Fractal Fract., Volume 6, Issue 12 (December 2022) – 59 articles

In this paper, by applying Petković’s iterative method to the Möbius conjugate mapping of a quadratic polynomial function, we attain an optimal eighth-order rational operator with a single parameter r and research the stability of this method by using complex dynamics tools on the basis of fractal theory. Through analyzing the stability of the fixed point and drawing the parameter space related to the critical point, the parameter family which can make the behavior of the corresponding iterative method stable or unstable is obtained. Lastly, the consequence is verified by showing their corresponding dynamical planes. Full article

We propose high-order schemes for nonlinear fractional initial value problems. We split the fractional integral into a history term and a local term. We take advantage of the sum of exponentials (SOE) scheme in order to approximate the history term. We also use a low-order quadrature scheme to approximate the fractional integral appearing in the local term and then apply a spectral deferred correction (SDC) method for the approximation of the local term. The resulting one-step time-stepping methods have high orders of convergence, which make adaptive implementation and accuracy control relatively simple. We prove the convergence and stability of the proposed schemes. Finally, we provide numerical examples to demonstrate the high-order convergence and adaptive implementation. Full article

The accurate determination of atmospheric temperature with telemetric platforms is an active issue, one that can also be tackled with the aid of multifractal theory to extract fundamental behaviors of the lower atmosphere, which can then be used to facilitate such determinations. Thus, in the framework of the scale relativity theory, PBL dynamics are analyzed through the aid of a multifractal hydrodynamic scenario. Considering the PBL as a complex system that is assimilated to mathematical objects of a multifractal type, its various dynamics work as a multifractal tunnel effect. Such a treatment allows one to define both a multifractal atmospheric transparency coefficient and a multifractal atmospheric reflectance coefficient. These products are then employed to create theoretical temperature profiles, which lead to correspondences with real results obtained by radiometer data (RPG-HATPRO radiometer), with favorable results. Such methods could be further used and refined in future applications to efficiently produce atmospheric temperature theoretical profiles. Full article

This paper explores the generalization of the fixed-point theorem for Fisher contraction on controlled metric space. The controlled metric space and Fisher contractions are playing a very crucial role in this research. The Fisher contraction on the controlled metric space is used in this paper to generate a new type of fractal set called controlled Fisher fractals (CF-Fractals) by constructing a system named the controlled Fisher iterated function system (CF-IFS). Furthermore, the interesting results and consequences of the controlled Fisher iterated function system and controlled Fisher fractals are demonstrated. In addition, the collage theorem on controlled Fisher fractals is established as well. The newly developing IFS and fractal set in the controlled metric space can provide the novel directions in the fractal theory. Full article

This paper presents numerical modeling and investigation for the Ripa system. This model is derived from a shallow water model by merging the horizontal temperature gradients. We applied the non-homogeneous Riemann solver (NHRS) method for solving the Ripa model. This scheme contains two stages named predictor and corrector. The first one is made up of a control parameter that is responsible for the numerical diffusion. The second one recuperates the balance conservation equation. One of the main features of the NHRS scheme, it can determine the numerical flux corresponding to the real state of solution in the non-attendance of Riemann solution. Various test cases of physical interest are considered. These case studies display the high resolution of the NHRS scheme and emphasize its ability to produce accurate results for the Ripa model. The presented solutions are very critical in superfluid applications of energy and many others. Finally, the NHRS technique can be used to solve a wide range of additional models in applied research. Full article

This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional ($\kappa ,\varphi$)-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space $C\left(\right[1,\mathtt{T}\left]\right)$ arising in biological population dynamics. The key findings of the article are based on theoretical concepts pertaining to the fractional calculus and the Hausdorff measure of non-compactness (MNC). To obtain this goal, we employ Darbo’s fixed-point theorem (DFPT) in the Banach space. In addition, we provide two numerical examples to demonstrate the applicability of our findings to the theory of fractional integral equations. Full article

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of the fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using Banach’s and Schauder’s fixed point theorems, we derive some sufficient conditions to guarantee the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs. Full article

Polymers and polymer matrix composites are commonly used materials with applications extending from packaging materials to delicate electronic devices. Epoxy resins and fiber-reinforced epoxy-based composites have been used as adhesives and construction parts. Fractal analysis has been recognized in materials science as a valuable tool for the microstructural characterization of composites by connecting fractal characteristics with composites’ functional properties. In this study, fractal reconstructions of different microstructural shapes in an epoxy-based composite were performed on field emission scanning electron microscopy (FESEM) images. These images were of glass fiber reinforced epoxy as well as a hybrid composite containing both glass and electrospun polystyrene fibers in an epoxy matrix. Fractal reconstruction enables the identification of self-similarity in the fractal structure, which represents a novelty in analyzing the fractal properties of materials. Fractal Real Finder software, based on the mathematical affine fractal regression model, was employed to reconstruct different microstructure shapes and calculate fractal dimensions to develop a method of predicting the optimal structure–property relations in composite materials in the future. Full article

The study of hidden attractors plays a very important role in the engineering applications of nonlinear dynamical systems. In this paper, a new three-dimensional (3D) chaotic system is proposed in which hidden attractors and self-excited attractors appear as the parameters change. Meanwhile, asymmetric coexisting attractors are also found as a result of the system symmetry. The complex dynamical behaviors of the proposed system were investigated using various tools, including time-series diagrams, Poincaré first return maps, bifurcation diagrams, and basins of attraction. Moreover, the unstable periodic orbits within a topological length of 3 in the hidden chaotic attractor were calculated systematically by the variational method, which required six letters to establish suitable symbolic dynamics. Furthermore, the practicality of the hidden attractor chaotic system was verified by circuit simulations. Finally, offset boosting control and adaptive synchronization were used to investigate the utility of the proposed chaotic system in engineering applications. Full article

This article is aimed at reviewing and studying the effects of the 2d-3d crossover on the effective fractal and spatial dimensions, as well as on the critical exponents of the physical properties of bulk and bounded systems at criticality. Here we consider the following problems: (1) the two types of dimensional crossovers and the concept of the universality classes; (2) a smooth 2d-3d crossover and the calculation of the effective fractal and spatial dimensions, as well as the effective critical indices; (3) the fractal dimension, its connection with the random mean square order-parameter fluctuations and a new phase formation; (4) the fractal nuclei of a new phase and the medical consequences of carcinogenesis and nucleation isomorphism. Full article

In this paper, we mainly establish Liouville-type theorems for the elliptic semi-linear equations involving the fractional Laplacian on the upper half of Euclidean space. We employ a direct approach by studying an equivalent integral equation instead of using the conventional extension method. Applying the method of moving planes in integral forms, we prove the non-existence of positive solutions under very weak conditions. We also extend the results to a more general equation. Full article

Dopaminergic treatment (DT), the standard therapy for Parkinson’s disease (PD), alters the dynamics of functional brain networks at specific time scales. Here, we explore the scale-free functional connectivity (FC) in the PD population and how it is affected by DT. We analyzed the electroencephalogram of: (i) 15 PD patients during DT (ON) and after DT washout (OFF) and (ii) 16 healthy control individuals (HC). We estimated FC using bivariate focus-based multifractal analysis, which evaluated the long-term memory (H(2)) and multifractal strength (ΔH₁₅) of the connections. Subsequent analysis yielded network metrics (node degree, clustering coefficient and path length) based on FC estimated by H(2) or ΔH₁₅. Cognitive performance was assessed by the Mini Mental State Examination (MMSE) and the North American Adult Reading Test (NAART). The node degrees of the ΔH₁₅ networks were significantly higher in ON, compared to OFF and HC, while clustering coefficient and path length significantly decreased. No alterations were observed in the H(2) networks. Significant positive correlations were also found between the metrics of H(2) networks and NAART scores in the HC group. These results demonstrate that DT alters the multifractal coupled dynamics in the brain, warranting the investigation of scale-free FC in clinical and pharmacological studies. Full article

The focus on renewable energy is increasing globally to lessen reliance on conventional sources and fossil fuels. For renewable energy systems to work at their best and produce the desired results, precise feedback control is required. Microbial electrochemical cells (MEC) are a relatively new technology for renewable energy. In this study, we design and implement a model-based robust controller for a continuous MEC reactor. We compare its performance with those of traditional methods involving a proportional integral derivative (PID), H-infinity (H${}_{\infty }$) controller and PID controller tuned by intelligent genetic algorithms. Recently, a dynamic model of a MEC continuous reactor was proposed, which describes the complex dynamics of MEC through a set of nonlinear differential equations. Until now, no model-based control approaches for MEC have been proposed. For optimal and robust output control of a continuous-reactor MEC system, we linearize the model to state a linear time-invariant (LTI) state-space representation at the nominal operating point. The LTI model is used to design four different types of controllers. The designed controllers and systems are simulated, and their performances are evaluated and compared for various operating conditions. Our findings show that a structured linear fractional transformation (LFT)-based H${}_{\infty }$ control approach is much better than the other approaches against various performance parameters. The study provides numerous possibilities for control applications of continuous MEC reactor processes. Full article

This paper focuses on studying a random effects semiparametric regression model (RESPRM) with separable space-time filters. The model cannot only capture the linearity and nonlinearity existing in a space-time dataset, but also avoid the inefficient estimators caused by ignoring spatial correlation and serial correlation in the error term of a space-time data regression model. Its profile quasi-maximum likelihood estimators (PQMLE) for parameters and nonparametric functions, and a generalized F-test statistic for checking the existence of nonlinear relationships are constructed. The asymptotic properties of estimators and asymptotic distribution of test statistic are derived. Monte Carlo simulations imply that our estimators and test statistic have good finite sample performance. The Indonesian rice farming data are used to illustrate our methods. Full article

Unconventional shale reservoirs and typical fine-grained rocks exhibit complicated, oriented features at various scales. Due to the complex geometry, combination and arrangement of grains, as well as the substantial heterogeneity of shale, it is challenging to analyze the oriented structures of shale accurately. In this study, we propose a model that combines both multifractal and structural entropy theory to determine the oriented structures of shale. First, we perform FE–SEM experiments to specify the microstructural characteristics of shale. Next, the shape, size and orientation parameters of the grains and pores are identified via image processing. Then fractal dimensions of grain flatness, grain alignment and pore orientation are calculated and substituted into the structural entropy equation to obtain the structure-oriented entropy model. Lastly, the proposed model is applied to study the orientation characteristic of the Yan-Chang #7 Shale Formation in Ordos Basin, China. A total of 1470 SEM images of 20 shale samples is analyzed to calculate the structure-oriented entropy (SOE) of Yan-Chang #7 Shale, whose values range from 0.78 to 0.96. The grains exhibit directional arrangement (SOE ≥ 0.85) but are randomly distributed (SOE < 0.85). Calculations of samples with different compositions show that clay and organic matters are two major governing factors for the directivity of shale. The grain alignment pattern diagram analyses reveal three types of orientation structures: fusiform, spider-like and eggette-like. The proposed model can quantitatively evaluate the oriented structure of shale, which helps better understand the intrinsic characteristics of shale and thereby assists the successful exploitation of shale resources. Full article

The aim of this paper is to obtain some new results about common fixed points. Our results use weaker conditions than those previously used. We have relaxed the conditions for commutating pair mappings and compatible mappings of the type $\left(A\right)$, which were introduced in 1976. The theorems are enriched by using the concept of $WC$ and various types of weakly commuting pairs of maps in metric spaces. To discuss the existence and uniqueness of the common solutions, we have obtained an application to the functional equations in dynamic programming. Full article

We establish sufficient conditions for the existence of solutions of an integral boundary value problem for a $\mathsf{\Psi }$-Hilfer fractional integro-differential equations with non-instantaneous impulsive conditions. The main results are proved with a suitable fixed point theorem. An example is given to interpret the theoretical results. In this way, we generalize recent interesting results. Full article

In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. Since the Herglotz-type variational problem is a generalization of the classical variational problem, our main results generalize several results from the fractional calculus of variations. To illustrate the theoretical developments included in this paper, we provide some examples. Full article

We present the existence of solutions for sequential Caputo–Hadamard fractional differential equations (SC-HFDE) with fractional boundary conditions (FBCs). Known fixed-point techniques are used to analyze the existence of the problem. In particular, the contraction mapping principle is used to investigate the uniqueness results. Existence results are obtained via Krasnoselkii’s theorem. An example is used to illustrate the results. In this way, our work generalizes several recent interesting results. Full article

In this article, starting from a benchmark represented by a Direct Current-to-Direct Current (DC-DC) three-phase power electronic converter used as an interface and interconnection between the grid and a DC microgrid, we compare the performances of a series of control structures—starting with the classical proportional integrator (PI) type and continuing with more advanced ones, such as sliding mode control (SMC), integer-order synergetic, and fractional-order (FO) controllers—in terms of maintaining the constant DC voltage of the DC microgrid. We present the topology and the mathematical modeling using differential equations and transfer functions of the DC-DC three-phase power electronic converter that provides the interface between the grid and a DC microgrid. The main task of the presented control systems is to maintain the DC voltage supplied to the microgrid at an imposed constant value, regardless of the total value of the current absorbed by the consumers connected to the DC microgrid. We present the elements of fractional calculus that were used to synthesize a first set of FO PI, FO tilt-integral-derivative (TID), and FO lead-lag controllers with Matlab R2021b and the Fractional-order Modeling and Control (FOMCON) toolbox, and these controllers significantly improved the control system performance of the DC-DC three-phase power electronic converter compared to classical PI controllers. The next set of proposed and synthesized controllers were based on SMC, together with its more general and flexible synergetic control variant, and both integer-order and FO controllers were developed. The proposed control structures are cascade control structures combining the SMC properties of robustness and control over nonlinear systems for the outer voltage control loop with the use of properly tuned synergetic controllers to obtain faster response time for the inner current control loop. To achieve superior performance, this type of cascade control also used a properly trained reinforcement learning-twin delayed deep deterministic policy gradient (RL-TD3) agent, which provides correction signals overlapping with the command signals of the current and voltage controllers. We present the Matlab/Simulink R2021b implementations of the synthesized controllers and the RL-TD3 agent, along with the results of numerical simulations performed for the comparison of the performance of the control structures. Full article

The fractal dimension $D_f$ has been widely used to describe the structural and morphological characteristics of aggregates. Box-counting (BC) and power law (PL) are the most common methods to calculate the fractal dimension of aggregates. However, the prefactor $k$, as another important fractal property, has received less attention. Furthermore, there is no relevant research about the BC prefactor ($k_{BC}$). This work applied a tunable aggregation model to generate a series of three-dimensional aggregates with different input parameters (power law fractal properties: $D_{f,PL}$ and $k_{PL}$, and the number of primary particles $N_P$). Then, a projection method is applied to obtain the 2D information of the generated aggregates. The fractal properties ($k_{BC}$ and $D_{f,BC}$) of the generated aggregates are estimated by both, for 2D and 3D BC methods. Next, the relationships between the box-counting fractal properties and power law fractal properties are investigated. Notably, 2D information is easier achieved than 3D data in real processes, especially for aggregates made of nanoparticles. Therefore, correlations between 3D BC and 3D PL fractal properties with 2D BC properties are of potentially high importance and established in the present work. Finally, a comparison of these correlations with a previous one (not considering $k$) is performed, and comparison results show that the new correlations are more accurate. Full article

The effect of track geometry on vehicle vibration is a major concern in high-speed rail (HSR) operation from the perspectives of ride comfort and safety. However, how to quantitatively characterize the relation between them remains a problem to be solved in track quality assessment. By using fractal analysis, this paper studies the detailed correlation between track surface and alignment irregularities and car body vertical and lateral acceleration in various wavelength ranges. The time-frequency features of the track irregularity and car-body acceleration are first analyzed based on empirical mode decomposition (EMD). Then, the fractal features of the inspection data are determined by calculating the Hurst exponent of their intrinsic mode functions (IMFs). Finally, the fractal dimensions of the track irregularity and car-body acceleration are obtained, and the correlation between their fractal dimensions with respect to different IMFs is revealed using regression analysis. The results show that the fractal dimension is only related to the roughness of the IMF waveforms of the track irregularity and car-body vibration and is irrelevant to the amplitude of the time series of the data; the correlation coefficient of the fractal dimension of the track irregularity and car-body acceleration is greater than 0.7 for wavelengths greater than 30 m, indicating that the relationship between track irregularity and car-body vibration acceleration is more obvious for long wavelengths. The findings of this research could be used for optimizing HSR track maintenance work from the viewpoint of the ride quality of high-speed trains. Full article

In this paper, a new type of convexity is defined, namely, the left–right-(k,h-m)-p IVM (set-valued function) convexity. Utilizing the definition of this new convexity, we prove the Hadamard inequalities for noninteger Katugampola integrals. These inequalities generalize the noninteger Hadamard inequalities for a convex IVM, (p,h)-convex IVM, p-convex IVM, h-convex, s-convex in the second sense and many other related well-known classes of functions implicitly. An apt number of numerical examples are provided as supplements to the derived results. Full article

The Langevin system is an important mathematical model to describe Brownian motion. The research shows that fractional differential equations have more advantages in viscoelasticity. The exploration of fractional Langevin system dynamics is novel and valuable. Compared with the fractional system of Caputo or Riemann–Liouville (RL) derivatives, the system with Mittag–Leffler (ML)-type fractional derivatives can eliminate singularity such that the solution of the system has better analytical properties. Therefore, we concentrate on a nonlinear Langevin system of ML-type fractional derivatives affected by time-varying delays and differential feedback control in the manuscript. We first utilize two fixed-point theorems proposed by Krasnoselskii and Schauder to investigate the existence of a solution. Next, we employ the contraction mapping principle and nonlinear analysis to establish the stability of types such as Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) as well as generalized UH and UHR. Lastly, the theoretical analysis and numerical simulation of some interesting examples are carried out by using our main results and the DDESD toolbox of MATLAB. Full article

In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well. Full article

In this paper, we study the existence and uniqueness of solutions for nabla fractional systems. By using the properties of bijective functions, we obtain a necessary and sufficient condition ensuring the existence and uniqueness of solutions for a class of fractional discrete systems. Furthermore, we derive two sufficient conditions guaranteeing the existence of solutions by means of a nonlinear functional analysis method. In addition, the above conclusions are extended to high-dimensional delayed systems. Finally, two examples are given to illustrate the validity of our results. Full article

Most of the fractal functions studied so far run through numerical values. Usually they are supported on sets of real numbers or in a complex field. This paper is devoted to the construction of fractal curves with values in abstract settings such as Banach spaces and algebras, with minimal conditions and structures, transcending in this way the numerical underlying scenario. This is performed via fixed point of an operator defined on a b-metric space of Banach-valued functions with domain on a real interval. The sets of images may provide uniparametric fractal collections of measures, operators or matrices, for instance. The defining operator is linked to a collection of maps (or iterated function system, and the conditions on these mappings determine the properties of the fractal function. In particular, it is possible to define continuous curves and fractal functions belonging to Bochner spaces of Banach-valued integrable functions. As residual result, we prove the existence of fractal functions coming from non-contractive operators as well. We provide new constructions of bases for Banach-valued maps, with a particular mention of spanning systems of functions valued on C*-algebras. Full article

This paper gives the null controllability for nonlocal stochastic differential inclusion with the Hilfer fractional derivative and Clarke subdifferential. Sufficient conditions for null controllability of nonlocal Hilfer fractional stochastic differential inclusion are established by using the fixed-point approach with the proof that the corresponding linear system is controllable. Finally, the theoretical results are verified with an example. Full article

Actuators made of dielectric elastomers are used in soft robotics for a variety of applications. However, due to their mechanical properties, they exhibit viscoelastic behaviour, especially in the initial phase of their performance, which can be observed in the first cycles of dynamic excitation. A fully fractional generalised Maxwell model was derived and used for the first time to capture the viscoelastic effect of dielectric elastomer actuators. The Laplace transform was used to derive the fully fractional generalised Maxwell model. The Laplace transform has proven to be very useful and practical in deriving fractional viscoelastic constitutive models. Using the global optimisation procedure called Pattern Search, the optimal parameters, as well as the number of branches of the fully fractional generalised Maxwell model, were derived from the experimental results. For the fully fractional generalised Maxwell model, the optimal number of branches was determined considering the derivation order of each element of the branch. The derived model can readily be implemented in the simulation of a dielectric elastomer actuator control, and can also easily be used for different viscoelastic materials. Full article