Multifractal analysis of polyalanines time series

Multifractal properties of the energy time series of short α-helix structures, specifically from a polyalanine family, are investigated through the MF-DFA technique (multifractal detrended fluctuation analysis). Estimates for the generalized Hurst exponent h(q) and its associated multifractal exponents τ(q) are obtained for several series generated by numerical simulations of molecular dynamics in different systems from distinct initial conformations. All simulations were performed using the GROMOS force field, implemented in the program THOR. The main results have shown that all series exhibit multifractal behavior depending on the number of residues and temperature. Moreover, the multifractal spectra reveal important aspects of the time evolution of the system and suggest that the nucleation process of the secondary structures during the visits on the energy hyper-surface is an essential feature of the folding process.     

The time-singularity multifractal spectrum distribution

Although the multifractal singularity spectrum revealed the distribution of singularity exponent, it failed to consider the temporal information, therefore it is hard to describe the dynamic evolving process of non-stationary and nonlinear systems. In this paper, we aim for a multifractal analysis and propose a time-singularity multifractal spectrum distribution (TS-MFSD), which will hopefully reveal the spatial dynamic character of fractal systems. Similar to the Wigner–Ville time-frequency distribution, the time-delayed conjugation of fractal signals is selected as the windows function. Furthermore, the time-varying Holder exponent and the time-varying wavelet singularity exponent are deduced based on the instantaneous self-correlation fractal signal. The time-singularity exponent distribution i.e. TS-MFSD is proposed, which involves time-varying Hausdorff singularity spectrum distribution, time-varying large deviation multifractal spectrum and time-varying Legendre spectrum distribution, which exhibit the singularity exponent distribution of fractal signal at arbitrary time. Finally, we studied the algorithm of the TS-MFSD based on the wavelet transform module maxima method, analyzed and discussed the characteristic of TS-MFSD based on Devil Staircase signal, stochastic fractional motion and real sea clutter.     

Real-space renormalisation approach to the Anderson localisation

A real-space renormalisation method is proposed for random systems. The equation for the Green function in real space is reduced to that for the Green function in renormalised space after the nth decimation transformation to obtain the renormalised Hamiltonian.     

Comment on ‘multifractal diffusion entropy analysis on stock volatility in financial markets’ [Physica A 391 (2012) 5739–5745]

In their recent article ‘multifractal diffusion entropy analysis on stock volatility in financial markets’ Huang, Shang and Zhao (2012) [6] suggested a generalization of the diffusion entropy analysis method with the main goal of being able to reveal scaling exponents for multifractal times series. The main idea seems to be replacing the Shannon entropy by the Rényi entropy, which is a one-parametric family of entropies. The authors claim that based on their method they are able to separate long range and short correlations of financial market multifractal time series. In this comment I show that the suggested new method does not bring much valuable information in obtaining the correct scaling for a multifractal/mono-fractal process beyond the original diffusion entropy analysis method. I also argue that the mathematical properties of the multifractal diffusion entropy analysis should be carefully explored to avoid possible numerical artefacts when implementing the method in analysis of real sequences of data.     

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.     

Multifractal Formalism for Almost all Self-Affine Measures

We conduct the multifractal analysis of self-affine measures for “almost all” family of affine maps. Besides partially extending Falconer’s formula of L ^q -spectrum outside the range 1 < q ≤ 2, the multifractal formalism is also partially verified.     

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     

Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.     

Direct Evidence for Inversion Formula in Multifractal Financial Volatility Measure

The inversion formula for conservative multifractal measures was unveiled mathematically a decade ago, which is however not well tested in real complex systems. We propose to verify the inversion formula using high-frequency turbulent financial data. We construct conservative volatility measure based on minutely S＆P 500 index from 1982 to 1999 and its inverse measure of exit time. Both the direct and inverse measures exhibit nice multifractal nature, whose sealing ranges are not irrelevant. Empirical investigation shows that the inversion formula holds in financial markets.     

Linearization effect in multifractal analysis: Insights from the Random Energy Model

The analysis of the linearization effect in multifractal analysis, and hence of the estimation of moments for multifractal processes, is revisited borrowing concepts from the statistical physics of disordered systems, notably from the analysis of the so-called Random Energy Model. Considering a standard multifractal process (compound Poisson motion), chosen as a simple representative example, we show the following: (i) the existence of a critical order q^∗ beyond which moments, though finite, cannot be estimated through empirical averages, irrespective of the sample size of the observation; (ii) multifractal exponents necessarily behave linearly in q, for q>q^∗. Tailoring the analysis conducted for the Random Energy Model to that of Compound Poisson motion, we provide explicative and quantitative predictions for the values of q^∗ and for the slope controlling the linear behavior of the multifractal exponents. These quantities are shown to be related only to the definition of the multifractal process and not to depend on the sample size of the observation. Monte Carlo simulations, conducted over a large number of large sample size realizations of compound Poisson motion, comfort and extend these analyses.     

Mixed Local-Nonlocal Vector Schrödinger Equations and Their Breather Solutions

To the best of our knowledge, all nonlinearities in the known nonlinear integrable systems are either local or nonlocal. A natural problem is whether there exist some nonlinear integrable systems with both local and nonlocal nonlinearities, and how to solve this kinds of spectral nonlinear integrable systems with both local and nonlocal nonlinearities. Recently, some novel mixed local-nonlocal vector Schrödinger equations are presented, which are different from the single local and nonlocal coupled Schrödinger equation. We investigate the Darboux transformation of mixed local-nonlocal vector Schrödinger equations with a spectral problem. Starting from a special Lax pairs, the mixed localnonlocal vector Schrödinger equations are constructed. We obtain the one- and two- and N-soliton solution formulas of the mixed local-nonlocal vector Schrödinger equations with N-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-solitons are exhibited, the overtaking elastic interactions among the two-breather solitons are considered. We find that unlike the local and nonlocal cases, the mixed local-nonlocal vector Schrödinger equations have some novel results. The results in this paper might be helpful for understanding some physical phenomena described in plasmas.     

Multifractal properties of the Indian financial market

We investigate the multifractal properties of the logarithmic returns of the Indian financial indices (BSE & NSE) by applying the multifractal detrended fluctuation analysis. The results are compared with that of the US S&P 500 index. Numerically we find that qth-order generalized Hurst exponents h(q) and τ(q) change with the moments q. The nonlinear dependence of these scaling exponents and the singularity spectrum f(α) show that the returns possess multifractality. By comparing the MF-DFA results of the original series to those for the shuffled series, we find that the multifractality is due to the contributions of long-range correlations as well as the broad probability density function. The financial markets studied here are compared with the Binomial Multifractal Model (BMFM) and have a smaller multifractal strength than the BMFM.     

A Wigner-function formulation of finite-state quantum mechanics

For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N×N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.     

Transport in polygonal billiards

We present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain. This billiard is a system with no exponential instability. Our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion. Our conclusions are drawn from the numerical evaluation of the mean square displacement, the velocity autocorrelation function and its spectral analysis. Furthermore we analyze the properties of the incoherent scattering function. The super Burnett coefficient seems to be ill defined in all systems. The multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported. Finally, we study the heat conduction in our polygonal chain.     

基于小波leaders的海杂波时变奇异谱分布分析

海杂波的奇异谱分析不仅能从理论上揭示海洋表面的动力学机理,同时也是对海探测雷达的关键技术之一.本文提出基于小波leaders的海杂波时变奇异谱分析方法,将时间信息引入海杂波的奇异谱分析之中,从而实现动态的解析描述海杂波随时间变化的奇异谱特性.在理论上,通过信号自身加窗,将时间信息引入传统的奇异谱(或称多重分形谱),实现了对海杂波时变奇异谱分布分析;在算法上,充分利用了小波leaders技术对于多种奇异性的提取能力(包括chirp奇异性和cusp奇异性),通过对时变奇异性指数和时变尺度函数的Legendre变换,实现对海杂波时变奇异谱分布的计算;在应用部分,采用经典的多重分形模型——随机小波序列(RWC)以及三级海态条件下连续波多普勒体制雷达海杂波进行仿真分析,实验结果表明:1)基于小波leaders的奇异谱分布能跟踪海杂波的时变尺度特性,有效展示其时变奇异性谱分布;2)算法具有较好的负矩特性和统计收敛性.该方法能为复杂非线性系统及随机多重分形信号分析提供参考.     

Identifying financial crises in real time

Following the thermodynamic formulation of a multifractal measure that was shown to enable the detection of large fluctuations at an early stage, here we propose a new index which permits us to distinguish events like financial crises in real time. We calculate the partition function from which we can obtain thermodynamic quantities analogous to the free energy and specific heat. The index is defined as the normalized energy variation and it can be used to study the behavior of stochastic time series, such as financial market daily data. Famous financial market crashes–Black Thursday (1929), Black Monday (1987) and the subprime crisis (2008)–are identified with clear and robust results. The method is also applied to the market fluctuations of 2011. From these results it appears as if the apparent crisis of 2011 is of a different nature to the other three. We also show that the analysis has forecasting capabilities.     

Nonparametric identification of nonlinear dynamic systems using a synchronisation-based method

The present study proposes an identification method for highly nonlinear mechanical systems that does not require a priori knowledge of the underlying nonlinearities to reconstruct arbitrary restoring force surfaces between degrees of freedom. This approach is based on the master–slave synchronisation between a dynamic model of the system as the slave and the real system as the master using measurements of the latter. As the model synchronises to the measurements, it becomes an observer of the real system. The optimal observer algorithm in a least-squares sense is given by the Kalman filter. Using the well-known state augmentation technique, the Kalman filter can be turned into a dual state and parameter estimator to identify parameters of a priori characterised nonlinearities. The paper proposes an extension of this technique towards nonparametric identification. A general system model is introduced by describing the restoring forces as bilateral spring-dampers with time-variant coefficients, which are estimated as augmented states. The estimation procedure is followed by an a posteriori statistical analysis to reconstruct noise-free restoring force characteristics using the estimated states and their estimated variances. Observability is provided using only one measured mechanical quantity per degree of freedom, which makes this approach less demanding in the number of necessary measurement signals compared with truly nonparametric solutions, which typically require displacement, velocity and acceleration signals. Additionally, due to the statistical rigour of the procedure, it successfully addresses signals corrupted by significant measurement noise. In the present paper, the method is described in detail, which is followed by numerical examples of one degree of freedom (1DoF) and 2DoF mechanical systems with strong nonlinearities of vibro-impact type to demonstrate the effectiveness of the proposed technique.     

Estimation of Critical Point in Branching Reactions: Further Examination of Gel Point Formula

The theory of gel point in real polymer solutions is examined with the empirical correlation between the reciprocal of the percolation threshold and the coordination number given by the percolation theory. Applying a larger value of the relative frequency of cyclization, an excellent agreement is obtained between the present theory and the percolation result. This suggest that while the ring distribution on lattices is similar to that in real systems, ring production is more frequent in the lattice model than in real systems. To confirm this conjecture, we derive the ring distribution function of the lattice model as a limiting case of d→∞, and show that the solution is in fact identical to the asymptotic formula of C→∞ in real systems except for the coefficient C, which has a maximum at d = 5, in support of the above conjecture. To examine the validity of the asymptotic solution for the lattice model, we apply it to the critical point problem of the percolation theory, showing that the solution works well in high dimensions greater than six.