Fractals

We investigate how various linear and nonlinear filters affect the scaling properties of long-range power-law multivariate synthetic series quantified by multivariate multifractal detrended fluctuation analysis (MV-MFDFA). We consider four types of transforms which are often encountered in physical and physiological processes: linear, nonlinear polynomial, logarithmic and power-law filters. The effect of filters is analyzed by numerical simulation of synthetic series generated by ARFIMA process and binomial multifractal model. The representation of auto-correlation properties of synthetic series before and after the transforms is illustrated by 3D Hurst surface, and the difference of effect is quantified by the proposed generalized mean distance. We find that the linear filters do not change the scaling properties of both synthetic series, while the effect of nonlinear polynomial is correlated with the power of the polynomial filter. For logarithmic and exponential filter, the scaling behavior is not affected for some values of the parameters.

Based on the theories of multivariate multiscale dispersion entropy, we propose an improved new model — dual-embedded dimensional multivariate multiscale dispersion entropy (mvMDE), and generalize the new model to fractional order (GmvMDE). The mvMDE and GmvMDE simultaneously consider the cross-correlation between multiple channels, and they provide a dynamic complexity measure for measuring the multivariable data observed in one system. We introduce dual embedded dimensions to construct subsequences, which makes the model more flexible and applicable to a variety of application scenarios. Through empirical analysis of simulated data and stock market data, this paper verifies that both mvMDE and GmvMDE can effectively measure and distinguish the sequence complexity, and GmvMDE can better capture the small evolution of sequences, with higher stability and accuracy. On the basis of mvMDE research, we also propose the multiscale system dispersion entropy (MSDE) and its fractional order form (GMSDE). Using the vector-to-vector distance quantization method, the relationship between individuals in the system can be better extracted, and the systemic complexity of financial portfolio can be measured from the time dimension and the space dimension simultaneously. The empirical analysis based on the data of major industries in Chinese stock market shows that MSDE and GMSDE can accurately capture the structural information in the system and effectively distinguish the complexity of financial portfolios.

In this paper, we study the solution theory of the nonlinear $q$-fractional differential equation of Caputo type ${}^c{D}_q^{\alpha }y(t)=f(t,y(t))$ with given initial values $D_q^{k}y(0_{+})=d_k,k=0,1,\dots ,n-1$ where $\alpha >0$ is the order, $n=[\alpha ]$ and $0<q<1$ is the scale index. For $0\le \beta <\alpha -n+1$, by assuming that function $t^{\beta }f(t,y)$ is bounded and satisfies the Lipschitz condition on variable $y$, we prove that this problem admits a unique solution in the $q$-integrable function space ${{𝒟}}_q^{(n)}(0,b)$ and this solution is absolutely stable in the $L_{\infty }$-norm. This unique existence condition allows that $f(t,y)$ is singular at $t=0$ and discontinuous for $t\in (0,b]$. Finally, a successive approximation method is presented to find out the analytic approximation solution of this problem.

In this paper, a new notion of super coalescence hidden-variable fractal interpolation function (SCHFIF) is introduced. The construction of SCHFIF involves choosing an IFS from a pool of several non-diagonal IFS at each level of iteration. Further, the integral of a SCHFIF is studied and shown to be a SCHFIF passing through a different set of interpolation data.

A model of particle-cluster aggregation on the rewired small-world network is studied. In this model, particles can adjoin to the end of the rewired long-range edges. The rewired long-range edge is controlled by two parameters, the reconnecting probability and distance controlling parameter. The simulation results indicate that, with the increasing of the length and the number of rewired long-range edges, the patterns become diffuse and the fractal dimensions of the aggregates decrease. The range of the value of dimension and the feature of patterns obtained in the model are consistent with the amyloid deposits.

The pulsating airflow through human bronchial tree is of great significance for understanding its function and morphology. Fractal theory and numerical simulation are applied in this paper to study the global and local flow characteristics in the bronchial tree under unstable conditions. First, the pulsating flow impedance of fractal bronchial tree is derived, and the structure of bronchial tree is optimized by minimizing flow impedance. It has been shown that the optimal structure depends on the physical law governing the airflow. The optimized diameter ratio between parent and daughter branches for pulsating flow is different from Murray’s law, and the fractal dimension for branch diameter lies in 2 and 3. Afterwards, the local pulsating flow field by three-dimensional (3D) numerical simulation on a symmetrical bronchial model is compared with the global flow characteristics by fractal analysis. The numerical results show that asymmetrical airflow characteristics can be found at high Reynolds number, and the velocity distribution of the main bronchus is more irregular and the turbulence phenomenon is more evident. This work can help to understand the association between function and structure of the bronchial tree, and it may shed light on the physical mechanisms and drugs targeting of pulmonary disease.

In this paper, we study infinite families of orthogonal exponentials of some self-affine measures. The digit set

Fractal Dimension (FD) estimation in digital image analysis has received much attention due to its dimensional significance and therefore has become an active area of research over the year. The earlier FD-based techniques often followed traditional box-counting and its different variation of differential box-counting (DBC) paradigms, in which the proper choice of box count has remained a major concern. However, most of the state-of-the-art DBC variants suffer from considerable limitations like over-counting (OC), under-counting (UC), and limited their application only to square-shaped images, and it is still a major research problem! In this backdrop, the current investigation proposes a generalized box-counting (graylevel invariant DBC); and compares it with other state-of-the-art techniques. The proposed model is evaluated on five benchmark texture datasets (which include real and generated synthetic images) and obtained better results than the existing methods and achieved all desired outcomes by eliminating both OC and UC problems. This algorithm works for any arbitrarily sized (both squared and rectangular) images. It gives a higher rate of accuracy in terms of less fitting error in detecting exact surface roughness from given datasets.

In this study, we examine and using Laplace Transform as a way to find approximate solutions to multi-dimensional space and time fractional order problems and propose numerical algorithm for solving ($n+1$)-dimensional time and space fractional order heat like and time and space fractional wave-like equations. This method is a combination of Laplace Transform and iterative method. The fractional derivative is described in the Caputo sense. The results obtained by proposed scheme are compared with different other schemes. This scheme is found to be very efficient and effective for linear and nonlinear time and space fractional order problems.

Uncertain fractional differential equation driven by Liu process plays a significant role in depicting the memory effects of uncertain dynamical systems. This paper mainly investigates the stability problems for the Caputo type of uncertain fractional differential equations with the order $0<p\le 1$. The concept of stability in measure of solutions to uncertain fractional differential equation is proposed based on uncertainty theory. Several sufficient conditions for ensuring the stability of the solutions are derived, respectively, in which the systems are divided into two cases with order $\frac{1}{2}<p\le 1$ and $0<p\le \frac{1}{2}$. Some illustrative examples are performed to display the effectiveness of the proposed results.

In this paper, we define partial conformable fractional derivative (PCFD) which is based on the fractional derivative definition proposed by Abdeljawad. At the same time, we present some simple properties about the definition, the properties are useful for us when we carry out further research. In addition, taking advantage of Hölder’s inequality, we establish Opial-type inequalities for the partial conformable fractional (PCF) integral. As an application, the uniqueness of the partial system with initial value is proved.

In this paper, the Burgers–Benjamin–Bona–Mahony (Burgers–BBM) equation is described according to the use of the fractal derivative. The variational principle of the fractal Burgers–BBM equation is successfully established by adopting the fractal semi-inverse method, which is very helpful to research the structures of fractal solution for the solitary waves and construct the conservation law. Based on the proposed variational theory, a novel approximate method is successfully proposed for the fractal Burgers–BBM equation, which is called Fractal Variational Iteration Transform Method (FVITM). The example illustrates the FVITM is an excellent and fascinating tool for fractal models.

In this study, two different fractional models are developed for a second grade fluid coupled with the energy equation. These two fractional models are based on the definitions of Caputo–Fabrizio (CF) and Atangana–Baleanu (ABC) fractional derivatives. The fluid transport is considered over an infinite upright plate, that is nested in a permeable medium under the influence of an imposed magnetic field. The variable wall velocity and Newtonian heating boundary conditions are simultaneously applied for the very first time. Based on two different fractional derivatives, double fractional analysis is accomplished to evaluate the resulting two models. The momentum and energy solutions are separately established for each model by employing Laplace transform and Durbin’s numerical Laplace inversion. The verification analysis of these obtained solutions is performed with the aid of Zakian’s and Stehfest’s algorithms. The rise and fall in momentum and energy profiles due to variation of incipient parametric values are elucidated graphically and physical arguments behind these behaviors are interpreted. The velocity and temperature gradients are evaluated at the boundary to estimate the wall shear stress and heat transfer rate in terms of the skin friction coefficient and Nusselt number. The noteworthy physical impacts of incipient parameters on shear stress and heat transfer rate are analyzed through tabular study. In comparison, it is found that the energy profile for the ABC model is higher than CF model and ordinary model, respectively. Similarly, for velocity distribution, the ABC model exhibits the highest profile as compared to CF and ordinary models in the case of ramped velocity. However, an exactly opposite pattern of velocity profiles is observed for an isothermal velocity case. On the basis of this comparative analysis, it can be stated that the ABC model is the best choice to appropriately explain the memory effect of energy and momentum distributions.

Wythoff’s game as a classic combinatorial game has been well studied. In this paper, we focus on $(2n+1)$-dimensional Wythoff’s game; that is the Wythoff’s game with $(2n+1)$ heaps. We characterize their ${𝒫}$-positions explicitly and show that they have self-similar structures. In particular, the set of all ${𝒫}$-positions of 3-dimensional Wythoff’s game generates the well-known fractal set — the Sierpinski sponge.

In this study, a new fractal derivative is employed to describe the nonlinear oscillator model. A variational principle of the fractal model is successfully established by using the fractal semi-inverse transform method, and its approximate frequency is obtained by a new fractal transform frequency formulation. The numerical example shows that the proposed technique is simple and accurate for fractal nonlinear oscillators.

This paper focuses on the analysis of the system of equations that describes the blood alcohol concentration (BAC) involving two linked absorption processes, first in the stomach and then in the blood. Each process was modeled with conformable and Caputo derivatives; thus, in this way, the BAC was simulated via four different derivative combinations. In the analysis, the dimensional homogeneity of the equations was maintained. The efficiency (with respect to the classical integer order model) of the obtained solutions was estimated by fitting them to experimental data. The Caputo-conformable combination leads to the highest efficiency followed by the conformable-conformable combination.

A new fractional two-dimensional triangle function combination discrete chaotic map (2DTFCDM) with the discrete fractional difference is proposed in this paper. The chaos behaviors are observed through the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits. The proposed map is applied in color image encryption with the secret keys generated by Menezes–Vanstone Elliptic Curve Cryptosystem. The image encryption system is analyzed using six aspects indicating the superiority of the proposed algorithm compared to the other algorithms.

In this paper, we investigate partial integrals and partial derivatives of bivariate fractal interpolation functions (FIFs). We also prove that the mixed Riemann–Liouville fractional integral and derivative of order $\gamma =(p,q);p>0,q>0$, of bivariate FIFs are again bivariate interpolation functions corresponding to some iterated function system (IFS). Furthermore, we discuss the integral transforms and fractional order integral transforms of the bivariate FIFs.

In this paper, for renormalized Lindström Snowflake networks, we discuss the geodesic metric of their limit space which is quite different from the Euclidean metric on the Lindström Snowflake. Using the dihedral group, we simplify the geodesic patterns and obtain 10,850 patterns and thus calculate the limit of average distances on renormalized Lindström Snowflake networks.

In order to commercially develop shale gas reservoirs, it is necessary to understand the gas transport mechanisms in shale matrix and how matrix permeability evolves during reservoir depletion. In this work, improved apparent permeability models to describe gas transport through microstructures (i.e. micropores and microfractures) in organic matter (OM) and inorganic matter (iOM) of shale matrix are proposed. The models are able to consider the combined effects of poromechanics, non-Darcy flow, gas sorption and fractal dimension of microstructures on gas flow behavior. The obtained results indicate that microfracture aperture declines by a larger margin than micropore diameter when reservoir depletes. A greater microstructure size fractal dimension and maximum microstructure size lead to a larger apparent permeability, while a greater tortuosity fractal dimension leads to a smaller apparent permeability. In shale matrix, the apparent permeability of inorganic microstructures is much larger than that of organic microstructures. In OM or iOM, when micropores and microfractures have the same cross-section area, the apparent permeability of micropores is larger than that of microfractures. Furthermore, for microstructures in OM, the contribution of different flow regimes to total gas flow varies with pore pressure and microstructure size. For microstructures in iOM, the contribution of slip flow dominates the gas transport. The obtained results provide new insights for understanding gas transport behavior in shale reservoirs from a microscopic perspective.

It is of crucial significance to study a class of complex dynamic planar and spatial systems with disparate coefficients by taking advantage of thoughts of Julia set. In this paper, we present theoretical control methods for the stable domain and Julia sets of systems. Then, we utilize symmetry to exhibit the forms of compositional Julia sets of complex dynamic planar systems. Furthermore, by selecting different coefficients, the location, size, area and shape of Julia set can be controlled, the spatial Julia sets also have certain characteristics, this supplies a possible reference for how to control the stable domain of specific complex dynamic systems. The simulation results are in good agreement with the effectiveness of the methods.

As a special kind of complex network, fractal complex networks have their unique characteristics such as fractal dimension, scale invariance, percolation threshold, etc. There have been some researches on the topology of fractal network, among which the calculation of fractal dimension, the construction of fractal model, the phase transition and the robustness of fractal network were the hotspots. But unlike the studies of the scale-free and small-world networks, there was little research on the relationship between the stability and the fractal feature, especially between the vulnerability and fractal dimension of the fractal networks. In this paper, the relationship between vulnerability and fractal dimension was discussed based on two kinds of fractal growth models and different vulnerability measurement methods. The same power functional relationships were obtained and a universal expression was proposed. Then, the effect of fractal feature on the vulnerability was exploring through the fitting error when fractal dimensions were calculated. Finally, the vulnerability of the real networks and their similarities to the fractal models in term of vulnerability showed the rationality of the fractal network models in establishing stable models. The results of this study have important reference for creating structurally stable artificial neural networks, transportation networks and information networks.

Fractional operators are the main tools to describe diffusion problems on fractal objects. This paper first shows the equivalence between conformable fractional derivative and fractal derivative, and then investigates various models for the diffusion on fractal objects by the conformable fractional derivative and its generalized form. The recommended models are solved by the mesh-free physics-informed neural network method with high computational effectiveness and universality.

This paper focuses on the Hölder continuity and the Box Dimension to the $p$th Hadamard Fractional Integral (HFI) on a given interval $[\alpha ,\beta ]$. We use $H_p\varphi (x)$ to denote it. In this paper, two different methods are used to study this problem. By using the approximation method, we obtain that for $\varphi (x)\in {{\mathscr{H}}}^q([\alpha ,\beta ])$ with $p\in (0,1)$ and $q\in (0,1-p)$, if $\alpha >0$, then $H_p\varphi (x)$ is $\gamma$th Hölder continuous in $(\alpha ,\beta ]$ with $\gamma =\frac{p}{(1-q)(1+p)-p^2}$, and is $\left(\right.\frac{p}{1+p}\left)\right.$th Hölder continuous on $[\alpha ,\beta ]$. Moreover, the Box Dimension of the graph of $H_p\varphi (x)$ on the interval $[\alpha ,\beta ]$ is less than or equal to $2-\frac{(1+2p)\gamma }{(1+p)\gamma +(1+2p)}$. If $\alpha =0$, then $H_p\varphi (x)$ is locally $\gamma$th Hölder continuous in $(0,\beta ]$ with the same $\gamma$, and the Box Dimension of $H_p\varphi (x)$ on $[0,\beta ]$ is less than or equal to $2-\frac{\gamma }{1+\gamma }$. By using another method, we imply that, for $\varphi (x)\in {{\mathscr{H}}}^q([\alpha ,\beta ])$ with $q\in (0,1-p)$ and $0<p<1$, if $\alpha >0$, then $H_p\varphi (x)$ is $(p+q)$th Hölder continuous, and thus the Box Dimension of the graph of $H_p\varphi (x)$ is no more than $2-(p+q)$; if $\alpha =0$, then $H_p\varphi (x)$ is locally $(p+q)$th Hölder continuous in $(0,\beta ]$, and is $q$th Hölder continuous at $0$. Then the Box Dimension of the graph to $H_p\varphi (x)$ on $[0,\beta ]$ is less than or equal to $2-\frac{(1+q)(p+q)}{1+2q+p}$. We also give two examples to show that the above Hölder indexes given by the second method are optimal.

This paper lucubrates the bifurcations of a fractional-order bidirectional associative memory neural network (FOBAMNN) with leakage and communication delays. Firstly, leakage delay is taken as a bifurcation parameter to detect the bifurcations of the proposed neural network (NN). Leakage delay-induced bifurcation results are commendably established. Then communication delay is regarded as a bifurcation parameter to investigate the bifurcations of the developed FOBAMNN, and the stability interval and bifurcation point are derived. It reveals that the devised FOBAMNN possesses outstanding stability performance if selecting a lesser value of them, while the bifurcation emerges of FOBAMNN and eventually leads to performance deterioration with an outsize one. Besides, the influence of fractional order on the bifurcation points is carefully studied. It perceives that a proper fractional order can heighten the stability performance of the developed FOBAMNN. Ultimately, numerical simulations are exploited to underpin the developed theory.

Sparse code multiple access (SCMA) is a promising technique of non-orthogonal multiple access technology (NOMA). SCMA has the potential to increase the Total Sum Rate (TSR) in 5G systems. Previous works have considered the Fixed and Dynamic resource allocation based on columns (user) criterion. In this paper, we propose resource allocation based on rows (subcarriers). The numerical results show that our approach improves the TSR in an SCMA uplink channel.

The well-known generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation is widely used in the fluid mechanics, but it becomes invalid under the non-smooth boundary. So this paper, for the first time ever, extends the gBBMB equation into the fractal form that still works under the non-smooth boundary. By using the semi-inverse method, we develop the fractal variational formulations for the problem, which can provide the conservation laws in an energy form, and reveal the possible solution structures of the equation. Furthermore, the two-scale transform method combined with the variational iteration method is used to solve the fractal gBBMB equation. The obtained results show a good agreement with the existed results.

Oxygen diffusion in porous media (ODPM) with rough surfaces (RS) under dry and wet conditions is of great interest. In this work, a novel fractal model for the oxygen effective diffusivity of porous media with RS under dry and wet conditions is proposed. The proposed fractal model is expressed in terms of relative roughness, the water saturation, fractal dimension for tortuosity of tortuous capillaries, fractal dimension for pores, and porosity. It is observed that the normalized oxygen diffusivity decreases with increasing relative roughness and fractal dimension for capillary tortuosity. It is found that the normalized oxygen diffusivity increases with porosity and fractal dimension for pore area. Besides, it is seen that that the normalized oxygen diffusivity under wet condition decreases with increasing water saturation. The determined normalized oxygen diffusivity is in good agreement with experimental data and existing models reported in the literature. With the proposed analytical fractal model, the physical mechanisms of oxygen diffusion through porous media with RS under dry and wet conditions are better elucidated. Every parameter in the proposed fractal model has clear physical meaning, with no empirical constant.

Sudden cardiac death (SCD) is deemed as one of the main causes of death in humans. Therefore, the prediction of an SCD event will help people to receive timely treatment, allowing saving their life. In this sense, this paper introduces a new methodology based on the adroit fusion of fractal dimension (FD) algorithms and a fuzzy logic system for predicting an SCD event automatically. Five FD implementations are investigated in this work: Katz’s FD, Sevcik’s FD, Box’s FD, Higuchi’s FD, and Petrosian’s FD, in order to evaluate the geometrical complexity in electrocardiogram signals of 20 patients with SCD and 18 patients with a normal cardiac rhythm offered by Boston’s Beth Israel Hospital. The results indicate that the FD-based methodology can predict an SCD event up to 60min before the onset, reaching an accuracy of 91.54%.

It is known that heart activity changes during aging. In this paper, we evaluated alterations of heart activity from the complexity point of view. We analyzed the variations of heart rate of patients with congestive heart failure that are categorized into four different age groups, namely 30–39, 50–59, 60–69, and 70–79 years old. For this purpose, we employed three complexity measures that include fractal dimension, sample entropy, and approximate entropy. The results showed that the trend of increment of subjects’ age is reflected in the trend of increment of the complexity of heart rate variability (HRV) since the values of fractal dimension, approximate entropy, and sample entropy increase as subjects get older. The analysis of the complexity of other physiological signals can be further considered to investigate the variations of activity of other organs due to aging.

The stress-dependent flow and transport behaviors of porous media are ubiquitous in various scientific and engineering applications. It has been shown that the change of effective stress has important effects on the permeability and porosity of porous media. In this paper, a new stress sensitivity model for porous media is developed based on the fractal theory and the elasto-plastic thick-walled cylinder model. The proposed model is able to predict the elasto-plastic deformation of the fractal porous media under loading–unloading stress cycles, which plays a crucial role on the permanent variations of the permeability and porosity. It is found that the permeability of stress-sensitivity porous media is related to the capillary fractal dimension, capillary fractal tortuosity dimension, minimum and maximum capillary diameters, Young’s modulus and Poisson’s ratio of capillary. Each parameter has a clear physical meaning. The validity of the developed fractal model is verified by comparing the model predictions with the available experimental data.

In this paper, we investigated the learning ability of students in normal versus virtual reality (VR) watching of videos by mathematical analysis of electroencephalogram (EEG) signals. We played six videos in the 2D and 3D modes for nine subjects and calculated the Shannon entropy of recorded EEG signals to investigate how much their embedded information changes between these modes. We also calculated the Hurst exponent of EEG signals to compare the changes in the memory of signals. The analysis results showed that watching the videos in a VR condition causes greater information and memory in EEG signals. A strong correlation was obtained between the increment of information and memory of EEG signals. These increments also have been verified based on the answers that subjects gave to the questions about the content of videos. Therefore, we can say that when subjects watch a video in a VR condition, more information is transferred to their brains that cause increments in their memory.