The templex is a recently introduced topological object bridging homologies and templates for chaotic attractors: its cell complex encodes the directionless properties of the attractor's branched manifold in phase space, and its directed graph captures the flow-compatible paths starting and ending in joining loci. Algebraic topology is deeply connected to category theory because it studies spaces by translating them into algebraic objects through structure-preserving mappings. The homology functor translates structural properties into a set of layered invariants called homology groups. The templex is shown here to play the same role for directed spaces that cell complexes play for spaces. The directed properties of a templex are found therewith to admit a functorial formulation. This formulation provides a rigorous foundation for a theory of chaos topology developed so far algorithmically, and establishes operationally a topological criterion for finite-time chaos. A climatic simulation and an experimental speech signal are analyzed as illustrative applications.

The templex is a topological object bridging homologies and templates for chaotic dynamics. This article places the templex within category theory, introducing a directed path algebra, an edge operator on directed paths, and an equivalence relation for directed cycles that is distinct from directed homologies. The resulting functorial invariants are of two kinds: Abelian-group invariants, namely the homology groups, and semigroup invariants, namely the generatex semigroups. These invariants are separable through forgetful functors and constitute a robust framework for identifying tipping points, disambiguating physical mechanisms, and benchmarking data-driven models against observations or simulations. The formulation sets forth a nonmetric criterion for chaos from finite-time data and reveals that the concatenable nature of topological modes of variability is a direct consequence of the semigroup structure of the directed path algebra. Two applications are presented: an experimental speech signal and a climatic numerical simulation.

This Focus Issue is, along with the one on Nonautonomous dynamical systems: Theory, methods, and applications, part of the Double Focus Issue on Nonautonomous dynamical systems in the sciences. We refer to the two twin Focus Issues as NDS-G and NDS-C, for short, where "G" stands for General and "C" for Climate. A key area of inquiry for understanding climate behavior in this century is the impact of anthropogenic and natural forcing on a system that is highly nonlinear and exhibits both chaotic and random aspects. The theory of NDSs is the perfect framework for exploring this impact. The 16 papers in this issue address several questions within this broad area: (i) various types of tipping that have arisen in the system's history or may occur in its future; (ii) the effects that one component of the climate system might have on another one; and (iii) how we may better learn from observations and model simulations about all that is going on within the system.

Templexes are topological objects that encode the branching organization of a flow in phase space. We build on these objects to introduce the concept of topological modes of variability (TMVs). TMVs are defined as dynamical manifestations of algebraically defined cycles, called generatexes, in the templex; they provide a concrete link between abstract topological invariants and time-dependent behavior in a model or in observations. We apply this approach to a low-order model of the wind-driven ocean circulation, subject to both periodic and aperiodic forcing, and show how TMVs emerge or vanish over time in nonautonomous settings. The analysis reveals that TMVs allow for a qualitatively new understanding of variability in complex systems where linear modes fail to describe the nonlinear dynamics.

Significant changes in a system’s dynamics can be understood through modifications in the topological structure of its flow in phase space. In the Earth’s climate system, such changes are often referred to as tipping points. One of the large-scale components that may pass a tipping point is the Atlantic Meridional Overturning Circulation. Our understanding of tipping points can be enhanced using a recently proposed mathematical concept—the templex—which enables the identification of dynamics of different classes. Unlike traditional topological invariants, templex properties describe not only the topology of the underlying structure of a set of points in phase space associated with a finite time series but also the non-equivalent pathways allowed by the flow around that structure. In this study, we investigate the dynamics produced by an idealized autonomous model and its nonautonomous counterpart to consider long-term climate changes and reproduce phenomena occurring during different epochs, such as glacial and interglacial intervals. In the nonautonomous system, the trajectory visits two distinct domains in phase space, one of which shares certain properties with those found in the autonomous case. A dissection of the templex and the definition of active templex properties improve our understanding of how the system tips from one regime to another. We also discuss the relationship between our results and the nonautonomous model’s pullback attractor.

This work explores the generation of non-mixing islands in fluid flows-regions where particles remain grouped and resist mixing with the surrounding chaotic motion. Understanding such structures is essential in industrial mixing processes and in geophysical contexts, such as the dispersion of contaminants or nutrients in oceans. Traditional visualization using instantaneous Eulerian fields may misrepresent transport properties, motivating the use of Lagrangian Coherent Structures (LCS) to identify dynamically distinct regions. Building on principles of Chaos Topology (R. Gilmore and M. Lefranc, The Topology of Chaos. John Wiley, New York, (2002)) we classify particles according to their dynamic traits to detect these regions from advection data, including both physical drifters and virtual particles in simulations. The method applied here extracts topological information from time series, following a delay embedding reconstruction of the dynamics and an approximation of the branched manifold underlying the attractor. This framework has been successfully applied to incompressible flow models such as the Driven Double Gyre, the Bickley Jet, and CFD simulations of cylinder wakes (G. D. Charó et al., Physica D: Nonlinear Phenomena, 405:132371, (2020); G. D. Charó et al., Journal of Fluid Mechanics, 923, A17, (2021)). In this study, we analyze flow past a backward-facing step, using both 2D numerical simulations and wind tunnel experiments at matched Reynolds numbers. The region of interest is the downstream recirculating zone, influenced by the oscillatory dynamics of the reattachment point. Time series of particle positions are embedded to reconstruct phase-space dynamics, from which cell complexes are constructed. Topological coloring is used to label particles according to their finite-time dynamical class. The results distinguish regions of distinct Lagrangian behavior and allow for a comparison of the lifetime of non-mixing islands across experimental and numerical data, highlighting the robustness of the approach.

The wind-driven ocean circulation comprises the oceanic currents that are visible at the surface. In this paper, we use algebraic topology concepts and methods to study a highly simplified model of the evolution of this circulation subject to periodic winds. The low-order spectral model corresponds to a midlatitude ocean basin. For steady forcing, the model's intrinsic oscillations undergo a bifurcation from small-amplitude harmonic ones to relaxation oscillations (ROs) of high amplitude as the forcing increases. The ROs, in turn, give rise to chaotic behavior under periodic forcing. Topological invariants help identify distinct flow regimes that ensemble simulations visit under the action of the underlying deterministic rule in such a nonautonomous framework. We introduce topological variability modes of this idealized ocean circulation, based on the previously defined invariants.

Theoretical and numerical studies have shown that transient atmospheric motions leading to weather extremes can be classified through the instantaneous dimension and stability of a state of a dynamical system [Faranda et al., Sci. Rep., 2017]. The asymptotic values of these quantities can be computed theoretically only for specific systems, while their numerical counterpart for climate observables provides information on the rarity, predictability, and persistence of specific states. In this work, we present a first attempt to relate the presence of extreme events with the elements that make up a templex of the system under study, both in the deterministic [Charó et al., Chaos, 2022] and stochastic frameworks [Charó et al., Chaos, 2023]. The templex provides the key characteristics of the topological structure underlying a dynamical system. This work will present results for the classical, deterministic Lorenz [JAS, 1963] attractor and for the Lorenz Random Attractor, dubbed LORA [Ghil & Sciamarella, NPG, 2023].

This work presents the first application of the templex approach to observational datasets, using Lagrangian trajectories obtained from satellite altimetry in the ocean. The templex is a recent topological construct that extends classical ideas from template theory to higher-dimensional systems. Unlike other methods in topological data analysis, which lack flow information, a templex encodes both the structure of phase space through a branched manifold analysis through homologies cell complex and the organization of flow cycles upon it through a directed graph defined on its highest-dimensional cells. As shown in earlier works, the description of flows in phase space indirectly enables a description of fluid flows in physical space, since particles sharing the same dynamics are known to move coherently. Particle sets in the Southwestern Atlantic Ocean are analyzed, revealing that the Lagrangian finite-time dynamics in this region can be related to those produced by a nonautonomous meandering jet model. We distinguished non-mixing (regular) islands from the chaotic sea. The results are also compared to those obtained from metric methods describing material transport in fluid flows and to the spatial organization of chlorophyll-a concentration. The seasonal variability of chaotic dynamics is also discussed.

Discriminating different types of chaos is still a very challenging topic, even for dissipative three-dimensional systems for which the most advanced tool is the template. Nevertheless, getting a template is, by definition, limited to three-dimensional objects based on knot theory. To deal with higher-dimensional chaos, we recently introduced the templex combining a flow-oriented BraMAH cell complex and a directed graph (a digraph). There is no dimensional limitation in the concept of templex. Here, we show that a templex can be automatically reduced into a "minimal" form to provide a comprehensive and synthetic view of the main properties of chaotic attractors. This reduction allows for the development of a taxonomy of chaos in terms of two elementary units: the oscillating unit (O-unit) and the switching unit (S-unit). We apply this approach to various well-known attractors (Rössler, Lorenz, and Burke-Shaw) as well as a non-trivial four-dimensional attractor. A case of toroidal chaos (Deng) is also treated.

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.

Random attractors are the time-evolving pullback attractors of stochastically perturbed, deterministically chaotic dynamical systems. These attractors have a structure that changes in time, and that has been characterized recently using BraMAH cell complexes and their homology groups (Chaos, 2021, doi:10.1063/5.0059461). A more complete description is obtained for their deterministic counterparts if the cell 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 (Chaos, 2022, doi:10.1063/5.0092933). The present work (Chaos, 2023, arXiv:2212.14450 [nlin.CD]) introduces the stochastic version of a templex. Stochastic attractors in the pullback approach, like the LOrenz Random Attractor (LORA), include sharp transitions in their branched manifold. These sharp transitions can be suitably described using what we call here a random templex. In a random templex, there is one cell complex per snapshot of the random attractor and the 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 topology of the branched manifold associated with the Lorenz model’s random attractor (LORA) evolves in time. LORA’s time-evolving branched manifold robustly supports the point cloud associated with the system’s invariant measure at each instant in time. This manifold undergoes not only continuous deformations — with branches that bend, stretch or compress — but also discontinuous deformations, with branches that intersect at discrete times. These discontinuities in the system's invariant measure manifest themselves in the decrease or increase of the number of 1-holes, thus producing abrupt changes in the branched manifold’s topology. Topological tipping points (TTPs) are defined as abrupt changes in the topology of a random attractor’s branched manifold. Branched Manifold Analysis through Homologies (BraMAH) is a robust method that allows one to detect these fundamental changes. The existence of such TTPs is being confirmed by careful statistical analysis of LORA’s time-evolving branched manifold, following up on Charó et al. (Chaos, 2021, doi:10.1063/5.0059461). Research is being pursued on early warning signals for these TTPs, concentrating on local fluctuations in the system’s invariant measure.

This work considers the two-dimensional flow field of an incompressible viscous fluid in a parallel-sided channel. In our study, one of the walls is fixed whereas the other one is elastically mounted, and sustained oscillations are induced by the fluid motion. The flow that forces the wall movement is produced as a consequence that one of the ends of the channel is pressurized, whereas the opposite end is at atmospheric pressure. The study aims at reducing the complexity of models for several physiological systems in which fluid-structure interaction produces large deformation of the wall. We report the experimental results of the observed self-sustained oscillations. These oscillations occur at frequencies close to the natural frequency of the system. The vertical motion is accompanied by a slight trend to rotate the moving mass at intervals when the gap height is quite narrow. We propose a simplified analytical model to explore the conditions under which this motion is possible. The analytical approach considers asymptotic solutions of the Navier–Stokes equation with a perturbation technique. The comparison between the experimental pressure measured at the midlength of the channel and the analytical result issued with a model neglecting viscous effects shows a very good agreement. Also, the rotating trend of the moving wall can be explained in terms of the quadratic dependence of the pressure with the streamwise coordinate that is predicted by this simplified model.

The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies: this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest-dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated-namely, the spiral and funnel Rössler attractors, the Lorenz attractor, the Burke and Shaw attractor, and a four-dimensional system. A link is established with their description in terms of templates.

Abstract This work describes the application of a technique that extracts branched manifolds from time series to study numerically generated fluid particle behaviour in the wake past a cylinder performing a rotary oscillation at low Reynolds numbers, and compares it with the results obtained for a paradigmatic analytical model of Lagrangian motion: the driven double gyre. The approach does not require prior knowledge of the underlying equations defining the dataset. The time series taken as input corresponds to the evolution of a position coordinate of an individual fluid particle. A delay embedding is used to reconstruct the dynamics in phase space, and a cell complex is built to characterize the topology of the embedding. Fluid particles are said to belong to the same topological class when the Betti numbers, orientability chains and weak boundaries of the associated cell complexes coincide. Topological colouring consists of labelling or ‘colouring’ advected particles with the topological class obtained in their finite-time analyses. The results suggest that topological colouring can be used to distinguish between regions of the flow where trajectories exhibit different finite-time dynamics.

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be "strange" but it is frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies-a technique originally introduced to characterize the topological structure of deterministically chaotic flows-which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA's evolution includes sharp transitions that appear as topological tipping points.