Fractals

As a kind of classical fractal network models, Sierpinski Gasket and its extended network models have attracted the attention of many scholars because of their many applications in real-world networks. Based on the classical Sierpinski Gasket, this paper proposes a joint Sierpinski Gasket model, which can be used not only to study the topology and dynamic properties of Sierpinski Gasket itself, but also to study the Sierpinski Gasket under special conditions, such as the residual Sierpinski Gasket after damage. Since the mean time to absorption (MTA) on the network is a basic dynamic property related to the random walk, in this paper, we study the expression of MTA on the proposed general joint Sierpinski Gasket model, and find that the MTA is determined by the number of iterations $g$ and two variables determined by the mode variable $S$, which can be solved by matrix algorithm. Therefore, using the expression of MTA on the general joint Sierpinski Gasket model, we calculate the analytical expressions of the MTA on a general example of the joint Sierpinski Gasket model and three Sierpinski Gaskets with varying degrees of damage and do the corresponding numerical simulation to verify the correctness of the conclusion.

In this paper, a novel direct scheme to solve a set of time-delay fractional optimal control problems is introduced. The method firstly uses a set of Dickson polynomials as basis functions to approximate the states and control variables of the system. Next, the context of these basis functions and the use of a collocation method allow to transform the problem into a system of nonlinear algebraic equations. Finally, the unknown coefficients and control parameters in the original problem can be easily estimated by resolving the new system of equations. Given the high efficiency of the Dickson polynomials to deal with time-delay fractional systems, the proposed strategy involves a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The convergence analysis of the proposed method is presented, along with some illustrative examples which demonstrate its most relevant features.

In this work, the fractal cubic–quintic Duffing’s equation analytical solution is obtained using the two-scale transform and elliptic functions. Then, the analytical solution is used to study wave propagation in a fractal medium. Since the value of the fractal parameter adjusts the pulse frequency and wavelength propagation velocity, depending upon the fractal medium physical properties, it is found that the information contained in the pulse can be carried out faster over long distances without distortion or loss of intensities.

This paper offers a new light on the applicability of the two-scale transform of fractal theory to comprehend natural phenomena.

Generalized synchronization is a typical dynamical phenomenon in nonlinear systems, for which the real-valued setting has been widely investigated. The complex-valued functions relationship in generalized synchronization is equally important for complex-valued dynamical systems, which however are seldom studied. Complex parameters identification on the synchronization manifold remains an open problem owing to the absence of the persistent excitation (PE) condition in the complex field. This paper investigates generalized synchronization via a complex-valued vector mapping (CGS) for different-dimensional complex-variable chaotic (hyper-chaotic) systems (CVCSs) with complex parameters identification. Based on Lyapunov stability theory in the complex field and using an adaptive control method, some sufficient criteria are established to achieve CGS for CVCSs. Moreover, some necessary and sufficient criteria are derived to ensure complex parameters identification. Finally, the theoretical results are verified and demonstrated by reduced-order and increased-order simulation examples.

This work is aimed at establishing a unified fractional thermoelastic model for piezoelectric structures, and shedding light on the influence of different definitions on the transient responses. Theoretically, based upon Cattaneo-type equation, a unified form of fractional heat conduction law is proposed by adopting Caputo Fabrizio fractional derivative, Atangana Baleanu fractional derivative and Tempered Caputo fractional derivative. Then, thermoelastic model of fractional order is formulated for piezoelastic materials by combining the unified heat conduction law and the governing equations of elastic and electric fields. Numerically, the present theoretical model is applied to study the transient responses of piezoelectric medium that is subjected to a thermal shock. The governing equations are analytically derived and numerically solved with the aids of Laplace transform method. The obtained results are graphically illustrated, and the influences of different definitions of fractional derivative and different fractional order are revealed. This work may be helpful for understanding the multi-coupling effect of elastic, thermal and electric fields, and for inspiring further developments of fractional calculus.

This study experimentally estimated the fractal pore size distribution of granites, which is then linked to the hydro-mechanical properties of rocks treated by temperatures that range from 25${}^{\circ }$C to 900${}^{\circ }$C. The mercury intrusion experiment was carried out to characterize the pore size distributions and the MTS815.02 triaxial testing system was used to investigate hydro-mechanical properties of rocks. Finally, the micro-CT scanning system and scanning electron microscope system were employed to exhibit the evolutions of microstructures of cracks that were then linked to the macroscopic hydro-mechanical properties. The results show that the pore size distribution of granites follows the fractal scaling law and the fractal dimension ranges from 2.45 to 2.94. The fractal dimension decreases significantly when the temperature increases from 25${}^{\circ }$C to 100${}^{\circ }$C and then holds a constant with continuously increasing the temperature to 400${}^{\circ }$C. The fractal dimension slightly increases as the temperature increases from 400${}^{\circ }$C to 500${}^{\circ }$C and decreases following a linear relationship until the temperature of 900${}^{\circ }$C. The porosity obtained by the mercury intrusion experiment shows an exponential relationship with the fractal dimension. Both the axial stain and peak total stain, as well as the initial permeability and the permeability at the peak axial stress, have quadratic functions with the fractal dimension. The mean aperture of fractures increases from 56.75$\mu$m to 76.10$\mu$m with increasing the temperature from 100${}^{\circ }$C to 900${}^{\circ }$C through micro-CT scanning. The scanning electron microscope test clearly shows that the fractures are generated when the temperature exceeds 300${}^{\circ }$C, which agrees well with the micro-CT scanning results. With increasing the temperature from 400${}^{\circ }$C to 900${}^{\circ }$C, both the number and aperture of fractures increase, which well interprets that both the strain and permeability increase as indicated by the triaxial stress-flow tests.

The main objective of this paper is to present weighted $k$-fractional integral and derivative operators of a function with respect to another function and to uncover their properties. In addition to this, we find the weighted Laplace transform of the newly defined operators. As applications of the weighted $k$-fractional operators in mathematical physics, we study the fractional forms of kinetic differintegral equation and the time-fractional heat equation involving the novel operators and find their solutions using weighted Laplace transform.

Cooperative behavior is an extensive mechanism in nature, and seen in several organisms, from bacterial cells to primates. In this paper, we have investigated a spatial predator–prey model with hunting cooperation in predators. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of control parameters. Using extensive numerical simulations, we obtain complex patterns, namely, spotted pattern, stripe pattern and mixed pattern in the Turing domain, by varying the hunting cooperation rate in predators and ratio of diffusion coefficient. The results focus on the effect of hunting cooperation rate in pattern dynamics of a diffusive predator–prey model and help us in better understanding of the dynamics of the predator–prey interaction in real environment.

In this paper, we derive the probability distribution of linear fractal interpolation function (FIF) for a random dataset on ${ℝ}^2$, where randomness in the data is induced, using $\alpha$-stable noise, in the second coordinate. In particular, we show that the distribution of linear FIF, at any point (on the first coordinate), is also stable, but with different parameters which can be estimated. Finally, we also provide statistical validity of our results.

We identify a countable infinity of parameter regimes with strange nonchaotic attractors (SNAs). At the edge of each arc parameter area, there is an uncountable infinity of SNAs with torus intermittency. The mechanism for the creation of SNAs in different regime is induced by an $n$-frequency quasiperiodic orbit through a quasiperiodic analog of saddle-node bifurcation (Type-$\mathrm{\text{I}}$ intermittent route). We describe the transition between tori and SNAs by the largest Lyapunov exponent and phase diagram. These SNAs are characterized by the phase sensitivity exponents, rational approximations, singular-continuous spectra, and distribution of finite-time Lyapunov exponents.

The apparent liquid permeability (ALP) of shale is challenging to be characterized due to complex wettability and nanopore size distribution. The nanopores in organic matter of shale are usually hydrophobic and the nanopores in inorganic matter are hydrophilic. The flow behaviors in these two different nanopores are quite different, and accurately predicting the ALP of shale is difficult. This paper proposes a fractal model for predicting the ALP of shale with dual wettability. The nonflowing boundary layer effect of water in inorganic pores and the slip effect in organic pores are considered, and the equations for describing the flow rate in single organic pore and inorganic pore are derived, respectively. With the assumption of the fractal distributions of organic pores and inorganic pores in shale, the analytical expression for predicting the ALP of shale is derived, and the key parameters influencing shale ALP are analyzed with sensitivity study. The research results show that the nonflowing boundary layer can reduce the ALP of inorganic pores, but slip effect will increase the ALP of organic pores. ALP of inorganic pores is affected by the thickness of nonflowing boundary layer, which is determined by the displacement pressure gradient, fluid viscosity, and pore size distribution. ALP of inorganic pores is more affected by contact angle and pore size distribution.

Coal failure behavior under dynamic loading is significant for dealing with the failure issues encountered in underground coal mines. In this study, the crack propagation and internal fracture process of coal under dynamic loading were investigated by a split Hopkinson pressure bar (SHPB) experimental system and numerical approach, respectively. The experimental and numerical results show that the coal crack propagation and internal fracture surface manifest significant multifractal features. Further, multifractal analysis suggests that the multifractal feature becomes more and more significant during the dynamic loading. The formation mechanism of the multifractal features was further discussed, and the result demonstrates that the crack propagation path within coal is essentially a multifractal structure, thus causing the total fracture behavior of coal to show multifractal features under dynamic loading. Moreover, the multifractal spectrum parameter $\mathrm{\Delta }{\alpha }_L$ was proved to be closely related to the brittle fracture property of coal, which transpires that the multifractal feature is feasible to evaluate the coal brittleness under dynamic loading and also applicable to predict the coal failure risk in underground coal mines.

Dürer’s pentagon is known to the artist Albrecht Dürer. In this paper, we consider directed networks generated by Dürer-type polygons. We aim to present a study on some properties of these networks, such as degree distribution, clustering coefficient and average path length. We show that such networks have the scale-free effect, but do not have the small-world effect.

The geodesic structures on self-similar fractals are interesting. One feature of geodesic structures is average distance on fractal. We investigate the Sierpinski-like carpet and obtain the average distance in terms of self-similar measure.

In this work, we analyze various stability (from stability to global uniform asymptotic stability) of Caputo fractional-order nonautonomous systems (CFONSs). With $f(t,x)$ being only continuous, it is proven that the Caputo fractional derivative (CFD) of a general Lyapunov function $V(t,x(t))$ along solutions is continuous and moreover,

In this work, the Schrödinger equation is described by the fractal derivative, and its variational principle is obtained by using the fractal semi-inverse method. The variational principle is helpful to research the construction of the solution. The approximate analytical solution of the fractal Schrödinger equation is obtained based on the proposed variational approach and the fractal two-scale transform method. Finally, an example shows that the proposed approach is very fascinating in deal with nonlinear fractal models.

We investigate a class of self-similar networks modeled on the three-dimensional Vicsek fractal, and obtain the asymptotic formula of node-weighted average geodesic distances, which is an integral of geodesic distance in terms of the self-similar measure.

Fractal analysis was used in the study to determine a set of feature descriptors which could be applied in the process of diagnosing bone damage caused by osteoporosis. The subject of the research was CT images of vertebrae on the thoraco-lumbar region. The dataset contained images of healthy patients and patients diagnosed with osteoporosis. On the basis of fractal analysis and feature selection by linear stepwise regression, three descriptors were obtained. These were two fractal dimensions calculated by the variation method and fractal lacunarity calculated by the box counting method. The first two descriptors were obtained as a result of the analysis of gray images, and the third was the result of analysis of binary images. The effectiveness of the descriptors was verified using six popular supervised classification methods: linear and quadratic discriminant analyses, naive Bayes classifier, decision tree, $K$-nearest neighbors ($K$-NN) and random forests. The best results were obtained using the $K$-NN classifier; they were as follows: overall classification accuracy: 81%, classification sensitivity: 78%, classification specificity: 90%, positive predictive value: 90% and negative predictive value: 77%. The results of the research have shown that fractal analysis can be a useful tool to extract features of spinal CT images in the diagnosis of osteoporotic bone defects.

We investigate the $n$-level Sierpinski carpets including the classical one for $n=3$, which are non-p.c.f. self-similar fractals. By using the finitely, many geometric patterns and the self-similar measure, we obtain the mean geodesic distance on the $n$-level Sierpinski carpets and their skeleton networks.

In this paper, the finite-time stability in mean for the uncertain fractional order linear time-invariant discrete systems is investigated. First, the uncertain fractional order difference equations with the nabla operators are introduced. Then, some conditions of finite-time stability in mean for the systems driven by the nabla uncertain fractional order difference equations with the fractional order $0<\nu <1$ are obtained by the property of Riemann–Liouville-type nabla difference and the generalized Gronwall inequality. Furthermore, based on these conditions, the state feedback controllers are designed. Finally, some examples are presented to illustrate the effectiveness of the results.

In this paper, the definition of generalized s-preinvex function on Yang’s fractal sets ${ℝ}^{\gamma }(0<\gamma \le 1)$ is proposed, and the generalized Hermite–Hadamard’s inequality for this class of functions is established. By using this convexity, some generalized Hermite–Hadamard type integral inequalities with parameters are established. For these inequalities, the absolute values of twice local fractional order derivative of the functions are generalized s-preinvex functions. Some special integral inequalities can be obtained by assigning special values to the obtained inequalities, and two examples are given to illustrate our results. Finally, we propose the applications of the results in numerical integration and error estimation.

Let $[a_1(x),a_2(x),\dots ,a_n(x),\dots ]$ be the continued fraction expansion of an irrational $x\in (0,1)$. For any $n\ge 1$, write $T_n(x)={max}_{1\le k\le n}\{a_k(x)\}$. This paper is concerned with the Hausdorff dimension of the set

Wave motions play an important role in fluid dynamics and engineering issues. In this paper, a systematic study to the generalized KdV–Burgers–Kuramoto equation is presented resort to symmetry method. First, based on the Lie point symmetries of the generalized KdV–Burgers–Kuramoto equation, we get invariants and invariant solutions. In particular, we obtain series solutions of generalized KdV–Burgers–Kuramoto equation. Meanwhile, we find that this equation just exists in Lie point symmetries. Then, we present a conservation law, and derive reciprocal Bäcklund transformations of conservation law for the first time. Furthermore, Bäcklund transformation of KdV–Burgers–Kuramoto equation associated with truncated Painlevé expansion is studied for the first time. Subsequently, mCK method is employed to study KdV–Burgers–Kuramoto equation. Lastly, we investigate symmetries of generalized time fractional KdV–Burgers–Kuramoto equation, and derive a conservation law. The obtained results illustrate that symmetry method is a very powerful method to deal with nonlinear partial differential equations. The results provide theoretical support for explaining complex nonlinear phenomena.

In this paper, we discuss a problem of multiplicative Diophantine approximation of the expansions under different bases restricted on a line. More specifically, let $\phi :\mathbb{N}\to [0,\infty )$ be a positive function. A dichotomy law of the Hausdorff measure for the following set:

In this work, we initially study the strange nonchaotic dynamics of a two-degree-of-freedom quasiperiodically forced vibro-impact system. It is shown that strange nonchaotic attractors (SNAs) occur between two chaotic regions, but not between the quasiperiodic region and the chaotic one. Subsequently, we mainly focus on the abundant multistability in the system, especially the coexistence of SNAs and quasiperiodic attractors. Besides, the coexistence of quasiperiodic attractors of different frequencies, and the coexistence of quasiperiodic attractors and chaotic attractors are also uncovered. The basins of attraction of these coexisting attractors are obtained. The quasiperiodic attractor can transform into a chaotic attractor directly through torus break-up without passing through an SNA. The nonchaotic property of SNAs is verified by its maximal Lyapunov exponent, and the strange property of SNAs is described by its phase sensitivity, power spectrum, fractal structure, and rational approximations.

We obtain the Assouad dimensions of two Moran classes with zero infimum contraction under suitable conditions.

When an oscillator vibrates in a porous medium or along an unsmooth surface, a fractal model can be adopted using the two-scale fractal derivative. This paper elucidates a simple but effective method for fast and exclusive insight into the asymptotically periodic property at the initial stage and the extremely low frequency property when time tends to infinity. Some examples are illustrated to show the one-step solution process and the accurate results.

In this paper, we investigate how many real numbers can be well approximated by their convergents in the Engel expansions. Furthermore, the relative growth rate of convergence speed of convergents in the Engel expansion of an irrational number is studied to the rate of growth of its digits. The Hausdorff dimension of exceptional sets of points with a given relative growth rate is established.

During the development of deep-water oil and gas resources, severe engineering problems or even marine geological hazards might be induced by the complex geological characteristics of deep-water weakly consolidated (DWC) formation. To solve these problems, the power-law cementing fluid was used to solidify the DWC formation during the well-cementing operation. However, the transportation process of cementing fluid in DWC formation is still unclear. To guide the solidification operation of DWC formation, a mathematical model for predicting the penetration distance of cementing fluid in DWC formation was proposed based on the generalized Darcy’s law and fractal theory. All the parameters in this model have clear physical meaning and no empirical parameter is involved. The numerical and experimental validation results show that the proposed model calculated penetration distance agrees well with both simulated and experimental results. The model capacity relates to the penetration pressure, the viscosity of cementing fluid, the permeability of formation, the effective radius of the flow path and the fractal dimension of formation. The maximum error of this model is less than 20%, which indicates that the mathematical model is reliable.

Since skin activity, like other organs, is controlled by the brain, we decoded the correlation among the brain and skin responses in auditory stimulation by complexity-based analysis of EEG and GSR signals. Three pieces of music were selected according to the difference in the fractal exponent and sample entropy of embedded noises in them. We calculated the fractal dimension and sample entropy of EEG and GSR signals for 11 subjects in rest and response to these music pieces. The correlation coefficients of 0.9525 and 0.9822 in the case of fractal dimension and sample entropy demonstrated a strong correlation between the complexities of the GSR and EEG signals. Therefore, we can state that the skin and brain responses are coupled. This method can be applied to evaluate the relationship between the human brain and other organs.