Attractor geometry and fractals

In the preceding chapter we discussed the dynamical side of chaos which manifests itself in the sensitive dependence of the evolution of a system on its initial conditions. This strange behaviour in time of a deterministically chaotic system has its counterpart in the geometry of the set in phase space formed by the (non-transient) trajectories of the system, the attractor.

Attractors of dissipative chaotic systems (the kind of systems we are interested in) generally have a very complicated geometry, which led people to call them strange. However, strange sets can also occur without dissipation in more general settings. As we have pointed out already in Chapter 3, a system described by autonomous differential equations (a flow) cannot be chaotic in less than three dimensions. With the same argument that trajectories are not allowed to intersect in a deterministic system we can conclude that not only the phase space but also the attractor of a chaotic flow must be more than two dimensional. However, slightly more than two dimensions is sufficient and the motion on a 2 + ∈ dimensional fractal can indeed be chaotic. As we will see, strange attractors with fractional dimensions are typical of chaotic systems. Map-like systems can of course show chaos with attractor dimensions less than two. Noninteger dimensions are assigned to geometrical objects which exhibit an unusual kind of self-similarity and which show structure on all length scales.

Example 6.1 (Self-similarity of the NMR laser attractor). Such self-similarity is demonstrated in Fig. 6.1 for an attractor reconstructed from the NMR laser time series, Appendix B.2.

Abstract

The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions Dq and the f(α) singularity spectrum can be readily determined from the scaling behaviour of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of ‘generalized boxes’. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent applications of the wavelet transform modulus maxima method to fully developed turbulence data. Then we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic information about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered from the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period- doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos.

The reader might want to jump right into the book, but I decided to give a guided tour (which one may leave and rejoin at will of course) through the chapters, to whet the reader's taste.

Antoine opens in Chapter 1 with a brief survey of the basic properties of wavelet transforms, both continuous (CWT) and discrete (DWT). In the latter case one learns about the intuitively very appealing concept of multiresolution analysis. Section 1.4 looks ahead to the two- and more-dimensional versions, and summarily brings out connections with well known symmetry groups of physics, and the theory of coherent states.

In the second chapter, also by Antoine, the 2-D wavelet transform is treated. Here the characterization as mathematical microscope must be further qualified, because it misses the new and important property of orientability of the 2-D wavelets, which the 1-D case lacks. A real-world microscope is not more sensitive in one direction than in another one, it is ‘isotropic’. But the mathematical microscope as embodied in 2-D wavelets has an extra feature: these wavelets can be designed in such a way that they are directionally selective. Apart from dilation and translation, one can now also rotate the wavelet, which makes possible a sensitive detection of oriented features of a signal (a 2- D image). In many texts the 2-D case is still limited to the DWT, and the wavelets are usually formed by taking tensor products of 1-D wavelets in the x and y-direction, thereby giving preference to horizontal, vertical and diagonal features in the plane.

Abstract

In the field of atomic and solid state physics, wavelet analysis has been applied so far in three different directions: (i) time-frequency analysis of harmonic generation in laser-atom interactions; (ii) ab initio electronic structure calculations in atoms and molecules; and (iii) construction of localized bases for the lowest Landau level of a 2-D electron gas submitted to a strong magnetic field. We survey these three types of applications, with more emphasis on methods than on precise results.

Introduction

There are two ways in which wavelets could play a role in atomic physics and possibly in solid state physics.

First one may envisage them as physical objects, namely quantum states or wave functions. It is commonplace to remark that coherent states (CS) have a privileged role in atomic physics. Laser-atom interactions, revival phenomena, Rydberg wave packets and various semi-classical situations are all instances in which a coherent state description is clearly well-adapted. Of course, what is implied here are canonical CS, associated to the harmonic oscillator or the electromagnetic field [36]. But wavelets are also coherent states, namely those associated to the affine groups in various space dimensions, as we have seen in Chapter 2 (see [1] for a review on coherent states). Thus wavelets could well be thought of as convenient substitutes for canonical CS. However, this suggestion is still speculative at the present moment, very little has been achieved in this direction.

Abstract

Two fundamental properties of turbulence are intermittency and non-linearity. They imply that the standard Fourier spectral techniques are inadequate for its analysis. Spectral analysis based on wavelets provides a means to handle intermittency. New tools are required to handle non-linearity.

In this chapter, we redesign spectral analysis in terms of wavelet methods, paying particular attention to statistical stability, error estimates and nonlinearity. The application to both computer simulations and measurements carried out in fusion plasmas provide some illustrative examples.

Introduction

Although the phenomenon of turbulence is only partially understood, there seems to be consensus on several aspects. First, that intermittency is a basic property of turbulence. This means that the characteristics of the turbulence (spectral distribution, amplitude etc.) vary on a short time scale. Analysis techniques that rely on the accumulation of data over time scales larger than this characteristic time scale will then average out much of the dynamics and obliterate relevant information (as may occur with Fourier analyses). Wavelet analysis provides an interesting starting point for redesigning the standard analysis techniques in order to tackle this problem. In this chapter we shall redefine some basic Fourier analysis techniques in terms of wavelets, such as cross coherence. We shall emphasize the need for statistical stability and provide noise level estimates. Finally, we provide some examples of these techniques.

Second, it is generally accepted that turbulence only arises in non-linear systems. Therefore, to understand the nature of turbulence, it is essential to employ analysis tools that are capable of handling this non-linearity.

Abstract

We have used wavelets to analyse, model and compute turbulent flows. The theory and open questions encountered in turbulence are presented. The wavelet-based techniques that we have developed to study turbulence are explained and the main results are summarized.

Introduction

In this chapter we will summarize the ten years of research we have done to try to better understand, model and compute fully developed turbulent flows using wavelets and wavelet packets. Fully developed turbulence is a highly nonlinear regime (very large Reynolds number tending to infinity) and is distinct from the transition to turbulence (low Reynolds number). We have chosen to present a personal point of view concerning the current state of our understanding of fully developed turbulence. It may not always coincide with the point of view of other researchers in this field because many issues we are addressing in this chapter are still undecided and highly controversial. This paper is a substantially revised and extended version of: Wavelets and Turbulence by Farge, Kevlahan, Perrier and Goirand which appeared in Proceedings of the IEEE, vol. 84, no. 4, April 1996, pp. 639–669.

After more than a century of turbulence study [30], [173], no convincing theoretical explanation has produced a consensus among physicists (for a historical review of various theories of turbulence see [160], [158], [72], [91]). In fact, a large number of ad hoc ‘phenomenological’ models exist that are widely used by fluid mechanicians to interpret experiments and to compute many industrial applications (in aeronautics, combustion, meteorology …) where turbulence plays a role.

Abstract

The wavelet transform is used in astrophysics for many applications. Its use is connected to different properties. The Time-Frequency analysis results from the two-dimensional feature of this transform. Some interesting applications were performed on nonstationary astrophysical signals. Many astrophysical results were obtained by this analysis, either on quasi regular variables, and on chaotic light curves. Solar time series have been also carefully analysed by the wavelet transform. New results have been obtained for series with identified periods (sunspots, diameter, irradiance, chromospheric oscillations) and for chaotic signals (magnetic activity).

Astronomers have exploited the wavelet transform for image compression. Many packages are proposed with significant gains. Some full sky surveys are available now with images compressed by the wavelet transform. Filtering and restorations are derived from this scale-space analysis. Some thresholding rules furnish adapted filtering. The restoration is connected to an approach for which we progressively extract the most energetic features. This may be related to the notion of multiscale support. Many applications were done for Hubble Space Telescope (HST) images or for astronomical aperture synthesis. The ability of the wavelet transform to localize an object in scale-space led also to applying this transform to the detection and to the analysis of astronomical sources. A multiscale vision model was developed by our group, which allows one to detect and to characterize all the sources of different sizes in an astronomical image. Many applications of image analysis were performed on different astrophysical sources, and specifically the ones having a power-law correlation, i.e. a fractal-like behaviour: molecular clouds, infrared cirrus, clumpy galaxies, comets, X-ray clusters, etc.

Abstract

We present a combined wavelet and analytic signal approach to study biological and physiological nonstationary time series. The method enables one to reduce the effects of nonstationarity and to identify dynamical features on different time scales. Such an approach can test for the existence of universal scaling properties in the underlying complex dynamics. We applied the technique to human cardiac dynamics and find a universal scaling form for the heartbeat variability in healthy subjects. A breakdown of this scaling is associated with pathological conditions.

Introduction

The central task of statistical physics is to study macroscopic phenomena that result from microscopic interactions among many individual components. This problem is akin to many investigations undertaken in biology. In particular, physiological systems under neuroautonomic regulation, such as heart rate regulation, are good candidates for such an approach, since: (i) the systems often include multiple components, thus leading to very large numbers of degrees of freedom, and (ii) the systems usually are driven by competing forces. Therefore, it seems reasonable to consider the possibility that dynamical systems under neural regulation may exhibit temporal structures which are similar, under certain conditions, to those found in physical systems. Indeed, concepts and techniques originating in statistical physics are showing promise as useful tools for quantitative analysis of complicated physiological systems. An unsolved problem in biology is the quantitative analysis of a nonstationary time series generated under free-running conditions [1-3].

Why should physicists bother about wavelets? Why not leave them to the mathematicians and engineers?

Physicists are sometimes reluctant to learn about wavelets because they cannot be interpreted in physical terms as easily as sines and cosines and their frequencies. This is understandable enough: the ‘harmonic oscillator’ has been with us for more than three centuries, and continues to play its important role. But as we hope to show in the chapters that follow, wavelets can also be of great help in uncovering the presence or absence of certain frequencies in a physical phenomenon. Wavelet analysis is not replacing frequency analysis, but is rather an important refinement and expansion of it: Fourier analysis analyses a signal globally, whereas wavelet analysis looks into the signal locally.

Let us illustrate this is in musical terms. If you listen to a classical symphony you hear several parts, usually three to four. Each of them has its own main key: e.g. C minor, Eb major, etc. The Fourier power spectrum of the symphony will of course reveal the dominating keys: groundtones, and their harmonics. Frequencies of other chords which occur more fleetingly during modulations and variations in the piece of music, will also show up. If you would play the parts in a different order, the power spectrum would not change at all, but to the listener it becomes a very different piece, and more so if you interchange parts within the parts, at an ever finer scale: you have changed the musical score drastically.

Since the hardback edition of this book was put together wavelets have continued to flourish both in mathematics and in applications in ever more diverse branches of science and engineering. A standard library electronic alert system now easily produces more than fourteen hundred references to papers per year, developing or using wavelet techniques. These are published in a very broad array of journals. Here we can point to only a few of the recently developed methods, in particular as they have been used in physics.

In recent years many variations on the wavelet theme have appeared. One tries to go ‘beyond wavelets’. In this context there is a whole family of new animals in the wavelet zoo. Its members carry names like bandelets, beamlets, chirplets, contourlets, curvelets, fresnelets, ridgelets … These are new bases or frames of functions, customized to handle 2D or 3D data processing better. In [23] for example, it is explained how ridgelets and curvelets can be used in astrophysics. It turns out that noise filtering, contrast enhancement and morphological component analysis of galaxy images are performed much better by a skilful combination of the new transforms than by mere wavelet transforms. More examples can be found on the ‘curvelet homepage’ [24], maintained by J.L. Starck.

It seems that the applications of the discrete wavelet transform (DWT) far outnumber those of the continuous wavelet transform (CWT), although the latter started the modern development of wavelet theory in the early eighties.

Abstract

Atmospheric blocking is an irregularly recurring anomalous state of the atmospheric circulation which is large and spatially localized. Atmospheric blocking during three unusual winter months is studied by multiresolution analysis and a new periodic wavelet-based adaptation of traditional Fourier series-based energetics. New forms of the transfer functions of kinetic energy with the mean and eddy parts of the atmospheric circulation are introduced. These quantify the zonally localized conversion of energy between scales. A new accounting method for wavelet-indexed transfers permits the introduction of a physically meaningful zonally localized scale flux function. These techniques are applied to National Meteorological Center data. Blocking is found to be largely described by just the second-largest scale part of the multiresolution analysis. New support is found for the hypothesis that blocking is partially maintained by a particular kind of upscale cascade. Specifically, in both Atlantic and Pacific blocking cases there is a downscale (upscale) cascade west (east) of the block.

Introduction

Although wavelet analysis in the time domain has been applied to atmospheric boundary layer turbulence (e.g. [8]) and climatic time series (e.g. [3, 15, 17]), and in the space domain to numerically simulated turbulence [7, 9, 18], there has not been any application to observed global synoptic meteorological data. A broad review of wavelets applied to turbulence is presented by Farge et al., this volume, Chapter 4. A collection of blocking studies is contained in [1]. During blocking, the normal progression of weather is locally inhibited.

Abstract

We review the general properties of the wavelet transform, both in its continuous and its discrete versions, in one or more dimensions. We also indicate some generalizations and applications in physics.

What is wavelet analysis?

Wavelet analysis is a particular time- or space-scale representation of signals which has found a wide range of applications in physics, signal processing and applied mathematics in the last few years. In order to get a feeling for it and to understand its success, let us consider first the case of one-dimensional signals.

It is a fact that most real life signals are nonstationary and usually cover a wide range of frequencies. They often contain transient components, whose apparition and disparition are physically very significant. In addition, there is frequently a direct correlation between the characteristic frequency of a given segment of the signal and the time duration of that segment. Low frequency pieces tend to last a long interval, whereas high frequencies occur in general for a short moment only. Human speech signals are typical in this respect: vowels have a relatively low mean frequency and last quite long, whereas consonants contain a wide spectrum, up to very high frequencies, especially in the attack, but they are very short.

Clearly standard Fourier analysis is inadequate for treating such signals, since it loses all information about the time localization of a given frequency component.

Abstract

The energy cascade found in fully developed fluid turbulence is believed to originate as large-scale organized motions called coherent structures. The process of detecting, locating, and tracking these coherent structures is therefore of central importance to the continued study of turbulence. A number of researchers have applied wavelet-based methods to the problem of coherent structure detection, and significant performance improvements over other existing methods have already been reported.

In this paper, we compare the performance of various conventional as well as wavelet-based detector algorithms for cylinder wake flow data. The resulting ROC curves quantitatively demonstrate the effectiveness of wavelet methods. The detections are then used to form conditional averages of the velocity time-series, revealing their underlying physical structure.

Introduction

Recently, advances in the theoretical understanding and implementation of wavelets have led to their increased use in analysis and signal processing. Wavelet methods can be very effective in the study of non-stationary phenomena [7], and have thus sparked a concerted interest in applying them to the analysis of turbulent flows in general, and to the detection of coherent structures in particular [8].

It is possible that some coherent structure detectors are better suited for certain types of turbulent flows, or operate most effectively under specific conditions. As the numbers and types of detectors grow, it becomes increasingly important to measure their relative performance in quantitatively meaningful ways. In this chapter, we describe an approach to the comparison of detector algorithms by means of the Receiver Operating Characteristic (ROC) curves, and demonstrate the utility of this method for the case of cylinder wake flow.

Abstract

We begin with a short review of the 2-D continuous wavelet transform (CWT) and describe a number of physical applications. Then we discuss briefly the mathematical background, namely coherent states derived from group representations, and we show how it allows a straightforward extension to more general situations, such as higher dimensions, wavelets on the sphere or time-dependent wavelets. We conclude with a short outline of the 2- D discrete wavelet transform, some generalizations and a few physical applications.

Introduction

As we have seen in Chapter 1, both the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT) may be extended to two dimensions. Here also, many applications have been developed, in various branches of physics and in image processing. As in the 1-D case, the CWT is better adapted to analysis, for instance the detection of specific features in an image. This is true, in particular, for oriented features, if one uses a wavelet which is directionally selective. On the other hand, the strong point of the DWT is data compression, notably in transmitting or reconstructing a 2-D signal after processing (e.g. denoising).

We will spend most of the present chapter discussing the 2-D CWT, for two reasons. First, it admits a number of interesting physical applications, that we will describe in Section 2.3. The second motivation is that its mathematical background, namely group representation theory (Section 2.4), suggests a straightforward extension to more general situations, such as wavelets in higher dimensions, or on manifolds (a sphere, for instance), or time- dependent wavelets, a promising tool for motion tracking (Section 2.5).