On a general concept of multifractality: Multifractal spectra for dimensions,entropies, and Lyapunov exponents. Multifractal rigidity

We introduce the mathematical concept of multifractality and describe various multifractal spectra for dynamical systems, including spectra for dimensions and spectra for entropies. We support the study by providing some physical motivation and describing several nontrivial examples. Among them are subshifts of finite type and one-dimensional Markov maps. An essential part of the article is devoted to the concept of multifractal rigidity. In particular, we use the multifractal spectra to obtain a "physical" classification of dynamical systems. For a class of Markov maps, we show that, if the multifractal spectra for dimensions of two maps coincide, then the maps are differentiably equivalent. (c) 1997 American Institute of Physics.     

Multifractal Analysis of Local Entropies for Expansive Homeomorphisms with Specification

In the present paper we study the multifractal spectrum of local entropies. We obtain results, similar to those of the multifractal analysis of pointwise dimensions, but under much weaker assumptions on the dynamical systems. We assume our dynamical system to be defined by an expansive homeomorphism with the specification property. We establish the variational relation between the multifractal spectrum and other thermodynamical characteristics of the dynamical system, including the spectrum of correlation entropies. Received: 22 September 1998 / Accepted: 11 December 1998     

Multifractal properties of return time statistics

The global statistics of the return times of a dynamical system can be described by a new spectrum of generalized dimensions. Comparison with the usual multifractal analysis of measures is presented, and the difference between the two corresponding sets of dimensions is established. Theoretical analysis and numerical examples of dynamical systems in the class of iterated functions are presented.     

Multifractal Analysis of Human Heartbeat in Sleep

We study the dynamical properties of heart rate variability （HRV） in sleep by analysing the scaling behaviour with the multifractal detrended fluctuation analysis method. It is well known that heart rate is regulated by the interaction of two branches of the autonomic nervous system： the parasympathetic and sympathetic nervous systems. By investigating the multifractal properties of light, deep, rapid-eye-movement （REM） sleep and wake stages, we firstly find an increasing multifractal behaviour during REM sleep which may be caused by augmented sympathetic activities relative to non-REM sleep. In addition, the investigation of long-range correlations of HRV in sleep with second order detrended fluctuation analysis presents irregular phenomena. These findings may be helpful to understand the underlying regulating mechanism of heart rate by autonomic nervous system during wake-sleep transitions.     

Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.     

Are crude oil markets multifractal? Evidence from MF-DFA and MF-SSA perspectives

In this article, we investigated the multifractality and its underlying formation mechanisms in international crude oil markets, namely, Brent and WTI, which are the most important oil pricing benchmarks globally. We attempt to find the answers to the following questions: (1) Are those different markets multifractal? (2) What are the dynamical causes for multifractality in those markets (if any)? To answer these questions, we applied both multifractal detrended fluctuation analysis (MF-DFA) and multifractal singular spectrum analysis (MF-SSA) based on the partition function, two widely used multifractality detecting methods. We found that both markets exhibit multifractal properties by means of these methods. Furthermore, in order to identify the underlying formation mechanisms of multifractal features, we destroyed the underlying nonlinear temporal correlation by shuffling the original time series; thus, we identified that the causes of the multifractality are influenced mainly by a nonlinear temporal correlation mechanism instead of a non-Gaussian distribution. At last, by tracking the evolution of left- and right-half multifractal spectra, we found that the dynamics of the large price fluctuations is significantly different from that of the small ones. Our main contribution is that we not only provided empirical evidence of the existence of multifractality in the markets, but also the sources of multifractality and plausible explanations to current literature; furthermore, we investigated the different dynamical price behaviors influenced by large and small price fluctuations.     

Statistical physics and physiology: monofractal and multifractal approaches.

Even under healthy, basal conditions, physiologic systems show erratic fluctuations resembling those found in dynamical systems driven away from a single equilibrium state. Do such "nonequilibrium" fluctuations simply reflect the fact that physiologic systems are being constantly perturbed by external and intrinsic noise? Or, do these fluctuations actually, contain useful, "hidden" information about the underlying nonequilibrium control mechanisms? We report some recent attempts to understand the dynamics of complex physiologic fluctuations by adapting and extending concepts and methods developed very recently in statistical physics. Specifically, we focus on interbeat interval variability as an important quantity to help elucidate possibly non-homeostatic physiologic variability because (i) the heart rate is under direct neuroautonomic control, (ii) interbeat interval variability is readily measured by noninvasive means, and (iii) analysis of these heart rate dynamics may provide important practical diagnostic and prognostic information not obtainable with current approaches. The analytic tools we discuss may be used on a wider range of physiologic signals. We first review recent progress using two analysis methods--detrended fluctuation analysis and wavelets--sufficient for quantifying monofractual structures. We then describe recent work that quantifies multifractal features of interbeat interval series, and the discovery that the multifractal structure of healthy subjects is different than that of diseased subjects.     

Empirical evidences of a common multifractal signature in economic, biological and physical systems

Recent developments in microcanonical multifractal formalism have lead to a sensible improvement in the numerical techniques for the determination of the multifractal characteristics of real signals. With the aid of these techniques, we have found empirical evidence of a common multifractal signature in six very different systems, ranging from stock market time series to sea surface temperature records. These systems are not only found to be multifractal, but their singularity spectra are coincident. We propose an explanation of this striking coincidence in terms of a cascade process and analyze its consequences.     

Multifractality in Quasienergy Space of Coherent States as a Signature of Quantum Chaos

We present the multifractal analysis of coherent states in kicked top model by expanding them in the basis of Floquet operator eigenstates. We demonstrate the manifestation of phase space structures in the multifractal properties of coherent states. In the classical limit, the classical dynamical map can be constructed, allowing us to explore the corresponding phase space portraits and to calculate the Lyapunov exponent. By tuning the kicking strength, the system undergoes a transition from regularity to chaos. We show that the variation of multifractal dimensions of coherent states with kicking strength is able to capture the structural changes of the phase space. The onset of chaos is clearly identified by the phase-space-averaged multifractal dimensions, which are well described by random matrix theory in a strongly chaotic regime. We further investigate the probability distribution of expansion coefficients, and show that the deviation between the numerical results and the prediction of random matrix theory behaves as a reliable detector of quantum chaos.     

Characteristic distributions of finite-time Lyapunov exponents

We study the probability densities of finite-time or local Lyapunov exponents in low-dimensional chaotic systems. While the multifractal formalism describes how these densities behave in the asymptotic or long-time limit, there are significant finite-size corrections, which are coordinate dependent. Depending on the nature of the dynamical state, the distribution of local Lyapunov exponents has a characteristic shape. For intermittent dynamics, and at crises, dynamical correlations lead to distributions with stretched exponential tails, while for fully developed chaos the probability density has a cusp. Exact results are presented for the logistic map, x-->4x(1-x). At intermittency the density is markedly asymmetric, while for "typical" chaos, it is known that the central limit theorem obtains and a Gaussian density results. Local analysis provides information on the variation of predictability on dynamical attractors. These densities, which are used to characterize the nonuniform spatial organization on chaotic attractors, are robust to noise and can, therefore, be measured from experimental data.     

Multifractal method for the instantaneous evaluation of the stream function in geophysical flows

Multifractal or multiaffine analysis is a promising new branch of methods in nonlinear physics for the study of turbulent flows and turbulentlike systems. In this Letter we present a new method based on the multifractal singularity extraction technique, the maximum singular stream-function method (MSSM), which provides a first order approximation to the stream function from experimental data in 2D turbulent systems. The essence of MSSM relies in relating statistical properties associated with the energy cascade in flows with geometrical properties. MSSM is a valuable tool to process sparse collections of data and to obtain instant estimates of the velocity field. We show an application of MSSM to oceanography as a way to obtain the current field from sea surface temperature satellite images; we validate the result with independent dynamical information obtained from sea level measurements.     

The formation and evolution of fractal structure within chaotic attractors

In this paper we examine in detail the formation and evolution of fractal structure in the chaotic attractors of nonlinear dynamical systems. We explicitly obtain the fractal structure of the underlying chaotic attractors of low-dimensional systems and study their evolution as a system parameter is varied. Using periodic enumeration, dimensional, andf() spectral techniques, we obtain a detailed characterization of the multifractal structure.     

Multifractal moving average analysis and test of multifractal model with tuned correlations

We address two common major problems in the study of time series characterizing fluctuations in complex systems: multifractal analysis and multifractal modeling. Specifically, we introduce a multi-fractal centered moving average (MF-CMA) analysis, which is computationally easier but equivalently performing compared with the well-established multi-fractal detrended fluctuation analysis (MF-DFA) with linear detrending. In addition, we study in detail a generalized binomial multi-fractal model (GB-MFM) to conveniently and reliably generate multifractal surrogate data with arbitrary singularity strengths and arbitrary long-term persistence. We use the data generated by this model as well as realistic, by construction monofractal data series with crossovers and trends to test and compare the multifractal analysis methods and discuss finite-size effects as well as limitations due to spurious multifractality.     

Fluctuation behaviors of financial time series by a stochastic Ising system on a Sierpinski carpet lattice

We develop a financial market model using an Ising spin system on a Sierpinski carpet lattice that breaks the equal status of each spin. To study the fluctuation behavior of the financial model, we present numerical research based on Monte Carlo simulation in conjunction with the statistical analysis and multifractal analysis of the financial time series. We extract the multifractal spectra by selecting various lattice size values of the Sierpinski carpet, and the inverse temperature of the Ising dynamic system. We also investigate the statistical fluctuation behavior, the time-varying volatility clustering, and the multifractality of returns for the indices SSE, SZSE, DJIA, IXIC, S&P500, HSI, N225, and for the simulation data derived from the Ising model on the Sierpinski carpet lattice. A numerical study of the model’s dynamical properties reveals that this financial model reproduces important features of the empirical data.     

On Multifractal Rigidity

We analyze when a multifractal spectrum can be used to recover the potential. This phenomenon is known as multifractal rigidity. We prove that for a certain class of potentials the multifractal spectrum of local entropies uniquely determines their equilibrium states. This leads to a classification which identifies two systems up to a change of variables.     

Multifractal Nonrigidity of Topological Markov Chains

Given a multifractal spectrum, we consider the problem of whether it is possible to recover the potential that originates the spectrum. The affirmative solution of this problem would correspond to a “multifractal” classification of dynamical systems, i.e., a classification solely based on the information given by multifractal spectra. For the entropy spectrum on topological Markov chains we show that it is possible to have both multifractal rigidity and multifractal “nonrigidity”, by appropriately varying the Markov chain and the potential defining the spectrum. The “nonrigidity” even occurs in some generic sense. This strongly contrasts to the usual opinion among some experts that it should be possible to recover the potential up to some equivalence relation, at least in some generic sense. Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, through FCT by Program POCTI/FEDER and the grant SFRH/BD/10154/2002.     

Multifractal Analysis of Lyapunov Exponent for Continued Fraction and Manneville–Pomeau Transformations and Applications to Diophantine Approximation

We extend some of the theory of multifractal analysis for conformal expanding systems to two new cases: The non-uniformly hyperbolic example of the Manneville–Pomeau equation and the continued fraction transformation. A common point in the analysis is the use of thermodynamic formalism for transformations with infinitely many branches. We effect a complete multifractal analysis of the Lyapunov exponent for the continued fraction transformation and as a consequence obtain some new results on the precise exponential speed of convergence of the continued fraction algorithm. This analysis also provides new quantitative information about cuspital excursions on the modular surface. Received: 13 October 1998 / Accepted: 19 April 1999     

Joint multifractal analysis based on wavelet leaders

Mutually interacting components form complex systems and these components usually have long-range cross-correlated outputs. Using wavelet leaders, we propose a method for characterizing the joint multifractal nature of these long-range cross correlations; we call this method joint multifractal analysis based on wavelet leaders (MF-X-WL). We test the validity of the MF-X-WL method by performing extensive numerical experiments on dual binomial measures with multifractal cross correlations and bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. Both experiments indicate that MF-X-WL is capable of detecting cross correlations in synthetic data with acceptable estimating errors. We also apply the MF-X-WL method to pairs of series from financial markets (returns and volatilities) and online worlds (online numbers of different genders and different societies) and determine intriguing joint multifractal behavior.     

Free Energy as a Geometric Invariant

The free energy plays a fundamental role in statistical and condensed matter physics. A related notion of free energy plays an important role in the study of hyperbolic dynamical systems. In this paper we introduce and study a natural notion of free energy for surfaces with variable negative curvature. This geometric free energy encodes a new type of marked length spectrum of closed geodesics, which lies between the well-known marked length spectrum (marked by the corresponding element of the fundamental group) and the unmarked length spectrum. We prove that the free energy parametrizes the boundary of the domain of convergence of a Poincaré series which also encodes this spectrum. We also show that this new length spectrum, or equivalently the geometric free energy, is not an isometry invariant. In the final section we use tools from multifractal analysis to effect a fine asymptotic comparison of word length and geodesic length of closed geodesics. We hope that our geometric understanding of free energy will provide new insight into this fundamental physical and dynamical quantity. The work of the second author was partially supported by a National Science Foundation grant DMS-0355180. This work was completed during a visit by the first author to Penn State as a Shapiro Fellow.