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Qualitative and quantitative properties of 'snapshot attractors' of random maps are considered. A snapshot attractor is the measure resulting from many iterations of a cloud of initial conditions viewed at a single instant. The multifractal properties of these snapshot attractors are studied, using the Liapunov-number partition function method to calculate the spectra of generalized dimensions and of scaling indices. Special attention is devoted to the numerical implementation of scaling indices. Special attention is devoted to the numerical implementation of the method and the evaluation of statistical errors due to the finite number of sample orbits. The work is relevant to problems in the convection of particles by chaotic fluid flows.

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... Les snapshot attractors sont l'outil qui va remplacer les orbites. Ils ont été introduits par Romeiras et al. (1990) pour les systèmes dynamiques ergodiques. Examinons dans un premier temps le cas du des anciens automnes, ce qui est compatible avec un glissement des attracteurs vers des valeurs plus « chaudes ». ...

... They are illustrated by two long trajectories in Fig. 2. To quantify the difference between the two attractors, it is first necessary to estimate the invariant measure of both attractors. We use the method of snapshot attractors (e.g., Romeiras et al., 1990;Chekroun et al., 2011) rather than considering one single long trajectory that could bias the sampling of some regions of the attractors, and requires the system to be ergodic. In the snapshot attractors, we draw N random initial conditions following a uniform distribution. ...

... We use the method of snapshot attractors (e.g. Romeiras et al., 1990;Chekroun et al., 2011) rather than considering one single long trajectory that could bias the sampling of 30 some regions of the attractors, and requires the system to be ergodic. In the snapshot attractors, we draw N random initial conditions following a uniform distribution. ...

Le système climatique génère un attracteur étrange, décrit par une distribution de probabilité, nommée la mesure SRB (Sinai-Ruelle-Bowen). Cette mesure décrit l'état et sa dynamique du système. Le but de cette thèse est d'une part de quantifier les modifications de cette mesure quand le climat change. Pour cela, la distance de Wasserstein, venant de la théorie du transport optimal, permet de mesurer finement les différences entre distributions de probabilités. Appliquée à un modèle jouet de Lorenz non autonome, elle a permis de détecter et quantifier l'altération due à un forçage similaire à celui du forçage anthropique. La même méthodologie a été appliquée à des simulations de scénarios RCP du modèle de l'IPSL. Des résultats cohérents avec les différents scénarios ont été retrouvés.D'autre part, la théorie du transport optimal fournit un contexte théorique pour la correction de biais dans un contexte stationnaire : une méthode de correction de biais est équivalente à une loi jointe de probabilité. Une loi jointe particulière est sélectionnée grâce à la distance de Wasserstein (méthode Optimal Transport Correction, OTC). Cette approche étend les méthodes de corrections en dimension quelconque, corrigeant en particulier les dépendances spatiales et inter-variables. Une extension dans le cas non-stationnaire a également été proposée (méthode dynamical OTC, dOTC). Ces deux méthodes ont été testées dans un contexte idéalisé, basé sur un modèle de Lorenz, et sur des données climatiques (une simulation climatique régionale corrigée avec des ré-analyses SAFRAN).

... The paper is organized as follows. In the next Section we show that a changing climate can be described by an extension of the traditional theory of chaotic attractors: in particular, the theory of snapshot/pullback attractors [8,9] appears to be an appropriate tool to handle the problem. This necessitates investigating ensembles of "parallel climate realizations" to be discussed in detail in Sect. ...

... From the 1990s, nevertheless, there has been a progress in the understanding of nonautonomous dynamics subjected to forcings of general time dependence: the concept of snapshot attractors emerged. The first article written by Romeiras et al. [8] drew attention to an interesting difference: a single long "noisy" trajectory traces out a fuzzy shape, while an ensemble of motions starting from many different initial conditions, using the same noise realization along each track, creates a structured fractal pattern at any instant, which changes in time. 1 With the help of this concept, phenomena that were not interpretable in the traditional view could be explained as snapshot attractors, e.g. a fractal advection pattern in an irregular surface flow which also appeared on the cover of Science [11]. It is worth noting that the concept of snapshot attractors has been used to understand a variety of time-dependent physical systems (see e.g. ...

... One sees indeed that this is constant up to the onset of the climate change at t = 0, after which a slightly increasing trend occurs with strong irregular vacillations superimposed on it. 8 The magenta curve is a single time series of x from one of the ensemble members. The vacillations here are so strong that hardly any trend can be extracted, but, as discussed earlier, such time series are ill-defined due to the unpredictability of the dynamics. ...

Based on the theory of “snapshot/pullback attractors”, we show that important features of the climate change that we are observing can be understood by imagining many replicas of Earth that are not interacting with each other. Their climate systems evolve in parallel, but not in the same way, although they all obey the same physical laws, in harmony with the chaotic-like nature of the climate dynamics. These parallel climate realizations evolving in time can be considered as members of an ensemble. We argue that the contingency of our Earth’s climate system is characterized by the multiplicity of parallel climate realizations rather than by the variability that we experience in a time series of our observed past. The natural measure of the snapshot attractor enables one to determine averages and other statistical quantifiers of the climate at any instant of time. In this paper, we review the basic idea for climate changes associated with monotonic drifts, and illustrate the large number of possible applications. Examples are given in a low-dimensional model and in numerical climate models of different complexity. We recall that systems undergoing climate change are not ergodic, hence temporal averages are generically not appropriate for the instantaneous characterization of the climate. In particular, teleconnections, i.e. correlated phenomena of remote geographical locations are properly characterized only by correlation coefficients evaluated with respect to the natural measure of a given time instant, and may also change in time. Physics experiments dealing with turbulent-like phenomena in a changing environment are also worth being interpreted in view of the attractor-based ensemble approach. The possibility of the splitting of the snapshot attractor to two branches, near points where the corresponding time-independent system undergoes bifurcation as a function of the changing parameter, is briefly mentioned. This can lead in certain climate-change scenarios to the coexistence of two distinct sub-ensembles representing dramatically different climatic options. The problem of pollutant spreading during climate change is also discussed in the framework of parallel climate realizations.

... Furthermore, comparison of solutions of our system (that only differ in their initial conditions) reveals that the framework of snapshot attractors 14,15 is the appropriate formulation of our prob-lem. Practically, this means that all investigations are carried out over an ensemble of climate realizations. ...

... However, these turn out to be not representative of the dynamics of all the possible climate histories under the given forcing. Instead, we turn to the framework of parallel climate realizations 29 , i.e. to the application of snapshot attractors 14,15,30,31 to the climate dynamical model. ...

Using an intermediate complexity climate model (Planet Simulator), we investigate the so-called Snowball Earth transition. For certain values of the solar constant, the climate system allows two different stable states: one of them is the Snowball Earth, covered by ice and snow, and the other one is today's climate. In our setup, we consider the case when the climate system starts from its warm attractor (the stable climate we experience today), and the solar constant is decreased continuously in finite time, according to a parameter drift scenario, to a state, where only the Snowball Earth's attractor remains stable. This induces an inevitable transition, or climate tipping from the warm climate. The reverse transition is also discussed. Increasing the solar constant back to its original value on individual simulations, we find that the system stays stuck in the Snowball state. However, using ensemble methods i.e., using an ensemble of climate realizations differing only slightly in their initial conditions we show that the transition from the Snowball Earth to the warm climate is also possible with a certain probability. From the point of view of dynamical systems theory, we can say that the system's snapshot attractor splits between the warm climate's and the Snowball Earth's attractor.

... In this study an investigation was carried out of the randomness of attractor dynamics resulting from solution of Lorenze's equations under the influence of a cloud of initial conditions based on the concepts by Romeiras et.al [5]. Since the system attractor may not provide sufficient information on the state of the dynamic system, hence the result concerning the state of the system is inconclusive as to whether the system is stable, periodic, or chaotic. ...

... It is evident from comparing figures (a) and (b) for each case that Lorenz's model is relatively stable for random effects.The proposed technique is equivalent to producing a snapshot attractor [5],selection of a large number of initial conditions and continuation of the integration process for a specific value of time t then plotting the resulting points which is equivalent to taking a snapshot of the evolution with time for a set of points in the mode space which results in a new attractor however the general shape is not different than Lorenz's known attractor,also the result of Lyapunov's spectrum indicate different behaviours according to the values given in table (1). ...

The effect of the choice of a cloud of initial conditions on the behavior of Lorenz model is studied. Attractors are generated by the usual method and by the snapshot, results of which prove the invariance of Lorenz model under the effect of randomness. Introduction:

... The pullback attractors (PBAs) of RDS theory 34,35 provide a natural framework for doing so and are the mathematically rigorous counterpart of the heuristically defined snapshot attractors of nonlinear physics. 36 Further theoretical details on PBAs and RDSs appear in Appendixes A and B. ...

... The subsets {A(t)} are often called snapshots. 5,36,84,85 According to (1) and (2) ...

Noise modifies the behavior of chaotic systems. Algebraic topology sheds light on the most fundamental effects involved, as illustrated here by using the Lorenz (1963) model. This model's attractor is "strange" but frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Here, we use Branched Manifold Analysis via Homologies (BraMAH) to describe LORA's coarse-grained branched manifold. BraMAH is thus extended from deterministic flows to noise-driven systems. LORA's homology groups change in time and differ from the deterministic one.

... The mathematical concepts underlying the ensemble view are snapshot (Romeiras et al., 1990) or pullback (Ghil et al., 2008) attractors. One might consider the ensemble of all permitted climate realizations over all times as the pullback attractor of the problem and the set of the permitted states of the climate at a given time instant as the snapshot attractor belonging to that time instant (their union over all time instants is the pullback attractor). ...

... Such an ensemble approach was shown to be the only method providing reliable statistical predictions in systems with underlying nonpredictable dynamics (since in this class the traditional approach based on a single time series is known to provide seriously biased results). A number of papers illustrate these statements within the physics literature (see, e.g., Romeiras et al., 1990;Lai, 1999;Serquina et al., 2008), as well as in low-order climate models (Chekroun et al., 2011;Bódai et al., 2011;Bódai and Tél, 2012;Bódai et al., 2013;Drótos et al., 2015), in general circulation models Kaszás et al., 2019;Pierini et al., 2018Pierini et al., , 2016Drótos et al., 2017;Herein et al., 2017;Bódai et al., 2020;Haszpra et al., 2020b, a), and also in experimental situations (Vincze, 2016;Vincze et al., 2017). ...

We develop a conceptual coupled atmosphere-phytoplankton model by combining the Lorenz'84 general circulation model and the logistic population growth model under the condition of a climate change due to a linear time dependence of the strength of anthropogenic atmospheric forcing. The following types of couplings are taken into account: (a) the temperature modifies the total biomass of phytoplankton via the carrying capacity; (b) the extraction of carbon dioxide by phytoplankton slows down the speed of climate change; (c) the strength of mixing/turbulence in the oceanic mixing layer is in correlation with phytoplankton productivity. We carry out an ensemble approach (in the spirit of the theory of snapshot attractors) and concentrate on the trends of the average phytoplankton concentration and average temperature contrast between the pole and Equator, forcing the atmospheric dynamics. The effect of turbulence is found to have the strongest influence on these trends. Our results show that when mixing has sufficiently strong coupling to production, mixing is able to force the typical phytoplankton concentration to always decay globally in time and the temperature contrast to decrease faster than what follows from direct anthropogenic influences. Simple relations found for the trends without this coupling do, however, remain valid; just the coefficients become dependent on the strength of coupling with oceanic mixing. In particular, the phytoplankton concentration and its coupling to climate are found to modify the trend of global warming and are able to make it stronger than what it would be without biomass.

... Such an attractor is called a pullback attractor (PBA; e.g., Ghil et al., 2008;Chekroun et al., 2011;Kloeden and Rasmussen, 2011;Carvalho et al., 2012) in the mathematical literature and a snapshot attractor (e.g., Romeiras et al., 1990;Bódai and Tél, 2012;Bódai et al., 2013) in the physical literature; it provides the natural extension to nonautonomous dissipative dynamical systems of the classical concept of an attractor that is fixed in time for autonomous systems. A global PBA is defined as a time-dependent set A(t) in the system's phase space that is invariant under its governing equations, along with the equally time-dependent, invariant measure µ(t) supported on this set, and to which all trajectories starting in the remote past converge (Arnold, 1998;Rasmussen, 2007;Kloeden and Rasmussen, 2011;Carvalho et al., 2012). ...

... Such an attractor is called a pullback attractor (PBA; e.g., Ghil et al., 2008;Chekroun et al., 2011;Kloeden and Rasmussen, 2011;Carvalho et al., 2012) in the mathematical literature and a snapshot attractor (e.g., Romeiras et al., 1990;Bódai and Tél, 2012;Bódai et al., 2013) in the physical literature; it provides the natural extension to nonautonomous dissipative dynamical systems of the classical concept of an attractor that is fixed in time for autonomous systems. A global PBA is defined as a time-dependent set A(t) in the system's phase space that is invariant under its governing equations, along with the equally time-dependent, invariant measure µ(t) supported on this set, and to which all trajectories starting in the remote past converge (Arnold, 1998;Rasmussen, 2007;Kloeden and Rasmussen, 2011;Carvalho et al., 2012). In the deterministic case, it is understood that A(t) depends also on the particular forcing, say F (t), that is being applied, but this dependence is usually not kept track of in the notation. ...

A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos induced by periodic forcing in systems that are nonchaotic in the autonomous limit. The analysis relies on the construction of the system's pullback attractors (PBAs) through ensemble simulations, based on a large number of initial states in the remote past. A preliminary analysis of the autonomous system is carried out by investigating its bifurcation diagram, as well as by calculating a metric that measures the mean distance between two initially nearby trajectories, along with the system's entropy. We find that nonchaotic attractors can still exhibit sensitive dependence on initial data over some time interval; this apparent paradox is resolved by noting that the dependence only concerns the phase of the periodic trajectories, and that it disappears once the latter have converged onto the attractor. The periodically forced system, analyzed by the same methods, yields periodic or chaotic PBAs depending on the periodic forcing's amplitude ε. A new diagnostic method – based on the cross-correlation between two initially nearby trajectories – is proposed to characterize the transition between the two types of behavior. Transition to chaos is found to occur abruptly at a critical value εc and begins with the intermittent emergence of periodic oscillations with distinct phases. The same diagnostic method is finally shown to be a useful tool for autonomous and aperiodically forced systems as well.

... However, the body of the presently available scientific analysis, albeit increasing (Lenton and Vaughan, 2013;Ferraro et al., 2014), is yet lacking the consideration of many more crucial aspects of the problem. For example, ours is the first attempt to conceptually categorize and frame geoengineering in terms of response theory (Kubo, 1966;Ruelle, 2009) and the theory of nonautonomous dynamical systems (Sell, 1967a, b;Romeiras et al., 1990;Crauel and Flandoli, 1994;Crauel et al., 1997;Arnold, 1998;Kloeden and Rasmussen, 2011;Carvalho et al., 2013), and then in turn as an inverse problem. This can be of little surprise, as these mathematical tools, although having been introduced to climate science for decades (Leith, 1975;Bell, 1980;Nicolis et al., 1985), are far from being exhausted, still finding many applications of tackling problems in climate science in general (Cionni et al., 2004;Gritsun and Branstator, 2007;Kirk-Davidoff, 2009;Majda 5 et al., 2010;Cooper et al., 2013;Lucarini and Sarno, 2011;Ragone et al., 2016;Lucarini et al., 2017;Herein et al., 2015Herein et al., , 2017Bódai and Tél, 2012;Drótos et al., 2015Drótos et al., , 2016. ...

... (1) the response of the system to an external forcing f (t) can be unambiguously defined in terms of the so-called snapshot attractor (Romeiras et al., 1990) of the system, and the natural probability distribution or the measure µ t (dx) supported by it. Both the attractor and the measure are unique objects; they are defined by an ensemble of trajectories initialized in the infinite past. ...

We investigate in an intermediate-complexity climate model (I) the applicability of linear response theory to assessing a geoengineering method, and (II) the success of the considered method. The geoengineering problem is framed here as a special optimal control problem, which leads mathematically to the following inverse problem. A given rise in carbon dioxide concentration [CO2] would result in a global climate change with respect to an appropriate ensemble average of the surface air temperature Ts]>. We are looking for a suitable modulation of solar forcing which can cancel out the said global change, or modulate it in some other desired fashion. It is rather straightforward to predict this solar forcing, considering an infinite time period, by linear response theory, and we will spell out an iterative procedure suitable for numerical implementation that applies to finite time periods too.
Regarding (I), we find that under geoengineering, i.e. the combined greenhouse and solar forcing, the actual response ΔTs]> asymptotically is not zero, indicating that the linear susceptibility is not determined correctly. This is due to a significant quadratic nonlinearity of the response under system identification achieved by a forced experiment. This nonlinear contribution can in fact be easily removed, which results in much better estimates of the linear susceptibility, and, in turn, in a five-fold reduction in ΔTs]> under geoengineering. Regarding (II), however, we diagnose this geoengineering method to result in a considerable spatial variation of the surface temperature anomaly, reaching more than 2 [K] at polar/high latitude regions upon doubling the [CO2] concentration, even in the ideal case when the geoengineering method was successful in canceling out the response in the global mean. In the same time, a new climate is realised also in terms of e.g. an up to 4 [K] cooler tropopause or drier/disrupted Tropics, relative to unforced conditions.

... The pullback attractors (PBAs) of RDS theory 34,35 provide a natural framework for doing so and are the mathematically rigorous counterpart of the heuristically defined snapshot attractors of nonlinear physics. 36 Further theoretical details on PBAs and RDSs appear in Appendixes A and B. ...

... The subsets {A(t)} are often called snapshots. 5,36,84,85 According to (1) and (2) ...

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.

... However, the body of the presently available scientific analysis, albeit increasing [39,24], is yet lacking the consideration of many more crucial aspects of the problem. For example, ours is the first attempt to conceptually categorize and frame geoengineering in terms of response theory [37,47] and the theory of nonautonomous dynamical systems [48,49,46,21,20,8,36,13], and then in turn as an inverse problem. This can be of little surprise, as these mathematical tools, although having been introduced to climate science for decades [38,9,44], are far from being exhausted, still finding many applications of tackling problems in climate science in general [15,28,35,43,17,40,45,42,31,30,11,22,23]. ...

... (1) the response of the system to an external forcing f (t) can be unambiguously defined in terms of the so-called snapshot attractor [46] of the system, and the natural probability distribution or the measure µ t (dx) supported by it. Both the attractor and the measure are unique objects; they are defined by an ensemble of trajectories initialized in the infinite past. ...

We investigate in an intermediate-complexity climate model (I) the applicability of linear response theory to a geoengineering problem and (II) the success of the considered geoengineering method. The geoengineering method is framed as an optimal control problem, which leads mathematically to the following inverse problem. A given rise in the CO2 concentration would result in a global climate change. We are looking for a suitable modulation of solar forcing which can cancel out the said global change, or modulate it in some other desired fashion. It is possible to predict this solar forcing, considering an infinite time period, by linear response theory, and we will spell out an iterative procedure suitable for numerical implementation that applies to finite time periods too. Regarding (I), we find that under geoengineering, i.e. the combined greenhouse and solar forcing, the actual response asymptotically is not zero, indicating that the linear susceptibility is not determined correctly. This is due to a significant quadratic nonlinearity of the response under system identification achieved by a forced experiment. This nonlinear contribution can in fact be easily removed, which results in much better estimates of the linear susceptibility, and, in turn, in a five-fold reduction in the temperature change under geoengineering. Regarding (II), however, we diagnose this geoengineering method to result in a considerable spatial variation of the surface temperature anomaly, reaching more than 2K at polar/high latitude regions upon doubling the CO2 concentration, even in the ideal case when the geoengineering method was successful in canceling out the response in the global mean. In the same time, a new climate is realised also in terms of e.g. an up to 4 K cooler tropopause or drier/disrupted Tropics, relative to unforced conditions.

... We study the complexity of the quasiperiodic driven dynamical systems considering the snapshot attractor [30] on the set of points, which are determined by the Poencare section. This object corresponds to a given time moment, which contains the points of the trajectories ensemble. ...

We consider the concept of statistical complexity to write the quasiperiodical damped systems applying the snapshot attractors. This allows us to understand the behaviour of these dynamical systems by the probability distribution of the time series making a difference between the regular, random and structural complexity on finite measurements. We interpreted the statistical complexity on snapshot attractor and determined it on the quasiperiodical forced pendulum.

... We study the complexity of the quasiperiodic driven dynamical systems con- sidering the snapshot attractor [30] on the set of points, which are determined by the Poencaré section. This object corresponds to a given time moment, which contains the points of the trajectories ensemble. ...

We consider the concept of statistical complexity to write the quasiperiodical damped systems applying the snapshot attractors. This allows us to understand the behaviour of these dynamical systems by the probability distribution of the time series making a difference between the regular, random and structural complexity on finite measurements. We interpreted the statistical complexity on snapshot attractor and determined it on the quasiperiodical forced pendulum.

... After a transient time the ensemble correctly characterizes the potential set of typical climate states permitted by the climate dynamics. An introduction to the concept of this ensemble approach can be found in refs 15,17,[19][20][21] , and an experimental implementation of this concept is also available 22 . ...

The intensity of the atmospheric large-scale spreading can be characterized by a measure of chaotic systems, called topological entropy. A pollutant cloud stretches in an exponential manner in time, and in the atmospheric context the topological entropy corresponds to the stretching rate of its length. To explore the plethora of possible climate evolutions, we investigate here pollutant spreading in climate realizations of two climate models to learn what the typical spreading behavior is over a climate change.An overall decrease in the areal mean of the stretching rate is found to be typical in the ensembles of both climate models. This results in larger pollutant concentrations for several geographical regions implying higher environmental risk. A strong correlation is found between the time series of the ensemble mean values of the stretching rate and of the absolute value of the relative vorticity. Here we show that, based on the obtained relationship, the typical intensity of the spreading in an arbitrary climate realization can be estimated by using only the ensemble means of the relative vorticity data of a climate model.

... PBAs is that of snapshot attractors; it was introduced in a more intuitive and less rigorous manner to the physical literature by Romeiras, Grebogi, and Ott (1990). Snapshot attractors have also been used for studying time-dependent problems of climatic relevance Tél, 2011,2013;Bódai and Tél, 2012;Drótos, Bódai, and Tél, 2015). ...

The climate is a forced, dissipative, nonlinear, complex, and heterogeneous system that is out of thermodynamic equilibrium. The system exhibits natural variability on many scales of motion, in time as well as space, and it is subject to various external forcings, natural as well as anthropogenic. This review covers the observational evidence on climate phenomena and the governing equations of planetary-scale flow and presents the key concept of a hierarchy of models for use in the climate sciences. Recent advances in the application of dynamical systems theory, on the one hand, and nonequilibrium statistical physics, on the other hand, are brought together for the first time and shown to complement each other in helping understand and predict the system’s behavior. These complementary points of view permit a self-consistent handling of subgrid-scale phenomena as stochastic processes, as well as a unified handling of natural climate variability and forced climate change, along with a treatment of the crucial issues of climate sensitivity, response, and predictability.

... the physical literature by Romeiras et al. (1990). Snapshot attractors have also been used for studying timedependent problems of climatic relevance (Bódai et al., 2013(Bódai et al., , 2011Bódai and Tél, 2012;Drótos et al., 2015). ...

The climate system is a forced, dissipative, nonlinear, complex and heterogeneous system that is out of thermodynamic equilibrium. The system exhibits natural variability on many scales of motion, in time as well as space, and it is subject to various external forcings, natural as well as anthropogenic. This paper reviews the observational evidence on climate phenomena and the governing equations of planetary-scale flow, as well as presenting the key concept of a hierarchy of models as used in the climate sciences. Recent advances in the application of dynamical systems theory, on the one hand, and of nonequilibrium statistical physics, on the other, are brought together for the first time and shown to complement each other in helping understand and predict the system's behavior. These complementary points of view permit a self-consistent handling of subgrid-scale phenomena as stochastic processes, as well as a unified handling of natural climate variability and forced climate change, along with a treatment of the crucial issues of climate sensitivity, response, and predictability.

... The overall idea is that the relevant information in the climate system must be derived from statistical analyses of an ensemble of different system trajectories, each corresponding to a different initial state, provided that the corresponding trajectories have converged to the system's time-dependent attractor. 20 Such an attractor is called a pullback attractor (PBA; e.g., Ghil et al., 2008;Chekroun et al., 2011;Kloeden and Rasmussen, 2011;Carvalho et al., 2012) in the mathematical literature and a snapshot attractor (e.g., Romeiras et al., 1990;Bódai and Tél, 2012;Bódai et al., 2013) in the physical literature; it provides the natural extension to nonautonomous dissipative dynamical systems of the classical concept of an attractor that is fixed in time for autonomous systems. A global PBA is defined as a time-dependent set A(t) in the system's phase space that is invariant under its governing equations, along with the equally 25 time-dependent, invariant measure µ(t) supported on this set, and to which all trajectories starting in the remote past converge (Arnold, 1998;Rasmussen, 2007;Kloeden and Rasmussen, 2011;Carvalho et al., 2012). ...

The transition to chaos induced by periodic forcing in systems that are not chaotic in the autonomous limit is studied with a four-dimensional nonlinear spectral ocean model. The analysis is based on the systematic construction of the system’s pullback attractors (PBAs) through ensemble simulations derived from a large number of initial states in the remote past. A preliminary analysis of the autonomous system is carried out by constructing its bifurcation diagram, as well as by calculating a metric measuring the mean distance between two initially nearby trajectories, along with the system’s entropy. We find that nonchaotic attractors can still exhibit sensitive dependence on initial data; this apparent paradox is resolved by noting that the dependence only concerns the phase of the periodic trajectories, and that it disappears once the latter have converged onto the attractor. The periodically forced system, analyzed by the same methods, yields periodic or chaotic PBAs depending on the periodic forcing’s amplitude ε. A new diagnostic method – based on the cross-correlation between two initially nearby trajectories – is proposed to characterize the transition between the two types of behavior. Transition to chaos is found to occur abruptly at a critical value εc and begins with the intermittent emergence of periodic oscillations with distinct phases. The same diagnostic method is finally shown to be a useful tool for autonomous and aperiodically forced systems as well.

... The mathematical concepts underlying this qualitative view are snapshot ( Romeiras et al, 1990) or pullback ( Ghil et al, 2008) attractors. One might consider the ensemble of all permitted climate realizations over all times as the pullback attractor of the problem, and the set of the permitted states of the climate at a given time instant as the snapshot attractor belonging to that time instant (their union over all time instants is the pullback attractor). ...

Abstract. We develop a conceptual coupled atmosphere–phytoplankton model by combining the Lorenz'84 general circulation model and the logistic population growth model under the condition of a climate change due to a linear time dependence of the strength of anthropogenic atmospheric forcing. The following types of couplings are taken into account: (a) the temperature modifies the total biomass of phytoplankton via the carrying capacity, (b) the extraction of carbon dioxide by phytoplankton slows down the speed of climate change, (c) the strength of mixing/turbulence in the oceanic mixing layer is in correlation with phytoplankton productivity. We carry out an ensemble approach (in the spirit of the theory of snapshot attractors) and concentrate on the trends of the average phytoplankton concentration and average temperature contrast between the pole and equator, forcing the atmospheric dynamics. The effect of turbulence is found to have the strongest influence on these trends. Our results show that (a) sufficiently strong mixing is able to force the typical phytoplankton concentration to always decay globally in time, and the temperature contrast to decrease faster than what follows from direct anthropogenic influences. Simple relations found for the trends without this coupling do, however, remain valid, just the coefficients become dependent on the strength of coupling with oceanic mixing. In particular, the phytoplankton concentration and its coupling to climate is found to modify the trend of global warming, and is able to make it stronger than what it would be without biomass.

... Therefore, in light of recent studies (Ghil et al. 2008;Chekroun et al. 2011;Drótos et al. 2015Drótos et al. , 2016Drótos et al. , 2017Herein et al. 2016;Lucarini et al. 2017), we turn to the so-called snapshot (Romeiras et al. 1990;Drótos et al. 2015) (also known as pullback; Ghil et al. 2008;Chekroun et al. 2011) attractor framework, which is based on ensemble climate simulations in practice. The snapshot attractor framework implies that after a transient time the ensemble members, which slightly differ in their initial conditions, forget their initial conditions and from this time on at each time instant the ensemble correctly characterizes the potential set of climate states permitted by the climate dynamics, that is, the permitted climate states under the external forcing scenario up to that time (Drótos et al. 2015. ...

The Arctic Oscillation (AO) and its related wintertime phenomena are investigated under climate change by 2099 in an ensemble approach using the CESM1 Large Ensemble and the MPI-ESM Grand Ensemble with different RCP scenarios. The loading pattern of the AO is defined as the leading mode of the empirical orthogonal function (EOF) analysis of sea-level pressure from 20–90° N. It is shown that the traditional AO index (AOI) calculation method, based on a base period in a single climate realization, brings subjectivity to the investigation of the AO-related phenomena. Therefore, if an ensemble is available the changes in the AO and its related phenomena should rather be studied by a reconsidered EOF analysis (snapshot EOF) introduced hereby. This novel method is based only on the instantaneous fields of the ensemble, hence it is capable of monitoring the time evolution of the AO’s pattern and amplitude. Furthermore, instantaneous correlation coefficient ( r) can objectively be calculated between the AOI and, e.g., the surface temperature, thus, the time dependence of the strength of these connections can also be revealed. Results emphasize that both the AO and the related surface temperature pattern are non-stationary and their time evolution depends on the forcing. The AO’s amplitude increases and the Pacific center strengthens considerably in each scenario. Additionally, there exist such regions (e.g. Northern Europe or western North America) where r shows remarkable change (0.2–0.4) by 2099. This study emphasizes the importance of the snapshot framework when studying changes in the climate system.

... If the invariant family A is attracting in the forward sense, it is called a forward attractor, and if it is attracting in the pullback sense, it is called a pullback attractor. Pullback attractors have been called snapshot attractors in the physics literature [29]. ...

A dynamical system with a plastic self-organising velocity vector field was introduced in Janson and Marsden (Sci Rep 7:17007, 2017) as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not model or explain any specific brain mechanisms or processes, but instead expresses a hypothesised principle possibly implemented by the brain. The hypothesis states that, by means of its plastic architecture, the brain creates a plastic self-organising velocity vector field, which embodies self-organising rules governing neural activity and through that the behaviour of the whole body. The model is represented by a two-tier dynamical system, in which the observable behaviour obeys a velocity field, which is itself controlled by another dynamical system. Contrary to standard brain models, in the new model the sensory input affects the velocity field directly, rather than indirectly via neural activity. However, this model was postulated without sufficient explication or theoretical proof of its mathematical consistency. Here we provide a more rigorous mathematical formulation of this problem, make several simplifying assumptions about the form of the model and of the applied stimulus, and perform its mathematical analysis. Namely, we explore the existence, uniqueness, continuity and smoothness of both the plastic velocity vector field controlling the observable behaviour of the system, and the of the behaviour itself. We also analyse the existence of pullback attractors and of forward limit sets in such a non-autonomous system of a special form. Our results verify the consistency of the problem and pave the way to constructing more models with specific pre-defined cognitive functions.

... For nonlinear climate models with general forcing, various studies (Chekroun et al., 2011;Ghil, 2015;Drótos et al., 2015;Ghil and Lucarini, 2019;Kaszás et al., 2019) have applied concepts from the theory of non-autonomous dynamical systems (Kloeden and Rasmussen, 2011) to understand aspects of behaviour. In this case the pullback attractor can consist of many trajectories that explore some timedependent subset of phase space, also called a snapshot attractor Romeiras et al. (1990). For nonlinear systems of the form (10) with stationary forcing, this pullback attractor may have positive Lyapunov exponents and sensitive dependence on initial conditions even in the unforced case. ...

This paper is currently in review for Global and Planetary Change. \\ The spectral view of variability is a compelling and adaptable tool for understanding variability of the climate. In the Mitchell (1976) seminal paper, it was used to express, on one graph with log scales, a very wide range of climate variations from millions of years to days. The spectral approach is particularly useful for suggesting causal links between forcing variability and climate response variability. However, (quasi-)periodic processes are also a natural part of the Earth system and a substantial degree of variability is intrinsic and responds to external forcing in a complex manner. There has been an enormous amount of work on understanding climate variability over the last decades. Hence in this paper, we address the question: Can we (after 40 years) update the Mitchell (1976) diagram in an essential way and provide it with a better interpretation? By reviewing both the extended observations available for such a diagram and new methodological developments in the study of the interaction between natural and forcing periodicity over a wide range of timescales, we give a positive answer to this question. In addition, we review alternative approaches to the spectral decomposition as in Mitchell and pose some challenges for a more detailed quantification of climate variability.

... In the case of perturbations not depending explicitly on time, response theory allows one to describe how the measure of the system changes differentiably with respect to small changes in the dynamics of the system. In the case of time-dependent perturbations, response theory makes it possible to reconstruct the measure supported on the pullback attractor [35][36][37] (see also the closely related concept of snapshot attractor [38,39]) of the non-autonomous system through a perturbative approach around a reference state, which, in the case of the climate studies referred to here, corresponds to the pre-industrial conditions. When we are nearing a critical transition, it is reasonable to expect a monotonic decrease of the spectral gap [40,41]. ...

For a wide range of values of the intensity of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak-noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, which provides a possible alternative with respect to the edge-tracking algorithm.

... See Refs. [22][23][24][25][26][27] for a more detailed discussion. While prediction of stationary systems by machine learning (ML) has received much recent attention, less progress has been made in applying ML to the problem of predicting the time evolution of nonstationary dynamical systems, particularly of their climate and of the tipping points they may experience. ...

In this paper we consider the machine learning (ML) task of predicting tipping point transitions and long-term post-tipping-point behavior associated with the time evolution of an unknown (or partially unknown), non-stationary, potentially noisy and chaotic, dynamical system. We focus on the particularly challenging situation where the past dynamical state time series that is available for ML training predominantly lies in a restricted region of the state space, while the behavior to be predicted evolves on a larger state space set not fully observed by the ML model during training. In this situation, it is required that the ML prediction system have the ability to extrapolate to different dynamics past that which is observed during training. We investigate the extent to which ML methods are capable of accomplishing useful results for this task, as well as conditions under which they fail. In general, we found that the ML methods were surprisingly effective even in situations that were extremely challenging, but do (as one would expect) fail when ``too much" extrapolation is required. For the latter case, we investigate the effectiveness of combining the ML approach with conventional modeling based on scientific knowledge, thus forming a hybrid prediction system which we find can enable useful prediction even when its ML-based and knowledge-based components fail when acting alone. We also found that achieving useful results may require using very carefully selected ML hyperparameters and we propose a hyperparameter optimization strategy to address this problem. The main conclusion of this paper is that ML-based approaches are promising tools for predicting the behavior of non-stationary dynamical systems even in the case where the future evolution (perhaps due to the crossing of a tipping point) includes dynamics on a set outside of that explored by the training data.

... Obviously, if one wants to model such a system (e.g. the changing climate with shifting atmospheric CO 2 concentration), this shortcoming has to be overcome. The mathematical concept of snapshot attractors 33 , known for many years in (theoretical and experimental) dynamical systems community [34][35][36][37][38][39][40][41] , fulfills entirely our wish. ...

Standard epidemic models based on compartmental differential equations are investigated under continuous parameter change as external forcing. We show that seasonal modulation of the contact parameter superimposed a monotonic decay needs a different description than that of the standard chaotic dynamics. The concept of snapshot attractors and their natural probability distribution has been adopted from the field of the latest climate-change-research to show the importance of transient effect and ensemble interpretation of disease spread. After presenting the extended bifurcation diagram of measles, the temporal change of the phase space structure is investigated. By defining statistical measures over the ensemble, we can interpret the internal variability of the epidemic as the onset of complex dynamics even for those values of contact parameter where regular behavior is expected. We argue that anomalous outbreaks of infectious class cannot die out until transient chaos is presented for various parameters. More important, that this fact becomes visible by using of ensemble approach rather than single trajectory representation. These findings are applicable generally in nonlinear dynamical systems such as standard epidemic models regardless of parameter values.

... Obviously, if one wants to model such a system this shortcoming has to be overcome. The mathematical concept of snapshot attractors introduced in the context of random dynamical systems has been known for many years in the (theoretical and experimental) dynamical systems community [34][35][36][37][38][39][40][41], and entirely fulfils our wish. Readers seeking a deeper understanding of mathematics are referred to appendix A and the references therein. ...

Standard epidemic models based on compartmental differential equations are investigated under continuous parameter change as external forcing. We show that seasonal modulation of the contact parameter superimposed upon a monotonic decay needs a different description from that of the standard chaotic dynamics. The concept of snapshot attractors and their natural distribution has been adopted from the field of the latest climate change research. This shows the importance of the finite-time chaotic effect and ensemble interpretation while investigating the spread of a disease. By defining statistical measures over the ensemble, we can interpret the internal variability of the epidemic as the onset of complex dynamics—even for those values of contact parameters where originally regular behaviour is expected. We argue that anomalous outbreaks of the infectious class cannot die out until transient chaos is presented in the system. Nevertheless, this fact becomes apparent by using an ensemble approach rather than a single trajectory representation. These findings are applicable generally in explicitly time-dependent epidemic systems regardless of parameter values and time scales.

... This concept was born in dynamical system theory as part of a progress made in understanding non-autonomous dynamics. The first article in this direction [73] drew attention to an interesting feature: a single trajectory leads to a washed out appearance on a map, while an ensemble of trajectories subjected to the same realization provides a structured pattern. This ensemble-related pattern, the snapshot chaotic attractor, changes its shape all the time in contrast to traditional chaotic attractors which are time-independent if observed on a stroboscopic map for periodic forcings [66]. ...

We argue that typical mechanical systems subjected to a monotonous parameter drift whose timescale is comparable to that of the internal dynamics can be considered to undergo their own climate change. Because of their chaotic dynamics, there are many permitted states at any instant, and their time dependence can be followed—in analogy with the real climate—by monitoring parallel dynamical evolutions originating from different initial conditions. To this end an ensemble view is needed, enabling one to compute ensemble averages characterizing the instantaneous state of the system. We illustrate this on the examples of (i) driven dissipative and (ii) Hamiltonian systems and of (iii) non-driven dissipative ones. We show that in order to find the most transparent view, attention should be paid to the choice of the initial ensemble. While the choice of this ensemble is arbitrary in the case of driven dissipative systems (i), in the Hamiltonian case (ii) either KAM tori or chaotic seas should be taken, and in the third class (iii) the best choice is the KAM tori of the dissipation-free limit. In all cases, the time evolution of the chosen ensemble on snapshots illustrates nicely the geometrical changes occurring in the phase space, including the strengthening, weakening or disappearance of chaos. Furthermore, we show that a Smale horseshoe (a chaotic saddle) that is changing in time is present in all cases. Its disappearance is a geometrical sign of the vanishing of chaos. The so-called ensemble-averaged pairwise distance is found to provide an easily accessible quantitative measure for the strength of chaos in the ensemble. Its slope can be considered as an instantaneous Lyapunov exponent whose zero value signals the vanishing of chaos. Paradigmatic low-dimensional bistable systems are used as illustrative examples whose driving in (i, ii) is chosen to decay in time in order to maintain an analogy with case (iii) where the total energy decreases all the time.

... Drótos et al. (2015) clarified that this ensemble owes its objectivity to the dissipative nature of the dynamical system, which gives rise to an attractor that supports a unique probability measure. Such an attractor is indeed well-defined also in the presence of external forcing, that is, under nonautonomous dynamics, and it is called the snapshot attractor (Romeiras et al., 1990). Therefore, the time evolution of the snapshot attractor and the probability measure, or any particular statistical quantity that is derived from it, including the mean, can be viewed to soundly represent climate change (Bódai and Tél, 2012;Tél et al., 2020). ...

Coupled ocean-atmosphere teleconnections are characteristics of internal variability which have a forced response just like mean states. It is not trivial how to correctly and optimally estimate the forced response and changes of the El Niño-Southern Oscillation (ENSO)–Indian summer monsoon (ISM) teleconnection under greenhouse gas forcing. Here we use two different approaches to address it. The first approach, based on the conventional temporal method applied to 30 model simulations in Coupled Model Intercomparison Project Phase 6, suggests no model consensus on changes in the teleconnection on interannual timescale in response to global warming. The second approach is based on a converged infinite single model initial condition large ensemble (SMILE) and defines the relationship in an instantaneous climatological sense. In view of several characteristics of the teleconnection, a robust long-term strengthening of the teleconnection is found in the MPI-GE but not in the CESM1-LE. We discuss appropriateness and limitations of the two methods.

... In the physics literature [26] there exists a concept of a snapshot attractor, which appears to be related to the pullback attractor, but is not as rigorously defined. If the invariant family A is attracting in the forward sense, i.e. ...

We investigate the robustness with respect to random stimuli of a dynamical system with a plastic self-organising vector field, previously proposed as a conceptual model of a cognitive system and inspired by the self-organised plasticity of the brain. This model of a novel type consists of an ordinary differential equation subjected to the time-dependent “sensory” input, whose time-evolving solution is the vector field of another ordinary differential equation governing the observed behaviour of the system, which in the brain would be neural firings. It is shown that the individual solutions of both these differential equations depend continuously over finite time intervals on the input signals. In addition, under suitable uniformity assumptions, it is shown that the non-autonomous pullback attractor and forward omega limit set of the given two-tier system depend upper semi-continuously on the input signal. The analysis holds for both deterministic and noisy input signals, in the latter case in a pathwise sense.

... This concept was born in dynamical system theory as part of a progress made in understanding nonautonomous dynamics. The first article in this direction [59] drew attention to an interesting feature: a single trajectory leads to a washed out appearance on a map, while an ensemble of trajectories subjected to the same noise realization provides a structured pattern. This ensemblerelated pattern, the snapshot chaotic attractor, changes its shape all the time in contrast to traditional chaotic attractors which are time-independent if observed on a stroboscopic map for periodic forcings [53]. ...

We argue that typical mechanical systems subjected to a monotonous parameter drift whose time scale is comparable to that of the internal dynamics can be considered to undergo their own climate change. Because of their chaotic dynamics, there are many permitted states at any instant, and their time dependence can be followed - in analogy with the real climate - by monitoring parallel dynamical evolutions originating from different initial conditions. To this end an ensemble view is needed, enabling one to compute ensemble averages characterizing the instantaneous state of the system. We illustrate this on the examples of (i) driven dissipative and (ii) Hamiltonian systems and of (iii) non-driven dissipative ones. We show that in order to find the most transparent view, attention should be paid to the choice of the initial ensemble. While the choice of this ensemble is arbitrary in the case of driven dissipative systems (i), in the Hamiltonian case (ii) either KAM tori or chaotic seas should be taken, and in the third class (iii) the best choice is the KAM tori of the dissipation-free limit. In all cases, the time evolution of the chosen ensemble on snapshots illustrates nicely the geometrical changes occurring in the phase space, including the strengthening, weakening or disappearance of chaos. Furthermore, we show that a Smale horseshoe (a chaotic saddle) that is changing in time is present in all cases. Its disappearance is a geometrical sign of the vanishing of chaos. The so-called ensemble-averaged pairwise distance is found to provide an easily accessible quantitative measure for the strength of chaos in the ensemble. Its slope can be considered as an instantaneous Lyapunov exponent whose zero value signals the vanishing of chaos. Paradigmatic low-dimensional bistable systems are used as illustrative examples whose driving in (i, ii) is chosen to decay in time in order to maintain an analogy with case (iii) where the total energy decreases all the time.

... The concept of the ensemble view was born in dynamical systems theory, where the seminal article [13] drew attention to the fact that in the case of changing parameters a single long trajectory traces out a fuzzy shape, while an ensemble of motions starting from many different initial conditions generates a structured pattern at any instant. In such cases individual trajectories turn out to be not representative, while the use of ensembles provides statistically well established results. ...

... In contrast to weather forecast, one focuses here on long-term properties, independent of initial conditions, in order to characterize the internal variability, as well as the forced response of the climate. The mathematical concept that provides an appropriate framework is that of snapshot (Romeiras et al., 1990;Drótos et al., 2015) or pullback attractors (Arnold, 1998;Ghil et al., 2008;Chekroun et al., 2011); for details see section Methods. ...

... For nonlinear climate models with general forcing, various studies (Chekroun et al., 2011;Ghil, 2015;Drótos et al., 2015;Ghil and Lucarini, 2019;Kaszás et al., 2019) have applied concepts from the theory of non-autonomous dynamical systems (Kloeden and Rasmussen, 2011) to understand aspects of behaviour. In this case the pullback attractor can consist of many trajectories that explore some timedependent subset of phase space, also called a snapshot attractor Romeiras et al. (1990). For nonlinear systems of the form (10) with stationary forcing, this pullback attractor may have positive Lyapunov exponents and sensitive dependence on initial conditions even in the unforced case. ...

The spectral view of variability is a compelling and adaptable tool for understanding variability of the climate. In Mitchell (1976) seminal paper, it was used to express, on one graph with log scales, a very wide range of climate variations from millions of years to days. The spectral approach is particularly useful for suggesting causal links between forcing variability and climate response variability. However, a substantial degree of variability is intrinsic and the Earth system may respond to external forcing in a complex manner. There has been an enormous amount of work on understanding climate variability over the last decades. Hence in this paper, we address the question: Can we (after 40 years) update the Mitchell (1976) diagram and provide it with a better interpretation? By reviewing both the extended observations available for such a diagram and new methodological developments in the study of the interaction between internal and forced variability over a wide range of timescales, we give a positive answer to this question. In addition, we review alternative approaches to the spectral decomposition and pose some challenges for a more detailed quantification of climate variability.

... The concept of the so-called "snapshot view" was introduced into dynamical system theory to understand how nonautonomous dynamics behave when subjected to general time dependent forcings. Romeiras et al. (1990) drew attention to an interesting feature of dissipative dynamical systems: the fact that a single long "noisy" trajectory traces out a fuzzy shape, while an ensemble of motions starting from many different initial conditions, using the same noise realization along each trajectory, creates a structured fractal pattern at any instant. This ensemble-related pattern, the snapshot chaotic attractor, continuously changes its shape, in contrast to traditional chaotic attractors, which are time-independent (Lorenz 1963;Ott 1993). ...

Geoengineering can control only some climatic variables but not others, resulting in side-effects. We investigate in an intermediate-complexity climate model the applicability of linear response theory (LRT) to the assessment of a geoengineering method. This application of LRT is twofold. First, our objective (O1) is to assess only the best possible geoengineering scenario by looking for a suitable modulation of solar forcing that can cancel out or otherwise modulate a climate change signal that would result from a rise in carbon dioxide concentration [CO2] alone. Here, we consider only the cancellation of the expected global mean surface air temperature Δ⟨[Ts]⟩. It is in fact a straightforward inverse problem for this solar forcing, and, considering an infinite time period, we use LRT to provide the solution in the frequency domain in closed form as fs(ω)=(Δ⟨[Ts]⟩(ω)−χg(ω)fg(ω))/χs(ω), where the χ’s are linear susceptibilities. We provide procedures suitable for numerical implementation that apply to finite time periods too. Second, to be able to utilize LRT to quantify side-effects, the response with respect to uncontrolled observables, such as regional averages ⟨Ts⟩, must be approximately linear. Therefore, our objective (O2) here is to assess the linearity of the response. We find that under geoengineering in the sense of (O1), i.e., under combined greenhouse and required solar forcing, the asymptotic response Δ⟨[Ts]⟩ is actually not zero. This turns out not to be due to nonlinearity of the response under geoengineering, but rather a consequence of inaccurate determination of the linear susceptibilities χ. The error is in fact due to a significant quadratic nonlinearity of the response under system identification achieved by a forced experiment. This nonlinear contribution can be easily removed, which results in much better estimates of the linear susceptibility, and, in turn, in a fivefold reduction in Δ⟨[Ts]⟩ under geoengineering practice. This correction dramatically improves also the agreement of the spatial patterns of the predicted linear and the true model responses. However, considering (O2), such an agreement is not perfect and is worse in the case of the precipitation patterns as opposed to surface temperature. Some evidence suggests that it could be due to a greater degree of nonlinearity in the case of precipitation.

Strange nonchaotic attractors (SNAs) have been identified and studied in the literature exclusively in quasiperiodically driven nonlinear dynamical systems. It is an interesting question to ask whether they can be identified with other types of forcings as well, which still remains as an open problem. Here, we show that robust SNAs can be created by a small amount of noise in periodically driven nonlinear dynamical systems by a single force. The robustness of these attractors is tested by perturbing the system with logical signals leading to the emulation of different logical elements in the SNA regions.

To characterize chaos in systems subjected to parameter drift, where a number of traditional methods do not apply, we propose viable alternative approaches, both in the qualitative and quantitative sense. Qualitatively, following stable and unstable foliations is shown to be efficient, which are easy to approximate numerically, without relying on the need for the existence of an analog of hyperbolic periodic orbits. Chaos originates from a Smale horseshoe-like pattern of the foliations, the transverse intersections of which indicate a chaotic set changing in time. In dissipative cases, the unstable foliation is found to be part of the so-called snapshot attractor, but the chaotic set is not dense on it if regular time-dependent attractors also exist. In Hamiltonian cases stable and unstable foliations turn out to be not equivalent due to the lack of time-reversal symmetry. It is the unstable foliation, which is found to correlate with the so-called snapshot chaotic sea. The chaotic set appears to be locally dense in this sea, while tori with originally quasiperiodic character might break up, their motion becoming chaotic as time goes on. A quantity called ensemble-averaged pairwise distance evaluated in relation to unstable foliations is shown to be an appropriate tool to provide the instantaneous strength of time-dependent chaos.

This book, based on a selection of invited presentations from a topical workshop, focusses on time-variable oscillations and their interactions. The problem is challenging, because the origin of the time variability is usually unknown. In mathematical terms, the oscillations are nonautonomous, reflecting the physics of open systems where the function of each oscillator is affected by its environment. Time-frequency analysis being essential, recent advances in this area, including wavelet phase coherence analysis and nonlinear mode decomposition, are discussed. Some applications to biology and physiology are described. Although the most important manifestation of time-variable oscillations is arguably in biology, they also crop up in, e.g. astrophysics, or for electrons on superfluid helium. The book brings together the research of the best international experts in seemingly very different disciplinary areas.

The maintenance of an adequate microvascular perfusion sufficient to meet the metabolic demands of the tissue is dependent on neuralNetwork, neural, humoral and local vaso-mechanisms that determine vascular tone and blood flowBlood flow patterns within a microvascularNetworknetworkMicrovascular network. It has been argued that attenuation of these flow patterns may be a major contributor to disease risk. Thus, quantitative information on the in vivo spatio-temporal behaviour of microvascular perfusion is important if we are to understand networkNetwork functionality and flexibility in cardiovascular diseaseCardiovascular disease. Time and frequency-domainTime and frequency analysis analysis has been extensively used to describe the dynamic characteristics of Laser Doppler flowmetryLaser Doppler flowmetry (or fluximetry) (LDF) signals obtained from superficial microvascularNetworknetworksMicrovascular network such as that of the skin. However, neither approach has provided definitive and consistent information on the relative contribution of the oscillatory components of flowmotionFlowmotion (or flowmotion) (endothelial, neurogenic, myogenic, respiratory and cardiac) to a sustained and adequate microvascular perfusion; nor advance our understanding of how such processes are collectively modified in disease. More recently, non-linear complexity-based approaches have begun to yield evidence of a declining adaptability of microvascular flow patterns as disease severity increases. In this chapter we review the utility and application of these approaches for the quantitative, mechanistic exploration of microvascular (dys)function.

Kloeden, Peter E.Yang, MeihuaThe nature of time in a nonautonomous dynamicalDynamical system system isDynamical system, autonomous very different from that in autonomous systemsAutonomous, which depend on the time that has elapsed since starting rather on the actual time. This requires new concepts of invariant sets and attractors. Pullback and forward attractors as well as forward omega-limit sets will be reviewed here in the simpler setting of nonautonomous difference equations. Both two-parameter semi-group and skew product flow formulations ofNon-autonomous dynamical system nonautonomous dynamical systemsAutonomous dynamical system are considered.

The disappearance and reappearance of chaos by adjusting the internal parameters of dynamics in Lorenz system are studied. We observe monotonous and periodic time-dependent changes of Rayleigh number. There exists relaxation time for the disappearance of chaos, when we use the snapshot attractors to observe the change of the system attractors. We show that the rate of disappearance and reappearance of chaos is positively correlated with the control parameters. To reflect the relaxation phenomenon of chaotic disappearance and the sensitivity of trajectory, the concept of finite-time Lyapunov exponent is used. Then the statistical characteristics of the system can be presented by standard deviation. The chaotic disappearance and reappearance are manifested in the decrease and increase of the standard deviation. The standard deviation decreases continuously during chaotic disappearance, but increases discontinuously during chaotic reappearance. A distinctive scenario is that no matter which parameter changes, when we use the same rate of change in the process of chaotic disappearance and reappearance, their paths are different.

Strange nonchaotic attractors (SNAs) have been identified and studied in the literature exclusively in quasiperiodically driven nonlinear dynamical systems. It is an interesting question to ask whether they can be identified with other types of forcings as well, which still remains an open problem. Here, we show that robust SNAs can be created by a small amount of noise in periodically driven nonlinear dynamical systems by a single force. The robustness of these attractors is tested by perturbing the system with logical signals, leading to the emulation of different logical elements in the SNA regions.

In the development of modeling the climate, two principal paths can be distinguished. On one side the effort to increase the resolution to eliminate as much as possible the parameterization of small-scale processes. This approach is an improvement over the “mechanistic” view of modeling climate mainly through the use of General Circulation Models (GCM). The other approach, which is born out of the theory of system dynamics, has as objective an improvement in the theoretical basis of climate science. A minor but promising development has to do with the direct statistical simulation (DSS) following a suggestion made by Edward Lorenz almost sixty years ago. The improvement in GCM is more oriented to the classical application of predicting the future climate and its impact on the human activities while the other two approaches have explained some specific phenomena like atmospheric jets, or El Nino but their use as predicting tools is not in the near future.

Tipping phenomena, i.e. dramatic changes in the possible long-term performance of deterministic systems subjected to parameter drift, are of current interest but have not yet been explored in cases with chaotic internal dynamics. Based on the example of a paradigmatic low-dimensional dissipative system subjected to different scenarios of parameter drifts of non-negligible rates, we show that a number of novel types of tippings can be observed due to the topological complexity underlying general systems. Tippings from and into several coexisting attractors are possible, and one can find fractality-induced tipping, the consequence of the fractality of the scenario-dependent basins of attractions, as well as tipping into a chaotic attractor. Tipping from or through an extended chaotic attractor might lead to random tipping into coexisting regular attractors, and rate-induced tippings appear not abruptly as phase transitions, rather they show up gradually when the rate of the parameter drift is increased. Since chaotic systems of arbitrary time-dependence call for ensemble methods, we argue for a probabilistic approach and propose the use of tipping probabilities as a measure of tipping. We numerically determine these quantities and their parameter dependence for all tipping forms discussed.

The changes in the El Niño–Southern Oscillation (ENSO) phenomenon and its precipitation-related teleconnections over the Globe under climate change are investigated in the Community Earth System Model's Large Ensemble from 1950 to 2100. For the investigation, a recently developed ensemble-based method, the snapshot empirical orthogonal function (SEOF) analysis is used. The instantaneous ENSO pattern is defined as the leading mode of the SEOF analysis carried out at a given time instant over the ensemble. The corresponding principal components (PC1s) characterize the ENSO phases. We find that the largest amplitude changes in the variability of the sea surface temperature fields occur in the June–July–August–September (JJAS) season, in the Niño3–Niño3.4 region and in the eastern part of the Pacific Ocean, however, the increase is also considerable along the Equator in December–January–February (DJF). The Niño3 amplitude shows also an increase of about 20 % and 10 % in JJAS and DJF, respectively. The strength of the precipitation-related teleconnections of the ENSO is found to be non-stationary, as well. For example, the anti-correlation with precipitation in Australia in JJAS and the positive correlation in Central and North Africa in DJF are predicted to be more pronounced by the end of the 21th century. Half-year-lagged correlations, aiming to predict precipitation conditions from ENSO phases, are also studied. The Australian, Indonesian precipitation and that of the eastern part of Africa in both JJAS and DJF seem to be well predictable based on ENSO phase, while the South Indian precipitation is in relation with the half-year previous ENSO phase only in DJF. The strength of these connections increases, especially from the African region to the Arabian Peninsula.

We develop and test machine learning techniques for successfully using past state time series data and knowledge of a time-dependent system parameter to predict the evolution of the “climate” associated with the long-term behavior of a non-stationary dynamical system, where the non-stationary dynamical system is itself unknown. By the term climate, we mean the statistical properties of orbits rather than their precise trajectories in time. By the term non-stationary, we refer to systems that are, themselves, varying with time. We show that our methods perform well on test systems predicting both continuous gradual climate evolution as well as relatively sudden climate changes (which we refer to as “regime transitions”). We consider not only noiseless (i.e., deterministic) non-stationary dynamical systems, but also climate prediction for non-stationary dynamical systems subject to stochastic forcing (i.e., dynamical noise), and we develop a method for handling this latter case. The main conclusion of this paper is that machine learning has great promise as a new and highly effective approach to accomplishing data driven prediction of non-stationary systems.

Analysis methods that have been developed in the field of nonlinear dynamics have provided valuable insights into the physics of turbulent flows, although their application to open flows is less well explored. The nonlinear dynamics of a turbulent jet with a low-to-moderate Reynolds number is investigated by using the single-trajectory framework and ensemble framework. We have used Lyapunov exponents to calculate the spectra of scaling indices of the attractor. First, we evaluated the frameworks on two theoretical models, one with a stationary attractor (Lorenz-63) and the other with time-varying characteristics (Lorenz-84). Theoretical studies showed that in dynamical systems with a stable attractor, both frameworks estimated the same largest Lyapunov exponent. The ensemble framework enables us to resolve the unsteady characteristics of a time-varying strange attractor. Second, we applied both frameworks to time-resolved planar velocity fields in a turbulent jet at local Reynolds numbers (Reδ) of 3000 and 5000. Time-resolved particle image velocimetry was utilized to measure streamwise and transverse velocity components. Results support the presence of a low-dimensional attractor in the reconstructed phase space with a chaotic characteristic. Despite considerable changes in the dynamics for the higher Reynolds number case, the system’s fractal dimension did not change significantly. We have used Lagrangian Coherent Structures (LCSs) to study the relationship between changes in the Lyapunov exponent with flow topological features. Results suggest that holes in the stable LCSs provide a path for the entrainment of the coflow, which is shown to be one of the main contributors to high Lyapunov exponents.

We study the teleconnection between the El Niño–Southern Oscillation (ENSO) and the Indian summer monsoon (IM) in large ensemble simulations, the Max Planck Institute Earth System Model (MPI-ESM) and the Community Earth System Model (CESM1). We characterize ENSO by the JJA Niño 3 box-average SST and the IM by the JJAS average precipitation over India, and define their teleconnection in a changing climate as an ensemble-wise correlation. To test robustness, we also consider somewhat different variables that can characterize ENSO and the IM. We utilize ensembles converged to the system’s snapshot attractor for analyzing possible changes in the teleconnection. Our main finding is that the teleconnection strength is typically increasing on the long term in view of appropriately revised ensemble-wise indices. Indices involving a more western part of the Pacific reveal, furthermore, a short-term but rather strong increase in strength followed by some decrease at the turn of the century. Using the station-based SOI as opposed to area-based indices leads to the identification of somewhat more erratic trends, but the turn-of-the-century “bump” is well-detectable with it. All this is in contrast, if not in contradiction, with the discussion in the literature of a weakening teleconnection in the late 20 th century. We show here that this discrepancy can be due to any of three reasons: ensemble-wise and temporal correlation coefficients used in the literature are different quantities; the temporal moving correlation has a high statistical variability but possibly also persistence; MPI-ESM does not represent the Earth system faithfully.

Based on the example of a paradigmatic area preserving low-dimensional mapping subjected to different scenarios of parameter drifts, we illustrate that the dynamics can best be understood by following ensembles of initial conditions corresponding to the tori of the initial system. When such ensembles are followed, snapshot tori are obtained, which change their location and shape. Within a time-dependent snapshot chaotic sea, we demonstrate the existence of snapshot stable and unstable foliations. Two easily visualizable conditions for torus breakup are found: one in relation to a discontinuity of the map and the other to a specific snapshot stable manifold, indicating that points of the torus are going to become subjected to strong stretching. In a more general setup, the latter can be formulated in terms of the so-called stable pseudo-foliation, which is shown to be able to extend beyond the instantaneous chaotic sea. The average distance of nearby point pairs initiated on an original torus crosses over into an exponential growth when the snapshot torus breaks up according to the second condition. As a consequence of the strongly non-monotonous change of phase portraits in maps, the exponential regime is found to split up into shorter periods characterized by different finite-time Lyapunov exponents. In scenarios with plateau ending, the divided phase space of the plateau might lead to the Lyapunov exponent averaged over the ensemble of a torus being much smaller than that of the stationary map of the plateau.

The changes in the El Niño–Southern Oscillation (ENSO) phenomenon and its precipitation-related teleconnections over the globe under climate change are investigated in the Community Earth System Model Large Ensemble from 1950 to 2100. For the investigation, a recently developed ensemble-based method, the snapshot empirical orthogonal function (SEOF) analysis, is used. The instantaneous ENSO pattern is defined as the leading mode of the SEOF analysis carried out at a given time instant over the ensemble. The corresponding principal components (PC1s) characterize the ENSO phases. By considering sea surface temperature (SST) regression maps, we find that the largest changes in the typical amplitude of SST fluctuations occur in the June–July–August–September (JJAS) season, in the Niño3–Niño3.4 (5∘ N–5∘ S, 170–90∘ W; NOAA Climate Prediction Center) region, and the western part of the Pacific Ocean; however, the increase is also considerable along the Equator in December–January–February (DJF). The Niño3 amplitude also shows an increase of about 20 % and 10 % in JJAS and DJF, respectively. The strength of the precipitation-related teleconnections of the ENSO is found to be nonstationary, as well. For example, the anticorrelation with precipitation in Australia in JJAS and the positive correlation in central and northern Africa in DJF are predicted to be more pronounced by the end of the 21th century. Half-year-lagged correlations, aiming to predict precipitation conditions from ENSO phases, are also studied. The Australian and Indonesian precipitation and that of the eastern part of Africa in both JJAS and DJF seem to be well predictable based on the ENSO phase, while the southern Indian precipitation relates to the half-year previous ENSO phase only in DJF. The strength of these connections increases, especially from the African region to the Arabian Peninsula.

Based on the example of a paradigmatic low-dimensional Hamiltonian system subjected to different scenarios of parameter drifts of non-negligible rates, we show that the dynamics of such systems can best be understood by following ensembles of initial conditions corresponding to tori of the initial system. When such ensembles are followed, toruslike objects called snapshot tori are obtained, which change their location and shape. In their center, one finds a time-dependent, snapshot elliptic orbit. After some time, many of the tori break up and spread over large regions of the phase space; however, one may find some smaller tori, which remain as closed curves throughout the whole scenario. We also show that the cause of torus breakup is the collision with a snapshot hyperbolic orbit and the surrounding chaotic sea, which forces the ensemble to adopt chaotic properties. Within this chaotic sea, we demonstrate the existence of a snapshot horseshoe structure and a snapshot saddle. An easily visualizable condition for torus breakup is found in relation to a specific snapshot stable manifold. The average distance of nearby pairs of points initiated on an original torus at first hardly changes in time but crosses over into an exponential growth when the snapshot torus breaks up. This new phase can be characterized by a novel type of a finite-time Lyapunov exponent, which depends both on the torus and on the scenario followed. Tori not broken up are shown to be the analogs of coherent vortices in fluid flows of arbitrary time dependence, and the condition for breakup can also be demonstrated by the so-called polar rotation angle method.

Using an intermediate complexity climate model (Planet Simulator), we investigate the so-called snowball Earth transition. For certain values (including its current value) of the solar constant, the climate system allows two different stable states: one of them is the snowball Earth, covered by ice and snow, and the other one is today’s climate. In our setup, we consider the case when the climate system starts from its warm attractor (the stable climate we experience today), and the solar constant is changed according to the following scenario: it is decreased continuously and abruptly, over one year, to a state, where only the Snowball Earth’s attractor remains stable. This induces an inevitable transition or climate tipping from the warm climate. The reverse transition is also discussed. Increasing the solar constant back to its original value in a similar way, in individual simulations, depending on the rate of the solar constant reduction, we find that either the system stays stuck in the snowball state or returns to warm climate. However, using ensemble methods, i.e., using an ensemble of climate realizations differing only slightly in their initial conditions we show that the transition from the snowball Earth to the warm climate is also possible with a certain probability, which depends on the specific scenario used. From the point of view of dynamical systems theory, we can say that the system’s snapshot attractor splits between the warm climate’s and the snowball Earth’s attractor.

We consider compositions of random diffeomorphisms and show that the dimension of sample measures equals Lyapunov dimension as conjectured in the nonrandom case by Yorke et al.

A general self-similarity relation is shown to exist, expressing the Renyi-dimension function in terms of local expansion rates both for flows and maps. For the particular case of the information dimension, such an implicit equation yields the well-known Kaplan-Yorke relation. Moreover, it can be explicitly solved in some interesting cases, among which are two-dimensional maps with constant Jacobian. Detailed measurements are performed for the Hénon attractor, with a very accurate estimate of its capacity. Finally, an expansion around the information dimension allows recovery of the Grassberger-Procaccia estimates in an easy way.

A new type of phase transition associated with a singularity in the spectrum of generalized entropies, or in a corresponding free energy, is shown to exist in dynamical systems. Candidates for exhibiting such transitions are intermittent chaotic systems where certain generalized entropies vanish. This new transition may coexist with one associated with the static properties of the system.

We present measurements of the curve of f vs α from two-dimensional sections of the "dissipation" field of concentration fluctuations, and from one-dimensional sections of the dissipation field of passive temperature fluctuations, in turbulent jets. The results confirm the universality of the dissipation rate χ of scalar fluctuations and the applicability of Taylor's hypothesis, and show that the curve of f vs α is the same for different components of χ, that the additive properties of f(α) apply to intersections, and that the intermittency exponent of χ is considerably higher than that for the turbulent kinetic-energy dissipation.

A new class of instabilities for a plane-wave intracavity field in an optical ring resonator is identified. Dynamical systems techniques are explained and applied to the map. A bifurcation diagram is given that organizes the important information, and global pictures are developed that describe the evolution of the attractor and its basin boundary. Anomalous behavior observed in earlier numerical studies is explained.

New relations of the generalized dimensions and entropies of strange attractors to the fluctuations of divergence rates of
nearby orbits and to the eigenvalues of the Jacobian matrices of unstable periodic points are obtained in order to derive
the sspectra of singularities from the dynamical viewpoints.

We investigate the meaning of the dimension of a strange attractor for systems with noise. More specifically, we investigate the effect of adding noise of magnitude ε to a deterministic system with D degrees of freedom. If the attractor has dimension d and d < D, then its volume is zero. The addition of noise may be an important physical probe for experimental situations, useful for determining how much of the observed phenomena in a system is due to noise already present. When the noise is added, the attractor Aε has positive volume. We conjecture that the generalized volume of Aε is proportional to εD − d for ε near 0 and show this relationship is valid in several cases. For chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly.

It is theoretically shown that the transmitted light from a ring cavity containing a nonlinear dielectric medium undergoes transition from a stationary state to periodic and nonperiodic states, where the intensity of the incident light is increased. The nonperiodic state is characterized by a chaotic variation of the light intensity and associated broadband noise in the power spectrum. The experimental possibility of observing such a transition is also discussed.

A path integral method is developed for the calculation of the statistical properties of a class of discrete, dissipative mappings which exhibit strange attractors. Exact analytic results are derived for the low order statistical moments. These non-trivial results are verified numerically.

A simple model for the study of the effect of noise on the structure of strange attractors is presented, and an explicit solution for the resulting time asymptotic distribution function is obtained. As the noise level is decreased, the distribution function exhibits the Cantor set structure of the strange attractor on successively finer scales. Letting the noise level approach zero, a formal expression for the distribution function on the strange attractor is obtained.

Kolmogorov’s “third hypothesis” asserts that in intermittent turbulence the average \(
\bar \varepsilon
\) of the dissipation ε, taken over any domain D, is ruled by the lognormal probability distribution. This hypothesis will be shown to be logically inconsistent, save under assumptions that are extreme and unlikely. A widely used justification of lognormality due to Yaglom and based on probabilistic argument involving a self-similar cascade, will also be discussed. In this model, lognormality indeed applies strictly when D is “an eddy,” typically a three-dimensional box embedded in a self-similar hierarchy, and may perhaps remain a reasonable approximation when D consists of a few such eddies. On the other hand, the experimental situation is better described by considering averages taken over essentially one-dimensional domains D.

We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularities; and f, which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the function f(α). We apply this formalism to the 2∞ cycle of period doubling, to the devil’s staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.

The forces on a small rigid sphere in a nonuniform flow are considered from first principles in order to resolve the errors in Tchen's equation and the subsequent modified versions that have since appeared. Forces from the undisturbed flow and the disturbance flow created by the presence of the sphere are treated separately. Proper account is taken of the effect of spatial variations of the undisturbed flow on both forces. In particular the appropriate Faxen correction for unsteady Stokes flow is derived and included as part of the consistent approximiation for the equation of motion.

A theory is presented for first order phase transitions of multifractal chaotic attractors of nonhyperbolic two-dimensional maps. (These phase transitions manifest themselves as a discontinuity in the derivative with respect to q (analogous to temperature) of the fractal dimension q-spectrum, Dq (analogous to free energy).) A complete picture of the behavior associated with the phase transition is obtained.

The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hnon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.

Several different dimensionlike quantities, which have been suggested as being relevant to the study of chaotic attractors, are examined. In particular, we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points. It is found that some are and some are not. It is suggested that the word dimension be reversed only for those quantities have this invariance property.

Let M be a compact (Riemann) manifold and (f
t
): M → M a differentiable flow. A closed (f
t
)-invariant set ∧ ⊂ M containing no fixed points is hyperbolic if the tangent bundle restricted to ∧ can be written as the Whitney sum of three (Tf
t
)-invariant continuous subbundles $${T_\Lambda }M = E + {E^s} + {E^u}$$where E is the one-dimensional bundle tangent to the flow, and there are constants c λ>0 so that $$(a)||T{f^t}(v)||\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } c{e^{ - \lambda t}}||v||forv \in {E^s},t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0and$$
$$(b)||T{f^{ - 1}}(v)||\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } c{e^{ - \lambda t}}||v||forv \in {E^u},t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0.

The strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction. In view of this we study a new definition of attractors based on ideas of Conley. We argue that the attractors observed in the presence of small random perturbations correspond to this new definition.

We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions Dq, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities Dq. We prove that lim q→0Dq = fractal dimension (D), limq→1Dq = information dimension (σ) and Dq=2 = correlation exponent (v). Dq with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > Dq for any q′ > q. For homogeneous fractals Dq = Dq. A particularly interesting dimension is Dq=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D∞.

Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the “dimension of the natural measure”, and all of the metric dimensions take on a common value, which we call the “fractal dimension”. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.

In the stationary situation the transmitted light by a ring cavity containing a homogeneously broadened two level absorber exhibits a multiple-valued response to a constant incident light. The stability of the stationary state is investigated in the fast limit of the atomic relaxation. The stationary state is not always stable even when it belongs to the branch with a positive differential gain. In some cases all the stationary states becomes unstable and the transmitted light exhibits a “chaotic” behavior.

It is pointed out that there exists an infinity of generalized dimensions for strange attractors, related to the order-q Renyi entropies. They are monotonically decreasing with q. For q = 0, 1 and 2, they are the capacity, the information dimension, and the correlation exponent, respectively. For all q, they are measurable from recurrence times in a time series, without need for a box-counting algorithm. For the Feigenbaum map and for the generalized Baker transformation, all generalized dimensions are finite and calculable, and depend non-trivially on q.

Nonanalyticities in the generalized dimensions of multifractal sets of physical interest are interpreted as phase transitions. The problem is mapped onto thermodynamics of one-dimensional spin models. The spin Hamiltonians are explicitly constructed and their phase transitions discussed. This mapping can provide insight in both directions. PACS numbers: 05.70.−a

Several definitions of generalized fractal dimensions are reviewed, generalized, and interconnected. They concern (i) different ways of averaging when treating fractal measures (instead of sets); (ii) “partial dimensions” measuring the fractility in different directions, and adding up to the generalized dimensions discussed before.

Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a “strange attractor”. We show that the same properties can be observed in a simple mapping of the plane defined by: \({x_{i + 1}} = {y_i} + 1 - ax_i^2,{y_{i + 1}} = b{x_i}\). Numerical experiments are carried out for a =1.4, b = 0.3. Depending on the initial point (x
0,y
0), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a onedimensional manifold.by a Cantor set.

A formulation giving the q dimension Dq of a chaotic attractor in terms of the eigenvalues of unstable periodic orbits is presented and discussed.

We consider three types of changes that attractors can undergo as a system parameter is varied. The first type leads to the sudden destruction of a chaotic attractor. The second type leads to the sudden widening of a chaotic attractor. In the third type of change, which applies for many systems with symmetries, two (or more) chaotic attractors merge to form a single chaotic attractor and the merged attractor can be larger in phase-space extent than the union of the attractors before the change. All three of these types of changes are termed crises and are accompanied by a characteristic temporal behavior of orbits after the crisis. For the case where the chaotic attractor is destroyed, this characteristic behavior is the existence of chaotic transients. For the case where the chaotic attractor suddenly widens, the characteristic behavior is an intermittent bursting out of the phase-space region within which the attractor was confined before the crisis. For the case where the attractors suddenly merge, the characteristic behavior is an intermittent switching between behaviors characteristic of the attractors before merging.

A new method is proposed to locate and analyze phase transitions in a thermodynamic formalism for the description of fractal sets. By studying order parameters appropriate to the transitions we get both an efficient numerical tool for locating phase transitions and an understanding of the structure of the ordered phase. With this method, we examine fractal sets generated by a class of maps of the interval close to the map x-->4x(1-x). We show that the existence of phase transitions is a persistent phenomenon and remains when the map is perturbed although the structure of the entropy function changes drastically. For strong perturbations the transition disappears and the entropy function becomes nonsingular. The phase transitions describe transitions in the distribution of the characteristic Lyapunov exponents.

This paper explores some implications of the observed multifractal nature of the turbulent energy-dissipation field and of velocity derivatives of increasing order for the near-singularities of the Navier-Stokes equations and the singularities of Euler equations. Although these singularities occur on fractal sets of dimension close to (and only marginally less than) three, it is shown that most of the energy dissipation is concentrated on a subset of fractal dimension about 2.87 and volume zero. Similar statements can be made with respect to velocity derivatives. In particular, it is shown that the higher the order of the velocity derivative, the less space filling the corresponding singularities become.

The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily fine-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. Attention is given to the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and chaotic repellers are considered.

Following an analogy to the formalism of statistical mechanics, an entropy function and a free energy are introduced for multifractals. These functions give a full description of the scaling behaviors of multifractals. The method of Halsey et al. (1986) for characterizing multifractals can naturally be interpreted by the use of these functions. For the invariant set of a dynamical system, these functions are furthermore related to the measure-theoretic (Kolmogolov-Sinai) entropy, the topological entropy, and the Lyapunov exponent.

Multifractal dimension spectra for the stable and unstable manifolds of invariant chaotic sets are studied for the case of invertible two-dimensional maps. A dynamical partition-function formalism giving these dimensions in terms of local Lyapunov numbers is obtained. The relationship of the Lyapunov partition functions for stable and unstable manifolds to previous work is discussed. Numerical experiments demonstrate that dimension algorithms based on the Lyapunov partition functions are often very efficient. Examples supporting the validity of the approach for hyperbolic chaotic sets and for nonhyperbolic sets below the phase transition (q<q/sub T/) are presented.

The passive convection of vector fields and scalar functions by a prescribed incompressible fluid flow v(x,t) is considered for the case where v(x,t) is chaotic. By chaotic v(x,t) it is meant that typical nearby fluid elements diverge from each other exponentially in time. It is shown that in such cases, as time increases, a convected vector field and the gradient of a convected scalar will generally concentrate on a set which is fractal. The present paper relates the stretching properties of the flow to the resulting fractal dimension spectrum. Motivation for these considerations is provided by the kinematic magnetic dynamo problem (in the vector case) and (in the scalar case) by recent experiments which demonstrate the possibility of measuring the fractal dimension of the gradient squared of convected passive scalars.

Nonanalyticities in the generalized dimensions of multifractal sets of physical interest are interpreted as phase transitions. The problem is mapped onto thermodynamics of one-dimensional spin models. The spin Hamiltonians are explicitly constructed and their phase transitions discussed. This mapping can provide insight in both directions.

The chaotic convection of passive scalars by an incompressible fluid is considered. (The convection is said to be chaotic if nearby fluid elements typically diverge from each other exponentially in time.) It is shown that during the time evolution the square of the gradient of chaotically convected passive scalars typically concentrates on a fractal set. Considerations of the local stretching properties of the flow lead to a partition function which yields the dimension spectra of the resulting fractal measure. Fractal structure is a result of the nonuniform stretching of typical chaotic flows.

- R. V. Jensen