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On the Existence of Low Dimensional Climatic Attractors
Abstract
Recent work has highlighted the possibility of using certain ideas from dynamical systems theory to study climate. We review the evidence for chaos in climatic changes based on estimates of the dimension of the reconstructed attractor from time series data. -Authors
... Estimates of the correlation dimension D of a time series are commonly used for quantifying the chaotic complexity of a physical system such as the atmosphere (e.g., Grassberger and Procaccia 1983a,b;Eckmann and Ruelle 1985;Fraedrich 1986;Hense 1987;Henderson and Wells 1988;Tsonis and Elsner 1988;Fraedrich and Leslie 1989;Fraedrich et al. 1990;Baker and Gollub 1990;Sharifi et al. 1990;Yang 1991;Tsonis 1992;Zeng et al. 1993;Poveda-Jaramillo and Puente 1993;Orcutt and Arritt 1995;Weber et al. 1995). However, owing to the presence of both intrinsic and extrinsic sources of error, the merits of this procedure have been debated considerably in the literature (e.g., Nicolis 1984, 1987;Grassberger 1986Grassberger , 1987Smith et al. 1986;Theiler 1986Theiler , 1988Procaccia 1988;Smith 1988;Tsonis and Elsner 1988, 1989Pool 1989;Nerenberg and Essex 1990;Ruelle 1990;Essex and Nerenberg 1991;Zeng et al. 1992;Islam et al. 1993;Tsonis et al. 1993;Weber et al. 1995). ...
... The method most commonly used to estimate the correlation dimension for atmospheric and other geophysical time series has been the mean slope method, which consists of approximating the dimension using mean slopes of the graph of ln C(r) versus ln r for small ranges of r, where C(r) is the correlation integral and r is the distance between time series points (e.g., Grassberger and Procaccia 1983a,b;Nicolis and Nicolis 1984;Eckmann and Ruelle 1985;Fraedrich 1986;Hense 1987;Grassberger 1986;Nese 1987;Simm et al. 1987;Henderson and Wells 1988;Fraedrich and Leslie 1989;Keppene and Nicolis 1989;Tsonis and Elsner 1989;Nerenberg and Essex 1990;Sharifi et al. 1990;Fraedrich et al. 1990;Albano et al. 1991;Cutler 1991;Lorenz 1991;Nerenberg et al. 1991;Zeng et al. 1992;Sundermeyer and Vallis 1993;Poveda-Jaramillo and Puente 1993;Weber et al. 1995). However, in actual applications, this method may be quite sensitive to undersampling or superimposed noise. ...
... The method most commonly used to estimate the correlation dimension for atmospheric and other geophysical time series has been the mean slope method, which consists of approximating the dimension using mean slopes of the graph of ln C(r) versus ln r for small ranges of r, where C(r) is the correlation integral and r is the distance between time series points (e.g., Grassberger and Procaccia 1983a,b;Nicolis and Nicolis 1984;Eckmann and Ruelle 1985;Fraedrich 1986;Hense 1987;Grassberger 1986;Nese 1987;Simm et al. 1987;Henderson and Wells 1988;Fraedrich and Leslie 1989;Keppene and Nicolis 1989;Tsonis and Elsner 1989;Nerenberg and Essex 1990;Sharifi et al. 1990;Fraedrich et al. 1990;Albano et al. 1991;Cutler 1991;Lorenz 1991;Nerenberg et al. 1991;Zeng et al. 1992;Sundermeyer and Vallis 1993;Poveda-Jaramillo and Puente 1993;Weber et al. 1995). However, in actual applications, this method may be quite sensitive to undersampling or superimposed noise. ...
The correlation dimension D is commonly used to quantify the chaotic structure of atmospheric time series. The standard algorithm for estimating the value of D is based on finding the slope of the curve obtained by plotting In C(r) versus In r, where C(r) is the correlation integral and r is the distance between points on the attractor. An alternative, probabilistic method proposed by Takens is extended and tested here. This method is based on finding the sample means of the random variable (r/ρ) p[ln(r/ρ)] k, expressed as the conditional expected value E((r/ρ) p[ln(r/ρ)] k : r < ρ), for p and k nonnegative numbers. The sensitivity of the slope method and of the extended estimators D pk(ρ) for approximating D is studied in detail for three ad hoc correlation integrals and for integer values of p and k. The first two integrals represent the effects of noise or undersampling at small distances and the third captures periodic lacunarity, which occurs by definition when the ratio C(xρ)/C(ρ) fails to converge as ρ approaches zero. All the extended estimators give results that are superior to that produced by the most commonly used slope method. Moreover, the various estimators exhibit much different behavior in the two ad hoc cases: noise-contaminated signals are best diagnosed using D 11(ρ), and lacunar signals are best studied using D 0k(ρ), with k as large as possible in magnitude. Therefore, by using a wide range of values of p and k, one can infer whether degradation arising from noise or arising from lacunarity is more pronounced in the time series being studied, and hence, one can decide which of the estimates most efficiently approximates the correlation dimension for the series. These ideas are applied to relatively coarse samplings of the Hénon, Lorenz convection, and Lorenz climate attractors that in each case are obtained by calculating the distances between pairs of points on two trajectories. As expected from previous studies, lacunarity apparently dominates the Hénon results, with the best estimate of D, D = 1.20 ± 0.01, given by the case D 03(ρ). In contrast, undersampling or noise apparently affects the Lorenz convection and climate attractor results. The best estimates of D are given by the estimator D 11(ρ) in both cases. The dimension of the convection attractor is D = 2.06 ± 0.005, and that of the climate attractor is D = 14.9 ± 0.1. Finally, lagged and embedded time series for the Lorenz convection attractor are studied to identify embedding dimension signatures when model reconstruction is employed. In the last part of this study, the above results are used to help identify the best possible estimate of the correlation dimension for a low-frequency boundary layer time series of low-level horizontal winds. To obtain such an estimate, Lorenz notes that an optimally coupled time series must be extracted from the data and then lagged and embedded appropriately. The specific kinetic energy appears to be more closely coupled to the underlying low-frequency attractor, and so more nearly optimal, than is either individual wind component. When several estimates are considered, this kinetic energy series exhibits the same qualitative behavior as does the lagged and embedded Lorenz convective system time series. The series is either noise contaminated or under-sampled, a result that is not surprising given the small number of time series points used, for which the best estimate is given by D 11(ρ). The obtained boundary layer time series dimension estimate, 3.9 ± 0.1, is similar to the values obtained by some other investigators who have analyzed higher-frequency boundary layer time series. Although this time series does not contain as many points as might be required to accurately estimate the dimension of the underlying attractor, it does illustrate the requirement that in any estimate of the correlation dimension, a function of the measured variables must be chosen that is strongly coupled to the attractor.
We enlarge on theoretical insights concerning the multiple scaling/multifractal behaviour of geodynamical fields in the space-time domain and the very singular behaviour of their observables which are usually obtained by averaging over scales much greater than that of the homogeneity. We render more direct the link between statistical singularities (divergence of high order statistical moments) and hierarchies of singularities per realization (small scale divergence of densities). In the case of “hard multifractals” (having not only singular realizations but also — contrary to “soft multifractals” — singular statistics), we insist on the importance of the existence of “wild” singularities which although extremely rare, create the statistical divergences, as well as on the need to distinguish between “bare” and “dressed” properties of these multifractal fields.
A general method for estimating the amount of data sufficient to reliably determine the correlation dimension from a time series is presented. The results of this method are discussed in terms of previous attempts to calculate this quantity from climatological data.KeywordsCorrelation DimensionStrange AttractorNeighbour DistanceControl SequenceScaling BehaviourThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
An algorithm proposed by Takens, which can determine the capacity (generalized dimensionality) of a dynamical system from the time series of a single observable, is tested numerically for several intrinsically stochastic models. The algorithm is found to converge too slowly (if at all) to be useful for the analysis of experimental data.
A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.
The transition to geostrophic turbulence in a rotating annulus experiment has characteristics which differ from the scenarios for the transition to turbulence which have been previously described. Estimates of dimension from the experimental data suggest a mechanism in which a discrete symmetry of the preturbulent state plays an important role. The techniques used for the dimension estimates are new and substantially more efficient than those used previously.
Recently, it has been claimed1 that the worldwide climate over the past million years follows a low-dimensional strange attractor. Contrary to that claim, I report here that there is no sign of such an attractor. This holds both for the worldwide climate of the past 1–2 Myr (averaged over periods of ~5,000 yr) and for a local climate of the past ~7,000 yr with yearly averages. In this context, I shall also discuss the general problem of dimension estimates from sparse data.
Much of our information on climatic evolution during the past million years comes from the time series describing the isotope record of deep-sea cores. A major task of climatology is to identify, from this apparently limited amount of information, the essential features of climate viewed as a dynamic system. Using the theory of nonlinear dynamic systems we show how certain key properties of climate can be determined solely from time series data.
1984: in Turbulence and Chaotic Phenomena in Fluids
- K R Sreenivasan
- P J Strykowski