Effect of shear-induced contact area and aperture variations on nonlinear flow behaviors in fractal rock fractures

This study experimentally analyzes the nonlinear flow characteristics and channelization of fluid through rough-walled fractures during the shear process using a shear-flow-visualization apparatus.A series of fluid flow and visualization tests is performed on four transparent fracture specimens with various shear displacements of 1 mm,3 mm,5 mm,7 mm and 10 mm under a normal stress of 0.5 MPa.Four granite fractures with different roughnesses are selected and quantified using variogram fractal dimensions.The obtained results show that the critical Reynolds number tends to increase with increasing shear displacement but decrease with increasing roughness of fracture surface.The flow paths are more tortuous at the beginning of shear because of the wide distribution of small contact spots.As the shear displacement continues to increase,preferential flow paths are more distinctly observed due to the decrease in the number of contact spots caused by shear dilation;yet the area of single contacts in-creases.Based on the experimental results,an empirical mathematical equation is proposed to quantify the critical Reynolds number using the contact area ratio and fractal dimension.

The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

Numerical study of electromagnetic scattering from one-dimensional nonlinear fractal sea surface

In recent years, linear fractal sea surface models have been developed for the sea surface in order to establish an electromagnetic backscattering model. Unfortunately, the sea surface is always nonlinear, particularly at high sea states. We present a nonlinear fractal sea surface model and derive an electromagnetic backscattering model. Using this model, we numerically calculate the normalized radar cross section （NRCS） of a nonlinear sea surface. Comparing the averaged NRCS between linear and nonlinear fractal models, we show that the NRCS of a linear fractal sea surface underestimates the NRCS of the real sea surface, especially for sea states with high fractal dimensions, and for dominant ocean surface gravity waves that are either very short or extremely long.

A two scale nonlinear fractal sea surface model in a one dimensional deep sea

Using the theory of nonlinear interactions between long and short waves, a nonlinear fractal sea surface model is presented for a one dimensional deep sea. Numerical simulation results show that spectra intensity changes at different locations （in both the wave number domain and temporal-frequency domain）, and the system obeys the energy conservation principle. Finally, a method to limit the fractal parameters is also presented to ensure that the model system does not become ill-posed,

Hausdorff Dimension and Fractal Dimension of the Global Attractor for the Higher-Order Coupled Kirchhoff-Type Equations

This paper mainly deals with the higher-order coupled Kirchhoff-type equations with nonlinear strong damped and source terms in a bounded domain. We obtain some results that are estimation of the upper bounds of Hausdorff dimension and Fractal dimension of the global attractor.

SUBCOOLED BOILING REGIME RECOGNITION BASED ON FRACTAL CHARACTERISTIC QUANTITIES OF SOUND PRES-SURE

Based on acquisition of sound pressure in subcooled boiling twophase system and through dynamic data processing methods, the dynamical behavior of the system is discussed. With the introduction of fractal concept, an analysis to the fractal feature of sound pressure signal is carried out. Moreover, the pseudo phase diagrams of typical time series of sound pressure are given. Finally, through dynamic clustering and on the basis of calculating correlation dimension and Hurst exponent of sound time series on different subcooling conditions, the recognition of developing regime of the twophase system is delivered, which provides a new practical approach of recognition and diagnosis for vaporliquid boiling system.

Calculating the Exact Experimental Density of the Dark Energy in the Cosmos Assuming a Fractal Speed of Light

Fractal speed of light theory is a variation of Magueijo-Smolin varying speed of light (VSL) theoretical modification of Einstein’s energy mass relation. We use this theory to derive an exact value for the missing dark energy which is found to be in astonishing agreement with the latest result of the WMAP measurement and the independent supernova analysis. Thus while Einstein’s formula predicts 95.5% more energy than found in highly precise astrophysical measurement, our VSL- based calculation indicates an exact theoretical value of only 4.508497% real energy. Consequently, the exact conjectured missing dark energy must be 95.491502%. By any standards, this is an astounding confirmation for both the cosmological measurement and the VSL theory.

Application of fractal characteristic quantities of pressure fluctuation in subcooled boiling regime recognition

The dynamical behavior of the subcooled boiling two-phase system was introduced and discussed. With the introduction of fractal concept, an analysis of the fractal feature of pressure wave signals from nonlinear dynamics point of view, was carried out. Meanwhile, the pseudo phase diagrams of typical time series of sound pressure were given. Finally, through dynamic clustering and on the basis of calculating correlation dimension and Hurst exponent of pressure wave time series on different subcooling conditions, the recognition of developing regime of the two-phase system was delivered, which might provide a promising approach of recognition and diagnosis of a boiling system.

Insights into transferal to fractal space modeling:delayed forced Helmholtz-Duffing oscillator with the non-perturbative approach

The damped Helmholtz-Duffing oscillator is a topic of great interest in many different fields of study due to its complex dynamics.By transitioning from conventional continuous differential equations to their fractal counterparts,one gains insights into the system's response under new mathematical frameworks.This paper presents a novel method for converting standard continuous differential equations into their fractal equivalents.This conversion occurs after the nonlinear system is transformed into its linear equivalent.Numerical analyses show that there are several resonance sites in the fractal system,which differ from the one resonance point found in the continuous system.One important finding is that the fractal system loses some of its stabilizing power when decaying behavior is transformed into a diffuse pattern.Interestingly,a decrease in the fractal order in resonance settings shows a stabilizing impact,highlighting the dynamics'complexity inside fractal systems.This endeavor to convert to fractals is a revolutionary technique that is being employed for the first time.

CONTROLLING EROSION OF SAFE BASIN IN NONLINEAR PARAMETRICALLY EXCITED SYSTEMS

This paper considers the dynamical behavior of a Duffing-Mathieu type system with a cubic single-well potential during the principal parametric resonance. Both the cases of constant and time-dependent excitation amplitude are used to observe the variation of the extent and the rate of the erosion in safe basins. It is evident that the appearance of fractal basin boundaries heralds the onset of the losing of structural integrity. The minimum value of control parameter to prevent the basin from erosion is given along with the excitation amplitude varying. The results show the time-dependence of excitation amplitude can be used to control the extent and the rate of the erosion and delay the first occurrence of heteroclinic tangency.

Research on nonlinear R／S method and its application in earthquake prediction

ＲｅｓｅａｒｃｈｏｎｎｏｎｌｉｎｅａｒＲ／ＳｍｅｔｈｏｄａｎｄｉｔｓｐｐｌｉｃａｔｉｏｎｉｎｅａｒｔｈｑｕａｋｅｐｒｅｄｉｃｔｉｏｎＢｉ－ＱｕａｎＷＡＮＧ（王碧泉）；Ｈａｎ－ＭｉｎｇＨＵＡＮＧ（黄汉明）；Ｈｏｎｇ－ＳｈｕｎＦＡＮ（范洪顺）；Ｃｈｕｅｎ...

Long memory and nonlinear dependence structure in crude oil futures returns and volatility

In order to investigate the nature of international crude oil futures and present evidence of long memory and nonlinear dependence for crude oil futures volatility as well as returns, a certain number of recent statistical tests, such as the powerful BDS test, the fractional integration test and other known statistics, are applied. The results show that though the returns themselves contain little serial correlation, the market volatility series have significant long-term dependence structures which may have important implications for volatility forecasts and derivative pricing. On the other hand, evidence of strong ARCH effect is also presented, and, moreover, the BDS statistics on the standardized residuals of the fitted GARCH model indicate that the ARCH-type process may generally explain the nonlinearities in the data. It seems that the crude oil futures market can be appropriately modeled by ARCH and fractal processes. These findings indicate that it would be beneficial to assess the behavior of the crude oil and price the oil derivative contracts by encompassing long memory and nonlinear structure.

USING NEURAL NETWORK BASED NONLINEAR IFS TO MODEL TIME SEQUENCES

A novel neural network based iterated function system (IFS) model is presented in this paper while the precondition to ensure the model is also explored. Applying it to some practical data, the given signal can be approximated exactly by the attractor generated by this model, which provides another way to resolve fractal inverse problem.

Nonlinear magnitude frequency relation and two types of seismicity systems

nonlinear magnitude frequency equation has been derived in this paper on the assumption that all seismicity systems hold fractal characteristics, and according to the differences of relevant coefficients in the equation, seis-micity systems are classified into two types: type I, the whole earthquake activity is controlled by only one great unified system; type II, the whole earthquake activity is controlled by more than one great system. One type of seismicity system may convert to the other type, generally. For example, a type I system will change to a type II system prior to the occurrence of a strong earthquake in North China. This change can be regarded as an index for earthquake trend estimation. In addition, the difference between b value in nonlinear magnitude frequency equation and that in linear equation and the term dΔM related to the coefficients of nonlinear terms obtained in this paper are proved to be a pair of available parameters for medium short term earthquake prediction.

An innovative technique to solve a fractal damping Duffing-jerk oscillator

The idea of the present article is to look into the nonlinear dynamics and vibration of a damping Duffing-jerk oscillator in fractal space exhibiting the non-perturbative approach.Using a new analytical technique,namely,the modification of a He’s fractal derivative that converts the fractal derivative to the traditional derivative in continuous space,this study provides an effective and easy-to-apply procedure that is dependent on the He’s fractal derivative approach.The analytic approximate solution has a significant match with the results of the numerical simulation as the fractal parameter is very closer to unity,which proves the reliability of the method.Stability behavior is discussed and illustrated graphically.These new powerful analytical tools are developed in an attempt to obtain effective analytical tools,valid for any fractal nonlinear problems.

INVESTIGATION OF THE FLOW IN FRACTAL RESERVOIR INCLUDING THE EFFECTS OF QUADRATIC GRADIENT TERM

According to the assumption of slightly compressible fluid, the quadraticgradient term in the nonlinear partial differential equations for the traditional well-test model isusually neglected. The linear partial differential equation is thus established. It is known thatneglecting the quadratic gradient term results in errors for long-time well tests. A nonlinear flowmodel for fractal medium is constructed and the quadratic gradient term is considered. The exactsolutions of the fractal reservoir models are obtained by Laplace transform and Weber transform in aconstant-rate and constant-pressure production for an infinitely large system. This paper addressesthe variation of pressure with fluid compressibility coefficient and fractal reservoir parameters.The plots of the typical pressure curves are constructed, and the results can be applied towell-test analysis.

Fractal Features and Computer Simulation of Fan-Shaped Sedimentary Bodies in Faulted Basins

Ｆａｎ－ｓｈａｐｅｄｓｅｄｉｍｅｎｔａｒｙｂｏｄｉｅｓａｒｅｔｈｏｓｅｍａｉｎｌｙｃｏｍ－ｐｏｓｅｄｏｆｔｅｒｉｇｅｎｏｕｓｃｏａｒｓｅｄｅｐｏｓｉｔｓａｎｄｂｅｉｎｇｆａｎ－ｓｈａｐｅｄｉｎｍｏｒｐｈｏｌｏｇｙ，ｉｎｃｌｕｄｉｎｇａｌｕｖｉａｌｆａ...

Fractal Dimension of Pore Structure of Combustible Cartridge Cases

A fractal pore structure model of combustible cartridge cases was established by virtue of the fractal geometry. Pore structure information, such as backbone fractal dimension and pore fractal dimension, of four kinds of combustible cartridge case were obtained by mercury intrusion porosimetry (MIP) . The formation mechanism of fractal pore structure of combustible cartridge was studied. The results show that the backbone fractal dimension consists of the component and influenced by the component number and size of components; the pore percolation fractal dimension reflects the pore structures of components; and the fractal dimension of pore structure is positively relative to the tensile strength of combustible cartridge case.

RESEARCH PROGRESS IN NONLINEAR ANALYSIS OF HEART ELECTRIC ACTIVITIES

Nonlinear science research is a hot point in the world. It has deepened our cognition of determinism and randomicity, simplicity and com-plexity, noise and order and it will profoundly influ-ence the progress of the study of natural science, including life science. Life is the most complex nonlinear system and heart is the core of lifecycle system. In the late more than 20 years, nonlinear research on heart electric activities has made much headway. The commonly used parameters are based on chaos and fractal theory, such as correlation dimension, Lyapunov ex-ponent, Kolmogorov entropy and multifractal singu-larity spectrum. This paper summarizes the commonly used methods in the nonlinear study of heart electric signal. Then, considering the shortages of the above tradi-tional nonlinear parameters, we mainly introduce the results on short-term heart rate variability (HRV) signal (500 R-R intervals) and HFECG signal (1-2s). Finally, we point out it is worthwhile to put emphasis on the study of the sensitive nonlinearity parameters of short-term heart electric signal and their dynamic character and clinical effectivity.

分形激光精密控制研究综述

介绍了单通光纤中的孤子分形、矢量孤子碰撞中的分形、自旋波孤子中的分形、呼吸子激光中的分形等激光器中存在的典型分形结构及相关调控方法,阐述了锁模激光器中分形激光频谱密度高、调谐精度高、能够实现高分辨力飞秒光纤光梳等特点。探讨了提升飞秒光纤光梳分辨力的问题,指出利用呼吸子激光中的魔鬼阶梯和法里树分形结构,结合智能调谐手段可提升飞秒光纤光梳的探测分辨力。展望了呼吸子分形光梳在分子光谱、任意光波合成等领域的发展方向,指出未来通过发展光频域高精度锁定技术、时频相位全息跟踪探测技术、分形频谱关联与相噪分析技术,并优化理论模型,有望进一步抑制光梳的噪声,并突破谱密度极限。