Nonlinear Time Series Analysis of Sunspot Data

This article deals with the analysis of sunspot number time series using the Hurst exponent. We use the rescaled range (R/S) analysis to estimate the Hurst exponent for 259-year and 11 360-year sunspot data. The results show a varying degree of persistence over shorter and longer time scales corresponding to distinct values of the Hurst exponent. We explain the presence of these multiple Hurst exponents by their resemblance to the deterministic chaotic attractors having multiple centers of rotation.     

A STUDY OF MAGNETIC COMPLEXITY USING HURST'S RESCALED RANGE ANALYSIS

A fractal analysis using the classical Hurst method has been applied to artificial data, simulated sunspot magnetic field data, and to data acquired with NASA/Marshall Space Flight Center's vector magnetograph. The main goals of this study are to quantify the complexity of an active region and to determine if significant changes in complexity are associated with flare activity. We tested the analysis using three basic types of two-dimensional synthetic data: (1) data composed of gaussians with various types of superimposed features, (2) random data, and (3) synthetic sunspots created from a basic, simple configuration on which are placed increasingly smaller structures. Our results confirm that the Hurst method of analysis is sensitive to the presence of large-scale structures within a two-dimensional image. When the large-scale structure has been removed, the value of the Hurst exponent is inversely proportional to increasing complexity in the image. The Hurst exponent of magnetograph data with the large-scale structure of the sunspot removed, shows a tantalizing variation in the shear parameter five minutes prior to a flare.     

类星体3C 446射电光变非线性特性分析

类星体有剧烈、大幅度的光变现象, 光变研究有助于建立与观测相符的理论模型. 这篇文章从密歇根大学射电天文台数据库收集了类星体3C 446射电4.8、8.0和14.5GHz波段的长期观测数据. 传统的线性方法难以分析复杂的光变现象, 文章采用了集合经验模态分解(Ensemble Empirical Mode Decomposition, EEMD)方法和非线性分析方法相结合, 从混沌动力学特性、分形特性和周期性多角度对类星体光变随时间演化的规律进行了较全面的分析, 并重点对比分析了除去周期成分或混沌成分前后, 光变的周期性和非线性特性是否存在明显区别. 分析结果表明, 类星体3C 446射电波段光变资料由周期成分、趋势成分和混沌成分组成, 光变具有周期性、混沌性和分形特性. 除去混沌成分和趋势成分后的光变周期与原始光变资料的周期完全相同, 而两者的混沌和分形特性有明显不同. 从饱和关联维数来看, 重构动力学系统时, 除去周期成分和趋势成分后的光变资料比原始光变资料需要更多的独立参量, Kolmogorov熵值表明前者信息的损失率比后者大, 系统的混沌程度更高, 系统也更复杂, Hurst值表明后者自相似性和长程相关性比前者略强.     

Three-body dynamics: intermittent chaos with strange attractor

We have studied the structure of chaos in three-body dynamics using the concept of intermittency, implying that violent states of a system alternate in time with quasi-regular states producing together a non-stationary and evolving pattern of unpredictable behaviour. Computer simulations are produced to demonstrate explicitly sporadic short violent bursts in quasi-regular hierarchical states of the systems. This is seen both in orbits and in the long time series generated by the system. The time series prove to be similar in shape to what is observed in various physical experiments with laboratory chaotic systems when they reveal the so-called type-III intermittency. The new effective methods of time series analysis enable us to discover a strange attractor with a fractal dimension slightly above 2. This shows that three-body dynamics has the same intrinsic qualitative structure and quantitative measure of chaos as the widely known chaotic system, the Lorenz attractor.     

Low-dimensional chaos and fractal properties of long-term sunspot activity

Two primary solar-activity indicators sunspot numbers(SNs)and sunspot areas(SAs)in the time interval from November 1874 to December 2012 are used to determine the chaotic and fractal properties of solar activity.The results show that(1)the long-term solar activity is governed by a low-dimensional chaotic strange attractor,and its fractal motion shows a long-term persistence on large scales;(2)both the fractal dimension and maximal Lyapunov exponent of SAs are larger than those of SNs,implying that the dynamical system of SAs is more chaotic and complex than SNs;(3)the predictions of solar activity should only be done for short-to mid-term behaviors due to its intrinsic complexity;moreover,the predictability time of SAs is obviously smaller than that of SNs and previous results.     

Long-term persistence of solar activity

We examine the question of whether or not the non-periodic variations in solar activity are caused by a white-noise, random process. The Hurst exponent, which characterizes the persistence of a time series, is evaluated for the series of¹⁴C data for the time interval from about 6000 BC to 1950 AD. We find a constant Hurst exponent, suggesting that solar activity in the frequency range from 100 to 3000 years includes an important continuum component in addition to the well-known periodic variations. The value we calculate,H 0.8, is significantly larger than the value of 0.5 that would correspond to variations produced by a white-noise process. This value is in good agreement with the results for the monthly sunspot data reported elsewhere, indicating that the physics that produces the continuum is a correlated random process and that it is the same type of process over a wide range of time interval lengths.     

Multifractality due to long-range correlation in the L-band ionospheric scintillation S 4 index time series

The earth’s ionosphere is well recognized as a dynamical system and non-linearly coupled with the magnetosphere above and natural atmosphere below. The shape and time variability of the ionosphere indeed shows chaos, pattern formation, random behaviour and self-organization. The present paper studies the propriety of Multifractal Detrended Fluctuation Analysis (MF-DFA) technique for the ionospheric scintillation index time series. MF-DFA is used to identify the scaling behaviour of the ionospheric scintillation time-series data of two different nature. The obtained results show the robustness and the relevancy of the MF-DFA technique for the ionospheric scintillation index time series. The comparison of the MF-DFA results of original data to those of shuffled and surrogate series shows that the multifractal nature of considered time-series is almost due to long-range correlations. Subsequently, the Hurst exponents derived from two parallel methods namely Rescaled range analysis (R/S) and Auto Correlation Function (ACF) are also suggesting the presence of long range correlation. The presented results in this work may be of assistance for future modeling and simulation studies.     

Rescaled range analysis of the asymmetry of solar activity

Previous studies of the north-south asymmetry of solar activity (e.g., Carbonell, Oliver, and Ballester, 1993; Oliver and Ballester, 1994) suggest that the asymmetry time series can be represented by means of a multicomponent model made up of a long-term trend, a weak sinusoidal component (with a period close to 12.1 years) and a dominant random process. Here, we have used the rescaled range analysis to study the valuation of the stochastic component of the asymmetry. To avoid the influence of the trend and the sinusoidal component on the result, we have removed both from the original time series. The value obtained for the Hurst exponent (0.717 ± 0.002) suggests that the non-periodic component is a correlated random process.     

Research on the Methods of Performance Monitoring and Evaluation for Active Hydrogen Maserstwo

Active hydrogen maser is the main frequency standard for establishing and maintaining the time scale. It has the characteristics of high short-term stability and low phase noise. At present, it plays an increasingly important role in the international atomic time (TAI) and various local time scales. Firstly, in combination with the internal state parameters of an active hydrogen maser, the correlation between the internal state parameters and the comparison data of a hydrogen maser is analyzed, and a method for the performance monitoring of a hydrogen maser is proposed. Secondly, according to the characteristics of hydrogen maser performance, a method for evaluating the performance of a hydrogen maser is given. The hydrogen maser performance includes mainly two aspects, namely the frequency stability and predictability. The performances of two types of active hydrogen masers (CH1-75 and MHM-2010) are evaluated by this method. The correlation analysis of the atomic clock state parameters and comparison data shows that the state parameter monitoring can effectively predict the variation of the hydrogen maser performance. The evaluation results of atomic clock frequency stability and predictability show that the atomic clock with a higher medium-and-long term frequency stability has a better predictability. There are two methods for the predictability evaluation, one is based on the data published by BIPM (Bureau International des Poids et Measures), and another one is based on the quadratic model. Both methods are consistent with the weights published by BIPM. Therefore, the two methods can be used as a quantitative method to evaluate the predictability of a hydrogen maser.     

The Intermediate-Term Quasi-Cycles of Wolf Number and Group Sunspot Number Fluctuations

I apply spectral and auto-correlation analyses to the monthly Wolf number fluctuations for 22 solar cycles and to the group sunspot number fluctuations for 18 solar cycles and find the existence of an 11-month quasi-periodicity in these data. Its strength correlates very well (ρ ⑈ 0.8) with the variance of fluctuations. Moreover, for both Wolf and group sunspot indexes I divide a stationary version of fluctuation time series into two parts: those from periods of low and high solar activity. I find statistically significant quasi-periodicity (9 months) in both high- and low-activity data sets. I also find the quasi-period of about 15 months in the time series of high-activity periods.     

A Fractal Structure of the Time Series of Global Indices of Solar Activity

The structure of time series of daily global indices of solar activity is investigated: the sunspot numbers for the time interval between the years 1854 and 1996, the Greenwich total sunspot area for 1874–1983, the radio-flux at 10.7 cm (F10.7) for 1964–1996, and the Stanford mean solar magnetic field for 1975–1996. The fractal dimensions are determined by two fractal and spectral methods. The identified three time-scale ranges, 2 days–2 months, 2 months–2 years, 2 years–8 and more years, with the fractal dimensions 1.4–1.6, 2, 1.2–1.6, respectively, show perhaps some fractal structure of time series of global indices. The first time-scale range may correspond to ordinary brownian noise and the second to flicker noise. The solar rotation influence of the value of the fractal dimensions at the time range close to the rotational period is studied.     

Surface roughness of alumina fragments caused by hypervelocity impact

Hypervelocity impact experiments are carried out with alumina ceramic targets at impact velocities of 2-4 km/s. The fractal dimension (or roughness exponent) of fragment surfaces is estimated by two different methods (different length scales). One is a divider method (large scale) and the other is a gas-absorption one (small scale). The obtained fractal dimensions are different: ∼2.2-2.4 for larger scale and ∼2.5-2.7 for smaller scale. These values are regardless of the degree of fragmentation.     

天体物理非线性问题初探-活动星系核的光变及引力的局部近似

展示活动星系核研究和引力研究中的一些难点,揭示了某些天体的光变存在分形结构的可能性。我们提出了活动星系核的中心黑洞具有某种随机微活动性的假设,该假设可解释活动星系核的一些观测现象。我们发现Ｗｅｉｅｒｓｔｒａｓ函数的分形图形可以用来很好地拟合ＯＪ２８７的短时标光变。一个用Ｗｅｉｅｒｓｔｒａｓ函数构造的分形曲线与ＯＪ２８７的历史光变的线性相关系数达到了０．７３。我们计算了ＯＪ２８７光变曲线的计盒维数,信息维数和关联维数,结果表明,这些维数均大于１。我们研究了两个活动星系核（ＯＪ２８７和３Ｃ３４５）的历史光变,发现在星等和这两个物体滞留在这个星等之下的时间长度的对数之间存在线性关系。这种关系可能源于光变的分形结构。我们提出,空时点及附于该点的一个标架与Ｍｉｎｋｏｗｓｋｉ空间上的一个３阶反对称张量场对应。场中的张量取自于一个固定的张量集合。我们得到了Ｒｉｅｍａｎｎ正则坐标下引力场的一个级数表示式。我们还计算了Ｒｉｃｃｉ张量的线性近似和一级非线性近似。这有利于强引力场下的局部非线性结构分析。期望对于引力的进一步研究,通过非线性科学这门新方法的桥粱,沟通极端天文现象与基本引力相互作用之间的鸿沟     

On multifractality and synchronization of nonlinear stellar pulsators

Simple models of nonlinear stellar pulsation, whose temporal behavior may reproduce some of the observed features of different classes of variable stars, have been studied. The temporal behavior of dynamical variables of these models exhibits a cascade of period doubling chaos, depending on the specific values of the various control parameters. A multifractal detrended fluctuation analysis (MFDFA) method is further used to identify the scaling behavior of such synthetic time series. The MFDFA of the considered time series, for various models of nonlinear stellar pulsation, shows that the observed multifractal nature is due to long-range correlations. The pulsating star with increased nonadiabaticity and the star with increased convective luminosity, as represented by the simulated data, is shown to possess a strange attractor with noninteger correlation dimension that lies between 2–3. Also the problem of synchronization in coupled nonlinear pulsation models has been investigated using permutation entropy—a complexity measure of the system. The computed order parameter, Γ, representing the correlation of computed permutation entropy for different segments of the simulated time series of displacement of two nonidentical oscillators, has been further used to find the critical coupling parameter for general synchronization of the oscillators.     

Hurst analysis of Mt. Wilson rotation measurements

I study the temporal variation of the solar rotation on time scales shorter than the 11-year cycle by analyzing the daily Mt. Wilson Doppler measurements from 1967 to 1992. The differential rotation is represented by the three coefficients,A, B, andC, of the following expansion: =A +B sin²() +C sin⁴(). TheA, B, andC time series show clearly the 11-year solar cycle and they also show high-frequency fluctuations. The Hurst analysis of these time series shows that a Gaussian random process such as observational noise can only account for fluctuations on time scales shorter than 20 days. For time scales from 20 days to 11 years, the variations of A give rise to a Hurst exponent ofH = 0.83, i.e., the variations ofA are persistent. The temporal variations ofB show the same behavior asC, which is different fromA. From one to 11 years, theB andC variations are dominated by the 11-year cycle, while for time lags shorter than about 250 days, theB andC fluctuations give rise to a Hurst exponent ofH = 0.66, which lies betweenH = 1/2, of a Gaussian random process, and the exponent of the persistent process shown byA. An analysis of the equivalent coefficients of the first three even Legendre polynomials, computed usingA, B, andC, provides additional information. For time scales between 100 and 1000 days, the ranges,R/S, of Legendre polynomial coefficients decrease with increasing order of the polynomials which suggests that the persistent process operates mainly on large spatial scales. The Hurst exponent ofH = 0.83 for variations inA is the same asH for monthly sunspot numbers with time scales between 6 months and 200 years and for¹⁴C radiocarbon data with time scales between 120 years and 3000 years, previously analyzed by other authors. The combined results imply that the underlying solar process shows the same persistent behavior for time scales as short as about 20 days up to time scales of a few thousand years.Operated by the Association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation.     

Application of Non-linear Analysis to Intensity Oscillations of the Chromospheric Bright Points

We have applied several nonlinear techniques to the intensity oscillations of the chromospheric bright points observed at the Vacuum Tower Telescope (VTT) of the Sacramento Peak Observatory. A 35-min time sequence obtained in the Caii H line over a quiet region at the center of the solar disk under high spatial, spectral, and temporal resolution has been used. A relatively new approach is used to detect the hidden periodicity and to extract the associated periodic component(s) from an apparently irregular time series. The unique feature of this approach is that the constituent component(s) can be non-sinusoidal in nature. The periodicity analysis shows that time series of intensity oscillations of most of the bright points can be composed of two non-sinusoidal periodic components with periodicity varying between 2.4 min and 5.8 min. With the help of a multivariate embedding technique, globally significant spatial nonlinear correlation is found. The identification of the nonlinear interaction between bright points is performed by using the methods of dynamical phase synchronization and the similarity index. The analysis indicates that bright points are interconnected in the sense that some bright points are more active and can influence the other relatively passive bright points.     

Short-term variations of cosmic-ray intensity and flare related data in 1981–1983

A statistical analysis of the cosmic-ray intensity (CR) daily means, registered at three Neutron Monitor stations with different cut-off rigidities (Deep River, Climax and Alma-Ata), as well as, of the solar hard X-ray flares fluence recorded by Venera-13, -14 space-probes, has been performed for the time interval 1981–1983. Various methods of time series spectrum analysis, such as Fast Fourier Analysis (FFT) and Maximum Entropy (MESA), accompanied by appropriate statistical tests, have been employed to detect periodicities, while the method of Successive Approximations (SA) is used independently in order to define the amplitude and the phase of each fluctuation. New short-term periodicities of 100, 70, 50 and 32 days, in addition to the known ones of 152, 27 and 14 days, appeared in cosmic ray data. During this particular time interval, similar spectral behaviour has been reported in the solar hard X-ray flares data. The influence of the solar hard X-ray flares variability in the energy range 50–500 keV, expressed by their fluence values, upon the cosmic-ray modulation, is discussed.     

Constraints on Titan’s topography through fractal analysis of shorelines

Titan’s north polar hydrocarbon lakes offer a unique opportunity to indirectly characterize the statistical properties of Titan’s landscape. The complexity of a shoreline can be related to the complexity of the landscape it is embedded in through fractal theory. We mapped the shorelines of 290 of the north polar titanian lakes in the Cassini synthetic aperture radar dataset. Out of these, we used a subset of 190 lake shorelines for our analysis. The fractal dimensions of the shorelines were calculated via two methods: the divider/ruler method and the box-counting method, at length scales of (1-10) km and found to average 1.27 and 1.32, respectively. The inferred power-spectral exponent of Titan’s topography (β) from theoretical and empirical relations is found to be ?2, which is lower than the values obtained from the global topography of the Earth or Venus. Some of the shorelines exhibit multi-fractal behavior (different fractal dimensions at different scales), which we interpret to signify a transition from one set of dominant surface processes to another. We did not observe any spatial variation in the fractal dimension with latitude; however we do report significant spatial variation of the fractal dimension with longitude. A systematic difference between the dimensions of orthogonal sections of lake shorelines is also noted, which signifies possible anisotropy in Titan’s topography. The topographic information thus gleaned can be used to constrain landscape evolution modeling to infer the dominant surface processes that sculpt the landscape of Titan.     

Spectral methods for analyzing the rotation of solar structures

The application of spectral analysis methods for studying the rotation of solar structures is considered. The time series characterizing the time variation of the solar He I 1083 nm emission in 5° latitude zones have been used. Three types of spectral analysis have been tested: the Welch method, the multitaper method, and the Schuster periodogram method. The first two methods have been chosen for the analysis of observing time intervals 26 and 3 years in length. The Schuster periodogram method is more suitable for the sliding spectral analysis in a 1-year-long temporal window with a shift by half a year. The chosen methods for analyzing the power spectra allow one to obtain the spectral densities, the powers of significant peaks in them and the corresponding periods, the total powers in the specified intervals of periods and to estimate the significance of the peaks found and the intervals in which the true periods corresponding to the peaks can be located.     

Center-limb observations of inhomogeneities in the solar atmosphere

Center-limb observations of line-center intensity and velocity fluctuations in the Magnesium b lines are described. Autocorrelation and power spectral analyses indicate small scale brightness structures having periodicities of 3000 km and 8000 km and large scale structures of 22000 km. Corresponding velocity structures are 6000 km and 30000 km.The relative rms fluctuation amplitude for the small scale bright features is found to be of order 12% and for the large scale features 8%. The variation of these rms values with heliocentric angle is also shown.At disk center some weak correlation is found between bright features and downward velocities in the large scale structures. Towards the limb there is a strong correlation in all three lines between line of sight motions and bright features. This indicates that the large scale bright features are closely associated with the supergranule motions.By inspecting the actual brightness and velocity fluctuation tracings it can be seen that, in some regions, the small scale structures show a significant negative correlation over a range of about 25000 km. Beyond this characteristic length, however, the correlation may decrease abruptly or even become positive for a similar distance. There is some evidence which suggests that this behaviour may also be related to the supergranule motions.