e.g.,and references therein, while in the former they may enter via stochastic forcing, whether additive or multiplicative, Gaussian or not e.g.,and references therein. recently drew attention to yet another mechanism of interaction between stochastic forcing and nonlinear regime dynamics that might modify the picture.

How might topological data analysis contribute to clarify this thicket of apparently contradictory descriptions of LFV? One hint is found in the work of , who showed that blocking can be studied by extracting from the complex high-dimensional dynamics of a model its essential building blocks, given by truly nonlinear modes. In this work, they abandoned the classic identification of weather regimes with fixed points, as in , and directly considered the chaotic nature of the atmosphere, using the unstable periodic orbits (UPOs) that are a key component of the topological analysis of the chaos program.

This UPO-based approach did confirm certain theoretical results of and the laboratory findings of – about the relative stability and persistence of blocked and zonal flows – as well as providing further insights into the waves-versus-particles quandary . UPOs can be very useful in characterizing a chaotic system, since the information about them can be obtained in a finite time, which is particularly useful in nonstationary systems, and because a single UPO can already provide substantial information . Still, found it quite hard to carry out the necessary computations of very numerous UPOs even for the relatively simple model.

As explained here in Sects.  and , BraMAH is crucially inspired by the program. Yet it is more powerful than the knots-and-braids methodology, which is limited by the dimensionality of the phase spaces that it can be applied to. Likewise, it is more computationally efficient than the UPO methodology, and, as shown at the beginning of this subsection, it provides considerably more information than the PH methodology for chaotic dynamics.

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 attractor context, stripexes can greatly help, beyond counting holes, to determine regime multiplicity.

Finally, as stated at the end of Sect. , minimal or tight holes in the cell complex of a random templex can be located in phase space using the coordinates of each hole's barycenter. Plotting the position of the barycenters of all the holes present in the analysis in phase space yields a constellation set as shown in Fig. . The topology of this constellation set deserves further exploration. It might lead, quite conceivably, to the generalization of a stripex for a random templex and therefore to the extraction of the nonequivalent paths that a nonlinear system follows when driven by multiplicative noise. Random stripexes should provide us with the stretching, squeezing, folding and tearing mechanisms that knead, mold and alter the topological structure of a noise-driven flow in phase space.

The extension of the templex from autonomous and deterministic systems to non-autonomous and stochastic ones opens the way to the exploration of key aspects of the LFV quandaries associated with Fig. . More broadly, it can facilitate exploring a plethora of climate problems that are strongly affected by time-dependent forcing, such as anthropogenic greenhouse gas and aerosol emissions, and stochastic components, such as cloud microprocesses. One can imagine, for instance, applying methods from network theory to investigate the presence of cyclicity in a given model's or dataset's digraph as well as issues of multimodality or multistability.

More broadly, complex networks have found numerous applications in the climate sciences in recent years and could provide other links between topology and the multivariate time series analysis of nonlinear phenomena. The field of complex networks shares many of the challenges that are faced by the topology of chaos. Algebraic topology is not mentioned in the Zou et al. (2019) paper, but there have been some papers applying PH methods to complex networks . The network approach is used to reconstruct the phase space, which is a preliminary and certainly necessary step for the analysis of the topological structure of flows from data.

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.