scholarly journals FRACTAL FLUCTUATIONS AND STATISTICAL NORMAL DISTRIBUTION

Fractals ◽  
2009 ◽  
Vol 17 (03) ◽  
pp. 333-349 ◽  
Author(s):  
A. M. SELVAM
Keyword(s):  
Power Law ◽  
Power Spectra ◽  
Space Time ◽  
Inverse Power ◽  
Data Sets ◽  
Power Law Distribution ◽  
Self Organized ◽  
Inverse Power Law ◽  
Self Organized Criticality ◽  
Fractal Fluctuations

Dynamical systems in nature exhibit self-similar fractal fluctuations and the corresponding power spectra follow inverse power law form signifying long-range space-time correlations identified as self-organized criticality. The physics of self-organized criticality is not yet identified. The Gaussian probability distribution used widely for analysis and description of large data sets underestimates the probabilities of occurrence of extreme events such as stock market crashes, earthquakes, heavy rainfall, etc. The assumptions underlying the normal distribution such as fixed mean and standard deviation, independence of data, are not valid for real world fractal data sets exhibiting a scale-free power law distribution with fat tails. A general systems theory for fractals visualizes the emergence of successively larger scale fluctuations to result from the space-time integration of enclosed smaller scale fluctuations. The model predicts a universal inverse power law incorporating the golden mean for fractal fluctuations and for the corresponding power spectra, i.e., the variance spectrum represents the probabilities, a signature of quantum systems. Fractal fluctuations therefore exhibit quantum-like chaos. The model predicted inverse power law is very close to the Gaussian distribution for small-scale fluctuations, but exhibits a fat long tail for large-scale fluctuations. Extensive data sets of Dow Jones index, human DNA, Takifugu rubripes (Puffer fish) DNA are analyzed to show that the space/time data sets are close to the model predicted power law distribution.

scholarly journals Universal spectrum for short period (days) variability in atmospheric total ozone

MAUSAM ◽  
2022 ◽  
Vol 46 (3) ◽  
pp. 297-302
Author(s):  
A. M. SELVAM ◽  
M. RADHAMANI
Keyword(s):  
Energy Spectrum ◽  
Power Law ◽  
Fractal Geometry ◽  
Total Ozone ◽  
Inverse Power ◽  
Atmospheric Flows ◽  
Temporal Correlations ◽  
Self Organized ◽  
Inverse Power Law ◽  
Self Organized Criticality

  Long-range spatio-temporal correlations manifested as the self-similar fractal geometry to the spatial pattern concomitant with inverse power law form for the power spectrum of temporal fluctuations are ubiquitous to real world dynamical systems and are recently identified as signatures of self-organized criticality Self-organised criticality in atmospheric flows is exhibited as the fractal geometry 10 the global cloud cover pattern and the inverse power law form for the atmospheric eddy energy spectrum, In this paper, a recently developed cell dynamical system model for  atmospheric flows is summarized. The model predicts inverse power law form of the statistical normal distribution for atmospheric eddy energy spectrum as a natural consequence of quantum-like mechanics governing atmospheric flows extending up to stratospheric levels and above, Model Predictions are in agreement with continuous periodogram analyses of atmospheric total ozone. Atmospheric total ozone variability (in days) exhibits the temporal signature of self-organized criticality, namely, inverse power law form for the power spectrum. Further, the long-range temporal correlations implicit to self-organized criticality can be quantified in terms of the universal characteristics  of the normal distribution. Therefore the total pattern of fluctuations of total ozone over a period of time is predictable.  

SELF-ORGANIZED CRITICALITY MODEL FOR EARTHQUAKES: QUIESCENCE, FORESHOCKS AND AFTERSHOCKS

1999 ◽  
Vol 09 (12) ◽  
pp. 2249-2255 ◽  
Author(s):  
S. HAINZL ◽  
G. ZÖLLER ◽  
J. KURTHS
Keyword(s):  
Power Law ◽  
B Value ◽  
Power Law Distribution ◽  
Aftershock Activity ◽  
Omori Law ◽  
Self Organized ◽  
Spatiotemporal Clustering ◽  
Self Organized Criticality ◽  
Clustering Of Events ◽  
Foreshock Activity

We introduce a crust relaxation process in a continuous cellular automaton version of the Burridge–Knopoff model. Analogously to the original model, our model displays a robust power law distribution of event sizes (Gutenberg–Richter law). The principal new result obtained with our model is the spatiotemporal clustering of events exhibiting several characteristics of earthquakes in nature. Large events are accompanied by a precursory quiescence and by localized fore- and aftershocks. The increase of foreshock activity as well as the decrease of aftershock activity follows a power law (Omori law) with similar exponents p and q. All empirically observed power law exponents, the Richter B-value, p and q and their variability can be reproduced simultaneously by our model, which depends mainly on the level of conservation and the relaxation time.

scholarly journals Interplanetary Scintillation Power Spectra

1976 ◽  
Vol 29 (3) ◽  
pp. 201 ◽  
Author(s):  
RG Milne
Keyword(s):  
Power Spectrum ◽  
Power Law ◽  
Scattering Theory ◽  
Power Spectra ◽  
Inverse Power ◽  
Radio Sources ◽  
Interplanetary Scintillation ◽  
Inverse Power Law ◽  
Power Law Index ◽  
Strong Scattering

Power spectrum measurements of interplanetary scintillation at 408 MHz show that an inverse power law spectrum provides the best description for all scintillating radio sources. The inverse power law index is reasonably constant at ~ 2�4 for solar elongation angles 8 > 10�, and this agrees well with spacecraft observations. For 8 < 10� the index apparently decreases with decreasing 8, and this appears to be consistent with recent strong scattering theory. A Bessel analysis attempted in order to detect Fresnel structure proved unsuccessful because of noise on the power spectra.

scholarly journals Transition from Self-Organized Criticality into Self-Organization during Sliding Si3N4 Balls against Nanocrystalline Diamond Films

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1055
Author(s):  
Bogatov ◽  
Podgursky ◽  
Vagiström ◽  
Yashin ◽  
Shaikh ◽  
...  
Keyword(s):  
Friction Force ◽  
Power Law ◽  
Early Stage ◽  
Nanocrystalline Diamond ◽  
The Self ◽  
Self Organization ◽  
Power Law Distribution ◽  
Reciprocating Sliding ◽  
Self Organized ◽  
Self Organized Criticality

The paper investigates the variation of friction force (Fx) during reciprocating sliding tests on nanocrystalline diamond (NCD) films. The analysis of the friction behavior during the run-in period is the focus of the study. The NCD films were grown using microwave plasma-enhanced chemical vapor deposition (MW-PECVD) on single-crystalline diamond SCD(110) substrates. Reciprocating sliding tests were conducted under 500 and 2000 g of normal load using Si3N4 balls as a counter body. The friction force permanently varies during the test, namely Fx value can locally increase or decrease in each cycle of sliding. The distribution of friction force drops (dFx) was extracted from the experimental data using a specially developed program. The analysis revealed a power-law distribution f-µ of dFx for the early stage of the run-in with the exponent value (µ) in the range from 0.6 to 2.9. In addition, the frequency power spectrum of Fx time series follows power-law distribution f-α with α value in the range of 1.0–2.0, with the highest values (1.6–2.0) for the initial stage of the run-in. No power-law distribution of dFx was found for the later stage of the run-in and the steady-state periods of sliding with the exception for periods where a relatively extended decrease of coefficient of friction (COF) was observed. The asperity interlocking leads to the stick-slip like sliding at the early stage of the run-in. This tribological behavior can be related to the self-organized criticality (SOC). The emergence of dissipative structures at the later stages of the run-in, namely the formation of ripples, carbonaceous tribolayer, etc., can be associated with the self-organization (SO).

Self-Organized Critical Models

2003 ◽  
Author(s):  
M. E. J. Newman ◽  
R. G. Palmer
Keyword(s):  
Computer Simulations ◽  
Power Law ◽  
Large Scale ◽  
Fitness Landscape ◽  
Fitness Landscapes ◽  
Power Law Distribution ◽  
Edge Of Chaos ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Phenotype Space

The models discussed in the last chapter are intriguing, but present a number of problems. In particular, most of the results about them come from computer simulations, and little is known analytically about their properties. Results such as the power-law distribution of extinction sizes and the system's evolution to the "edge of chaos" are only as accurate as the simulations in which they are observed. Moreover, it is not even clear what the mechanisms responsible for these results are, beyond the rather general arguments that we have already given. In order to address these shortcomings, Bak and Sneppen (1993; Sneppen et al. 1995; Sneppen 1995; Bak 1996) have taken Kauffman's ideas, with some modification, and used them to create a considerably simpler model of large-scale coevolution which also shows a power-law distribution of avalanche sizes and which is simple enough that its properties can, to some extent, be understood analytically. Although the model does not directly address the question of extinction, a number of authors have interpreted it, using arguments similar to those of section 1.2.2.5, as a possible model for extinction by biotic causes. The Bak-Sneppen model is one of a class of models that show "self-organized criticality," which means that regardless of the state in which they start, they always tune themselves to a critical point of the type discussed in section 2.4, where power-law behavior is seen. We describe self-organized criticality in more detail in section 3.2. First, however, we describe the Bak-Sneppen model itself. In the model of Bak and Sneppen there are no explicit fitness landscapes, as there are in NK models. Instead the model attempts to mimic the effects of landscapes in terms of "fitness barriers." Consider figure 3.1, which is a toy representation of a fitness landscape in which there is only one dimension in the genotype (or phenotype) space. If the mutation rate is low compared with the time scale on which selection takes place (as Kauffman assumed), then a population will spend most of its time localized around a peak in the landscape (labeled P in the figure).

Fractal properties in fetal breathing dynamics

1992 ◽  
Vol 263 (1) ◽  
pp. R141-R147 ◽  
Author(s):  
H. H. Szeto ◽  
P. Y. Cheng ◽  
J. A. Decena ◽  
Y. Cheng ◽  
D. L. Wu ◽  
...  
Keyword(s):  
Time Series ◽  
Power Law ◽  
Power Spectra ◽  
Breathing Rate ◽  
Power Law Distribution ◽  
Inverse Power Law ◽  
Beta Power ◽  
Fetal Breathing ◽  
Log Normal ◽  
Log Normal Distribution

The dynamic pattern of fetal breathing was studied in 17 fetal lambs with chronically implanted electromyographic electrodes in the diaphragm. The instantaneous breathing rate time series appeared similar on different time scales, with clusters of faster breathing rates interspersed with periods of relative quiescience, suggesting self-similarity. Distribution histograms of the interbreath intervals (IBIs) showed log-normal distribution for IBIs less than 1 s and inverse power-law distribution for IBIs greater than 1 s. The ratio of log-normal distribution to power-law distribution varied from approximately 2 at 102 days to approximately 30 by 130 days of gestation. Fast Fourier transform of the breathing rate time series revealed 1/f beta power spectra for all animals, with beta increasing linearly from 0.43 to 0.88 between 102 and 139 days. Studies in the newborn lamb showed further maturation in both the distribution characteristics of the IBIs, as well as in the 1/f power spectra, with beta approaching 1.0 at 2 days after birth. The inverse power-law relationship in the distribution of the IBIs, together with the 1/f beta power spectra, indicate scale invariance and suggest that fractal mechanisms are involved in the regulation of fetal breathing.

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