#### Fractal and Fractional

Journal Information
EISSN: 25043110
Total articles ≅ 1,478

#### Latest articles in this journal

Published: 27 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040289

Abstract:
In recent years, cellular neural networks (CNNs) have become a popular apparatus for simulations in neuroscience, biology, medicine, computer sciences and engineering. In order to create more adequate models, researchers have considered memory effects, reaction–diffusion structures, impulsive perturbations, uncertain terms and fractional-order dynamics. The design, cellular aspects, functioning and behavioral aspects of such CNN models depend on efficient stability and control strategies. In many practical cases, the classical stability approaches are useless. Recently, in a series of papers, we have proposed several extended stability and control concepts that are more appropriate from the applied point of view. This paper is an overview of our main results and focuses on extended stability and control notions including practical stability, stability with respect to sets and manifolds and Lipschitz stability. We outline the recent progress in the stability and control methods and provide diverse mechanisms that can be used by the researchers in the field. The proposed stability techniques are presented through several types of impulsive and fractional-order CNN models. Examples are elaborated to demonstrate the feasibility of different technologies.
Published: 27 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040288

Abstract:
We propose an adaptive radial basis (RBF) neural network controller based on Lyapunov stability theory for uncertain fractional-order time-delay chaotic systems (FOTDCSs) with different time delays. The controller does not depend on the system model and can achieve synchronous control under the condition that nonlinear uncertainties and external disturbances are completely unknown. Stability analysis showed that the error system asymptotically tended to zero in combination with the relevant lemma. Numerical simulation results show the effectiveness of the controller.
Published: 26 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040287

Abstract:
Image encryption is increasingly becoming an important area of research in information security and network communications as digital images are widely used in various applications and are vulnerable to various types of attacks. In this research work, a color image cryptosystem that is based on multiple layers is proposed. For every layer, an encryption key and an S-box are generated and utilized. These are based on a four-dimensional (4D) dynamical Chen system of a fractional-order, the Mersenne Twister, OpenSLL, Rule 30 Cellular Automata and Intel’s MKL. The sequential application of Shannon’s ideas of diffusion and confusion three times guarantees a total distortion of any input plain image, thereby, resulting in a totally encrypted one. Apart from the excellent and comparable performance to other state-of-the-art algorithms, showcasing resistance to visual, statistical, entropy, differential, known plaintext and brute-force attacks, the proposed image cryptosystem provides an exceptionally superior performance in two aspects: a vast key space of ${2}^{1658}$ and an average encryption rate of $3.34$ Mbps. Furthermore, the proposed image cryptosystem is shown to successfully pass all the tests of the NIST SP 800 suite.
Published: 26 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040286

Abstract:
Alumina ceramics were obtained from three different alumina sources, A1–A3, with various rare-earth dopants (La2O3–La, Nd2O3–Nd, and Y2O3–Y), concentration levels (500 and 1000 ppm) and synthesizing routes (1500 °C, 1815 °C and cold plasma-P). Absorption (A) and density (ρ in text, rho in images) were measured, resulting in a complex, multivariate database. Principal Component Analysis (PCA) was run with the aim of deducing relationships between variables (alumina source, dopant level, thermal processing route, A and ρ), observations, and between variables and observations. A total of 206 Scanning Electron Microscopy (SEM) micrographs were recorded at various scales and the corresponding images were processed to quantify the microstructural features. Two techniques of edge detection were used; Fractal Dimension (FD) was calculated for each micrograph and results were compared. Various scales of the micrographs prevented us from using any other approach, such as simply measuring the grains or obtaining shape parameters. The initial database was extended by including FDs and PCA was run again. We found that plasma processing is positively correlated to A and negatively correlated to both temperature (T) and ρ; La ceramics have an opposite behavior to Y and Nd ceramics. FD successfully explained observations being correlated, mainly, to Y, Nd and, to a lesser extent, to La. FD proved that it is a reliable and simple approach to quantifying microstructural features when comparing highly different, noisy micrographs.
Published: 25 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040285

Abstract:
In this paper, we propose to quantitatively compare the loss of human lung health under the influence of the illness with COVID-19, based on the fractal-analysis interpretation of the chest-pulmonary CT pictures, in the case of small datasets, which are usually encountered in medical applications. The fractal analysis characteristics, such as fractal dimension and lacunarity measured values, have been utilized as an effective advisor to interpretation of pulmonary CT picture texture.
Published: 25 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040284

Abstract:
The purpose of this paper is to investigate the optimal control for fractional stochastic integrodifferential systems of order 1 < γ < 2. To ensure the existence and uniqueness of mild solutions, we first gather a novel list of requirements. Further, the existence of optimal control for the stated issue is given by applying Balder’s theorem. Additionally, we extend our existence outcomes with infinite delay. The outcomes are obtained via fractional calculus, Hölder’s inequality, the cosine family, stochastic analysis techniques, and the fixed point approach. The theory is shown by an illustration, as well.
Published: 24 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040283

Abstract:
Global demand for fossil fuels has increased the importance of flow measurement in the oil sector. As a result, a new submarket in the flowmeter business has opened up. To improve the accuracy of gamma-based two-phase flowmeters, this study employs time-feature extraction methods, a particle swarm optimization (PSO) based feature selection system, and an artificial neural network. This article proposes a fraction detection system that uses a 137Cs gamma source, two NaI detectors for recording the photons, and a Pyrex-glass pipe between them. The Monte Carlo N Particle method was used to simulate the geometry mentioned above. Thirteen time-domain features were extracted from the raw data recorded by both detectors. Optimal characteristics were identified with the help of PSO. This procedure resulted in the identification of eight efficient features. The input-output relationship was approximated using a Multi-Layer Perceptron (MLP) neural network. The innovation of the present research is in the use of a feature extraction technique based on the PSO algorithm to determine volume percentages, with results such as: (1) introducing eight appropriate time characteristics in determining volume percentages; (2) achieving an accuracy of less than 0.37 in root mean square error (RMSE) and 0.14 in mean square error (MSE) while predicting the volume fraction of components in a gas-liquid two-phase flow; and (3) reducing the calculation load. Utilizing optimization-based feature selection techniques has allowed for the selection of meaningful inputs, which has decreased the volume of computations while boosting the precision of the presented system.
Published: 24 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040282

Abstract:
Forecasting the dynamical behaviors of nonlinear systems over long time intervals represents a great challenge for scientists and has become a very active area of research. The employment of the well-known artificial recurrent neural networks (RNNs)-based models requires a high computational cost, and they usually maintain adequate accuracy for complicated dynamics over short intervals only. In this work, an efficient reservoir-computing (RC) approach is presented to predict the time evolution of the complicated dynamics of a fractional order hyperchaotic finance model. Compared with the well-known deep learning techniques, the suggested RC-based forecasting model is faster, more accurate for long-time prediction, and has a smaller execution time. Numerical schemes for fractional order systems are generally time-consuming. The second goal of the present study is to introduce a faster, more efficient, and simpler simulator to the fractional order chaotic/hyperchaotic systems. The RC model is utilized in a proposed RC-based digital image encryption scheme. Security analysis is carried out to verify the performance of the proposed encryption scheme against different types of statistical, KPA, brute-force, CCA, and differential attacks.
Published: 24 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040281

Abstract:
In this paper, the fixed-time multi-switch combination–combination synchronization (FTMSCCS) of fractional-order chaotic systems with uncertainties and external disturbances is studied. The appropriate sliding mode surface and controller are proposed based on a Lyapunov theorem, and fixed-time multi-switching combination–combination synchronizations between four fractional-order chaotic systems are realized. The Lyapunov function is designed to prove the feasibility of the controller theoretically, and the effectiveness and robustness of the synchronization mechanism are further verified by numerical simulations. The advantage of this article is that it extends fixed-time synchronization to multi-switch combination–combination synchronization, enabling synchronization for a limited time, while increasing the complexity of the synchronization mechanism and improving its confidentiality in communication applications.
Published: 24 March 2023
by MDPI
Fractal and Fractional, Volume 7; https://doi.org/10.3390/fractalfract7040280

Abstract:
In this work, the probability of return for random walks on $\mathbb{Z}$, whose increment is given by the k-bonacci sequence, is determined. Additionally, the Hausdorff, packing and box-counting dimensions of the set of these walks that return an infinite number of times to the origin are given. As an application, we study the return for tribonacci random walks to the first term of the tribonacci sequence.