THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS

Fractals ◽  
1995 ◽  
Vol 03 (01) ◽  
pp. 217-229 ◽  
Author(s):  
FRANK B. TATOM
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Fractional Calculus ◽  
Fractal Dimensions ◽  
Linear Functions ◽  
Fractal Curve ◽  
Random Fractal ◽  
Fractal Functions ◽  
Decay Exponent ◽  
The Relationship

The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).

scholarly journals On Similarity of Seismo Magnetic Power Density and Pressure Head Fractal Dimension for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia

2020 ◽  
pp. 1-8
Author(s):  
Khalid Elyas Mohamed Elameen Alkhidir ◽  
Keyword(s):  
Fractal Dimension ◽  
Power Density ◽  
Functional Relationship ◽  
Fractal Dimensions ◽  
Pressure Head ◽  
Density Maximum ◽  
Phase Saturation ◽  
Head Pressure ◽  
Second Procedure ◽  
The Relationship

The quality and assessment of a reservoir can be documented in details by the application of seismo magnetic power density. This research aims to calculate fractal dimension from the relationship among seismo magnetic power density, maximum seismo magnetic power density and wetting phase saturation and to approve it by the fractal dimension derived from the relationship among inverse pressure head * pressure head and wetting phase saturation. Two equations for calculating the fractal dimensions have been employed. The first one describes the functional relationship between wetting phase saturation, seismo magnetic power density, maximum seismo magnetic power density and fractal dimension. The second equation implies to the wetting phase saturation as a function of pressure head and the fractal dimension. Two procedures for obtaining the fractal dimension have been utilized. The first procedure was done by plotting the logarithm of the ratio between seismo magnetic power density and maximum seismo magnetic power density versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was determined by plotting the logarithm (inverse of pressure head and pressure head) versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. On the basis of the obtained results of the fabricated stratigraphic column and the attained values of the fractal dimension, the sandstones of the Shajara reservoirs of the Shajara Formation were divided here into three units

scholarly journals Seismo Electric Transfer Function Fractal Dimension for Characterizing Shajara Reservoirs Of The Permo-Carboniferous Shajara Formation, Saudi Arabia

2018 ◽  
Vol 1 (1) ◽  
Keyword(s):  
Fractal Dimension ◽  
Transfer Function ◽  
Capillary Pressure ◽  
Fractal Dimensions ◽  
Pressure Profile ◽  
Wide Range ◽  
Phase Saturation ◽  
Second Procedure ◽  
Bottom To Top ◽  
The Relationship

The quality of a reservoir can be described in details by the application of seismo electric transfer function fractal dimension. The objective of this research is to calculate fractal dimension from the relationship among seismo electric transfer fuction, maximum seismo electric transfer function and wetting phase saturation and to confirm it by the fractal dimension derived from the relationship among capillary pressure and wetting phase saturation. In this research, porosity was measured on real collected sandstone samples and permeability was calculated theoretically from capillary pressure profile measured by mercury intrusion techniques. Two equations for calculating the fractal dimensions have been employed. The first one describes the functional relationship between wetting phase saturation, seismo electric transfer function, maximum seismo electric transfer function and fractal dimension. The second equation implies to the wetting phase saturation as a function of capillary pressure and the fractal dimension. Two procedures for obtaining the fractal dimension have been developed. The first procedure was done by plotting the logarithm of the ratio between seismo electric transfer function and maximum seismo electric transfer function versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was completed by plotting the logarithm of capillary pressure versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. On the basis of the obtained results of the constructed stratigraphic column and the acquired values of the fractal dimension, the sandstones of the Shajara reservoirs of the Shajara Formation were divided here into three units. The gained units from bottom to top are: Lower Shajara Seismo Electric Transfer Function Fractal Dimension Unit, Middle Shajara Seismo Electric Tranfser Function Fractal dimension Unit, and Upper Shajara Seismo Electric Transfer Function Fractal Dimension Unit. The results show similarity between seismo electric transfer tunction fractal dimension and capillary pressure fractal dimension. It was also noted that samples with wide range of pore radius were characterized by high values of fractal dimension due to an increase in their connectivity and seismo electric transfer function. In our case , and as conclusions the higher the fractal dimension, the higher the permeability, the better the shajara reservoir characteristics.

A Microcontact Model Developed for Asperity Heights With a Variable Profile Fractal Dimension, a Surface Fractal Dimension, Topothesy, and Non-Gaussian Distribution

2008 ◽  
Author(s):  
Jen Luen Liou ◽  
Jen Fin Lin
Keyword(s):  
Fractal Dimension ◽  
Cross Sections ◽  
Fractal Dimensions ◽  
Surface Fractal Dimension ◽  
Surface Fractal ◽  
Deformation Regime ◽  
Circular Contour ◽  
The Mean ◽  
Non Gaussian ◽  
The Relationship

The cross sections formed by the contact asperities of two rough surfaces at an interference are island-shaped, rather than having the commonly assumed circular contour. These island-shaped contact surface contours show fractal behavior with a profile fractal dimension Ds. The surface fractal dimension for the asperity heights is defined as D and the topothesy is defined as G. In the study of Mandelbrot, the relationship between D and Ds was given as D = Ds+1 if these two fractal dimensions are obtained before contact deformation. In the present study, D, G, and Ds are considered to be varying with the mean separation (or the interference at the rough surface) between two contact surfaces. The D-Ds relationships for the contacts at the elastic, elastoplastic, and fully plastic deformations are derived and the inceptions of the elastoplastic deformation regime and the fully plastic deformation regime are redefined using the equality of two expressions established in two different ways for the number of contact spots (N).

A NEW IRREGULARITY CRITERION FOR DISCRIMINATION OF STOCHASTIC AND DETERMINISTIC TIME SERIES

Fractals ◽  
2008 ◽  
Vol 16 (02) ◽  
pp. 129-140 ◽  
Author(s):  
AMIR H. OMIDVARNIA ◽  
ALI M. NASRABADI
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Probability Density Function ◽  
State Space ◽  
Probability Density ◽  
Density Function ◽  
Confidence Level ◽  
Space Versus ◽  
Fractal Dimensions ◽  
Linear Region

In this paper, a new irregularity criterion based on fractal dimensions is introduced. In fact, this criterion (which is called ONH criterion) is the state space version of Higuchi fractal dimension, which can discriminate stochastic and deterministic time series from each other. To compute this criterion, we have exploited "the amount of state vectors fluctuations" in embedding space. By varying the reconstruction delay in embedding space, one can obtain a logarithmic diagram of overall changes corresponding to reconstructed state space versus delays. The slope of the linear region of this diagram could be considered as a new irregularity criterion. To discriminate random and deterministic time series, this criterion is adopted as a new statistic for hypothesis testing. Probability density function of this statistic under H0 hypothesis is constructed and regarding to a certain confidence level, one can determine whether randomness is accepted or rejected.

Influences of carvedilol treatment on the effects of acetylcholine on regional haemodynamics in the spontaneously hypertensive rat

1998 ◽  
Vol 95 (3) ◽  
pp. 303-309 ◽  
Author(s):  
Sven-Olof GRANSTAM ◽  
Bengt FELLSTRÖM ◽  
Lars LIND
Keyword(s):  
Heart Rate ◽  
Time Series ◽  
Fractal Dimension ◽  
Spontaneously Hypertensive Rat ◽  
Fractal Dimensions ◽  
Approximate Entropy ◽  
Normal Subjects ◽  
Heart Period ◽  
Spontaneously Hypertensive ◽  
Rate Versus

1.Investigations that assess cardiac autonomic function include non-linear techniques such as fractal dimension and approximate entropy in addition to the common time and frequency domain measures of both heart period and heart rate. This article evaluates the differences in using heart rate versus heart period to estimate fractal dimensions and approximate entropies of these time series. 2.Twenty-four-hour ECG was recorded in 23 normal subjects using Holter records. Time series of heart rate and heart period were analysed using fractal dimensions, approximate entropies and spectral analysis for the quantification of absolute and relative heart period variability in bands of ultra low (< 0.0033 ;Hz), very low (0.0033–0.04 ;Hz), low (0.04–0.15 ;Hz) and high (0.15–0.5 ;Hz) frequency. 3.Linear detrending of the time series did not significantly change the fractal dimension or approximate entropy values. We found significant differences in the analyses using heart rate versus heart period between waking up and sleep conditions for fractal dimensions, approximate entropies and absolute spectral powers, especially for the power in the band of 0.0033–0.5 ;Hz. Log transformation of the data revealed identical fractal dimension values for both heart rate and heart period. Mean heart period correlated significantly better with fractal dimensions and approximate entropies of heart period than did corresponding heart rate measures. 4.Studies using heart period measures should take the effect of mean heart period into account even for the analyses of fractal dimension and approximate entropy. As the sleep–awake differences in fractal dimensions and approximate entropies are different between heart rate and heart period, the results should be interpreted accordingly.

Fractal dimension and approximate entropy of heart period and heart rate: awake versus sleep differences and methodological issues

1998 ◽  
Vol 95 (3) ◽  
pp. 295-301 ◽  
Author(s):  
Vikram K. YERAGANI ◽  
E. SOBOLEWSKI ◽  
V. C. JAMPALA ◽  
Jerald KAY ◽  
Suneetha YERAGANI ◽  
...  
Keyword(s):  
Heart Rate ◽  
Time Series ◽  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Approximate Entropy ◽  
Normal Subjects ◽  
Heart Period ◽  
Common Time ◽  
Rate Versus ◽  
Waking Up

1.Investigations that assess cardiac autonomic function include non-linear techniques such as fractal dimension and approximate entropy in addition to the common time and frequency domain measures of both heart period and heart rate. This article evaluates the differences in using heart rate versus heart period to estimate fractal dimensions and approximate entropies of these time series. 2.Twenty-four-hour ECG was recorded in 23 normal subjects using Holter records. Time series of heart rate and heart period were analysed using fractal dimensions, approximate entropies and spectral analysis for the quantification of absolute and relative heart period variability in bands of ultra low (< 0.0033 ;Hz), very low (0.0033–0.04 ;Hz), low (0.04–0.15 ;Hz) and high (0.15–0.5 ;Hz) frequency. 3.Linear detrending of the time series did not significantly change the fractal dimension or approximate entropy values. We found significant differences in the analyses using heart rate versus heart period between waking up and sleep conditions for fractal dimensions, approximate entropies and absolute spectral powers, especially for the power in the band of 0.0033–0.5 ;Hz. Log transformation of the data revealed identical fractal dimension values for both heart rate and heart period. Mean heart period correlated significantly better with fractal dimensions and approximate entropies of heart period than did corresponding heart rate measures. 4.Studies using heart period measures should take the effect of mean heart period into account even for the analyses of fractal dimension and approximate entropy. As the sleep–awake differences in fractal dimensions and approximate entropies are different between heart rate and heart period, the results should be interpreted accordingly.

scholarly journals On Similarity of Seismo Magentic Field and Pressure Head Fractal Dimension for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia

2020 ◽  
Vol 4 (1) ◽  
Keyword(s):  
Magnetic Field ◽  
Fractal Dimension ◽  
Functional Relationship ◽  
Fractal Dimensions ◽  
Pressure Head ◽  
Field Versus ◽  
Phase Saturation ◽  
Head Pressure ◽  
Second Procedure ◽  
The Relationship

The quality and assessment of a reservoir can be documented in details by the application of seismo magnetic field. This research aims to calculate fractal dimension from the relationship among seismo magnetic field, maximum seismo magnetic field and wetting phase saturation and to approve it by the fractal dimension derived from the relationship among inverse pressure head * pressure head and wetting phase saturation. Two equations for calculating the fractal dimensions have been employed. The first one describes the functional relationship between wetting phase saturation, seismo magnetic field, maximum seismo magnetic field and fractal dimension. The second equation implies to the wetting phase saturation as a function of pressure head and the fractal dimension. Two procedures for obtaining the fractal dimension have been utilized. The first procedure was done by plotting the logarithm of the ratio between seismo magnetic field and maximum seismo magnetic field versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was determined by plotting the logarithm (inverse of pressure head and pressure head) versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. On the basis of the obtained results of the fabricated stratigraphic column and the attained values of the fractal dimension, the sandstones of the Shajara reservoirs of the Shajara Formation were divided here into three units.

scholarly journals Thermo Electric Sensitivity Fractal Dimension for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia

2019 ◽  
Vol 2 (3) ◽  
Keyword(s):  
Fractal Dimension ◽  
Capillary Pressure ◽  
Functional Relationship ◽  
Fractal Dimensions ◽  
Thermo Electric ◽  
Sensitivity Maximum ◽  
Phase Saturation ◽  
Second Procedure ◽  
The Relationship ◽  
Electric Sensitivity

The quality and assessment of a reservoir can be documented in details by the application of thermo electric sensitivity. This research aims to calculate fractal dimension from the relationship among thermo electric sensitivity, maximum thermo electric sensitivity and wetting phase saturation and to approve it by the fractal dimension derived from the relationship among capillary pressure and wetting phase saturation. Two equations for calculating the fractal dimensions have been employed. The first one describes the functional relationship between wetting phase saturation, thermo electric sensitivity, maximum Thermo electric sensitivity and fractal dimension. The second equation implies to the wetting phase saturation as a function of capillary pressure and the fractal dimension. Two procedures for obtaining the fractal dimension have been utilized. The first procedure was done by plotting the logarithm of the ratio between thermo electric sensitivity and maximum thermo electric sensitivity versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was determined by plotting the logarithm of capillary pressure versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. On the basis of the obtained results of the fabricated stratigraphic column and the attained values of the fractal dimension, the sandstones of the Shajara reservoirs of the Shajara Formation were divided here into three units.

INFLUENCE OF SAMPLING LENGTH AND SAMPLING INTERVAL ON CALCULATING THE FRACTAL DIMENSION OF CHAOTIC ATTRACTORS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250145
Author(s):  
CUICUI JI ◽  
HUA ZHU ◽  
WEI JIANG
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Sampling Interval ◽  
Computational Time ◽  
Chaotic Attractors ◽  
Computation Efficiency ◽  
Univariate Time Series ◽  
Saturation Method ◽  
Different Chaotic Systems

This paper intends to study the influences of the sampling length and sampling interval of time series on the chaotic attractors' fractal dimension calculation. Four kinds of univariate time-series signals from different chaotic systems were chosen, and then fractal dimensions of attractors under different sampling lengths and sampling intervals were calculated by the method of correlation dimension. The results show clearly that the chaotic attractors' fractal dimension is related to both the sampling length and the sampling interval. With the increase of the sampling length, all attractors' fractal dimensions tend to increase gradually first and then become stable. However, the fractal dimension remains stable only in a suitable range of the sampling interval, in which the attractor of the chaotic system can be reconstructed from one univariate time-series signal; if the sampling interval is unusually large or small, the fractal dimension will be unstable and the reconstructed attractor will be seriously distorted. Therefore, the dimension saturation method and the delay-coordinate's time difference method for determining the sampling length and the sampling interval were proposed separately, which are significant for improving the calculation accuracy for the chaotic attractor's dimension, reflecting the dynamics of complicated systems correctly, saving computational time as well as enhancing the computation efficiency.

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