Fractal and Fractional

In this study, a model-free  $P{I}^{\phi }$-sliding mode control ( $P{I}^{\phi }$-SMC) methodology is proposed to synchronize a specific class of chaotic fractional-order memristive neural network systems (FOMNNSs) with delays and input saturation. The fractional-order Lyapunov stability theory is used to design a two-level  $P{I}^{\phi }$-SMC which can effectively manage the inherent chaotic behavior of delayed FOMNNSs and achieve finite-time synchronization. At the outset, an initial sliding surface is introduced. Subsequently, a robust  $P{I}^{\phi }$-sliding surface is designed as a second sliding surface, based on proportional–integral (PI) rules. The finite-time asymptotic stability of both surfaces is demonstrated. The final step involves the design of a dynamic-free control law that is robust against system uncertainties, input saturations, and delays. The independence of control rules from the functions of the system is accomplished through the application of the norm-boundedness property inherent in chaotic system states. The soft actor-critic (SAC) algorithm based deep Q-Learning is utilized to optimally adjust the coefficients embedded in the two-level  $P{I}^{\phi }$-SMC controller’s structure. By maximizing a reward signal, the optimal policy is found by the deep neural network of the SAC agent. This approach ensures that the sliding motion meets the reachability condition within a finite time. The validity of the proposed protocol is subsequently demonstrated through extensive simulation results and two numerical examples. Full article

This paper derives the sharp bounds for Hermite–Hadamard inequalities in the context of Riemann–Liouville fractional integrals. A generalization of Jensen’s inequality called the Jensen–Mercer inequality is used for general points to find the new and refined bounds of fractional Hermite–Hadamard inequalities. The existing Hermite–Hadamard inequalities in classical or fractional calculus have been proved for convex functions, typically involving only two points as in Jensen’s inequality. By applying the general points in Jensen–Mercer inequalities, we extend the scope of the existing results, which were previously proved for two points in the Jensen’s inequality or the Jensen–Mercer inequality. The use of left and right Riemann–Liouville fractional integrals in inequalities is challenging because of the general values involved in the Jensen–Mercer inequality, which we overcame by considering different cases. The use of the Jensen–Mercer inequality for general points to prove the refined bounds is a very interesting finding of this work, because it simultaneously generalizes many existing results in fractional and classical calculus. The application of these new results is demonstrated through error analysis of numerical integration formulas. To show the validity and significance of the findings, various numerical examples are tested. The numerical examples clearly demonstrate the significance of this new approach, as using more points in the Jensen–Mercer inequality leads to sharper bounds. Full article

The ultrasonic vibration rock-breaking method has been successfully applied to hard rock due to its high efficiency and controllable energy, providing a novel approach for the development of a more efficient, intelligent, safe, and environmentally friendly reconstruction method for coal and rock reservoirs. By subjecting the rock to ultra-high frequency (>15 kHz) vibration load, rapid fatigue damage can be induced within a short period of time, thereby enhancing the extent of cracking in hard rock. In order to investigate the impact of high-frequency vibration excitation on coal cracking, this study conducted exploratory tests using an independently designed ultrasonic vibration excitation system. These tests were combined with nuclear magnetic resonance (NMR) and permeability measurements to compare and analyze the pore fracture structure and permeability changes in coal samples under resonant and non-resonant conditions. Additionally, multifractal characteristics of the coal samples were investigated. The results demonstrate that high-frequency vibration excitation leads to significant expansion of micropores and mesopores in coal samples. Moreover, there is a strong exponential relationship between coal porosity/permeability and excitation time. After 40 s of stimulation, both porosity and permeability increase by 32.4% and over 8400%, respectively; these increases are five times higher for resonance-state compared to non-resonance-state conditions. Furthermore, water-saturated coal samples exhibit multifractal characteristics in their NMR T₂ spectrum distribution, and multifractal parameters ΔD(q)and Δα show positive correlations with the proportion of mesoporous/macropores but negative correlations with the proportion of micropores; conversely, Δf shows an opposite trend relative to pore proportions. The pore structure of coal exhibits intricate multi-scale characteristics, and the heterogeneity at various scales is quantified through multifractal analysis. This study confirms the feasibility of utilizing high-frequency vibration excitation for cracking coal rocks while also providing valuable insights for further expanding its application. Full article

Event-triggering mechanisms reported in the existing prescribed-time (PT) control do not adequately capture the dynamic nature of network environments, and are not applied to distributed consensus tracking problems with unknown time delays. Therefore, designing a dynamic event-triggering mechanism is crucial to ensuring PT stability, even in the presence of unknown time delays. This article focuses on developing a dynamic event-triggering mechanism to achieve adaptive practical PT output-feedback consensus tracking for nonlinear uncertain multiagent systems with unknown time delays. Firstly, a delay-independent PT state observer using a time-varying gain function is designed to estimate the immeasurable states. Following this, a novel distributed delay-independent PT consensus tracking scheme is constructed, incorporating a dynamic event-triggered mechanism through the command-filtered backstepping approach. In this design, dynamic variables based on a time-varying gain function are developed to support the event-triggering mechanism, ensuring practical stability within the prescribed settling time. Consequently, the proposed output-feedback control protocol can achieve practical PT stability, meaning that consensus tracking errors are constrained to a neighborhood around zero within a pre-specified time, regardless of the initial states of the agents or design parameters, while also avoiding the Zeno phenomenon. Finally, the effectiveness of the proposed strategy is validated through an illustrative example, which includes a comparative analysis. Full article

In this paper, a novel fractal model for the contact resistance based on axisymmetric sinusoidal asperity is proposed, which focuses on the resistance characteristics of the rough interface at a microscopic scale. By introducing the unique geometric shape of axisymmetric sinusoidal asperity, and combining it with a three-dimensional fractal theory, the micro-morphology characteristics of the rough interface can be characterized more precisely. Subsequently, by conducting a theoretical analysis and numerically solving the deformation mechanisms of asperities on the rough interface, a refined model for contact resistance is constructed. This research comprehensively employs theoretical analysis, numerical simulation, and experimental testing methods to deeply explore the current transmission mechanisms during the contact process of the rough interface. The findings suggest that the proposed model is capable of precisely capturing the intricate interplay of various factors, including contact area, contact load, and material properties, with the contact resistance. Compared to the existing models, the presented model demonstrates significant advantages in terms of prediction accuracy and practicality. This research provides an important theoretical basis and design guidance for optimizing the electrical performance of the rough interface, which has great significance for engineering applications. Full article

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem. Full article

This work aims to comprehend the dynamics of neurodegenerative disease using a mathematical model of fractional-order yeast prions. In the context of the Caputo fractional derivative, we here study and examine the solution of this model using the Predictor–Corrector approach. An analysis has been conducted on the existence and uniqueness of the selected model. Also, we examined the model’s stability and the existence of equilibrium points. With the purpose of analyzing the dynamics of the Sup35 monomer and Sup35 prion population, we displayed the graphs to show the obtained solutions over time. Graphical simulations show that the behaviour of the populations can change based on fractional orders and threshold parameter values. This work may present a good example of how biological theories and data can be better understood via mathematical modelling. Full article

In this article, we develop multiplicative fractional versions of Simpson’s and Newton’s formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature. Full article

The study investigates the nonlinear contagion, tail dependence, and Granger causality relations with TAR-TR-GARCH–copula causality methods for daily Bitcoin, Fintech, energy consumption, and CO₂ emissions in addition to examining these series for entropy, long-range dependence, fractionality, complexity, chaos, and nonlinearity with a dataset spanning from 25 June 2012 to 22 June 2024. Empirical results from Shannon, Rényi, and Tsallis entropy measures; Kolmogorov–Sinai complexity; Hurst–Mandelbrot and Lo’s R/S tests; and Phillips’ and Geweke and Porter-Hudak’s fractionality tests confirm the presence of entropy, complexity, fractionality, and long-range dependence. Further, the largest Lyapunov exponents and Hurst exponents confirm chaos across all series. The BDS test confirms nonlinearity, and ARCH-type heteroskedasticity test results support the basis for the use of novel TAR-TR-GARCH–copula causality. The model estimation results indicate moderate to strong levels of positive and asymmetric tail dependence and contagion under distinct regimes. The novel method captures nonlinear causality dynamics from Bitcoin and Fintech to energy consumption and CO₂ emissions as well as causality from energy consumption to CO₂ emissions and bidirectional feedback between Bitcoin and Fintech. These findings underscore the need to take the chaotic and complex dynamics seriously in policy and decision formulation and the necessity of eco-friendly technologies for Bitcoin and Fintech. Full article

Little research has been carried out in terms of modeling and control of analgesia. However, emerging new technology and recent prototypes paved the way for several ideas on pain modeling for control. Recently, such an idea has been proposed for measuring the Depth of Analgesia (DoA). In this paper, that solution is further exploited towards obtaining a novel fractional-order model and dedicated controller for DoA. First, clinical data from patients undergoing general anesthesia are used to determine a commensurate fractional-order model of the skin impedance at each sampling period. Second, we provide a proof of concept indicating that fractional order changes due to variations in the infused opioid drug (Remifentanil). Third, a fractional-order model for DoA is developed correlating the changes in the pain index (as the output signal) and the Remifentanil infusion rate (as the input signal). Standard optimization routines are used to estimate the parameters. A database of 19 real patients is used. Lastly, a preliminary fractional-order controller is designed and tested in simulation for the 19 patients. The closed-loop simulation results correspond to the expected clinical outcomes. Full article

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front. Full article

In this article, a diffusion component in an SIR model is introduced, and its impact is analyzed using fractional calculus. We have included the diffusion component in the SIR model. in order to illustrate the variations. Here, we have applied the general fractional derivative to analyze the impact. The Laplace decomposition technique is employed to obtain the numerical outcomes of the model. In order to observe the effect of the diffusion component in the SIR model, graphical solutions are also displayed. Full article

This study aims to predict the carbon sequestration capacity of Chinese grasslands to address climate change and achieve carbon neutrality goals. Grassland carbon sequestration is a crucial part of the global carbon cycle. However, its capacity is significantly impacted by climate change and human activities, making its dynamic changes complex and challenging to predict. This study adopts a fractional-order accumulation grey model, using 11 provinces in China as samples, to analyze and forecast grassland carbon sequestration. The study finds significant differences in grassland carbon sequestration trends across the sample regions. The carbon sequestration capacity of the grasslands in Xizang (Tibet) and Heilongjiang province is increasing, while it is decreasing in other provinces. The varying prediction results are influenced not only by regional climatic and natural conditions, but also by human interventions such as overgrazing, irrational reclamation, excessive mineral resource exploitation, and increased tourism development. Therefore, more region-specific grassland management and protection strategies should be formulated to enhance the carbon sequestration capacity of grasslands and promote the sustainable development of ecosystems. The significance of this study lies not only in providing scientific guidance for the protection and sustainable management of Chinese grasslands, but also in contributing theoretical and practical insights into global carbon sequestration strategies. Full article

In this work, some properties of the general convolutional operators of general fractional calculus (GFC), which satisfy analogues of the fundamental theorems of calculus, are described. Two types of general fractional (GF) operators on a finite interval exist in GFC that are conventionally called the L-type and T-type operators. The main difference between these operators is that the additivity property holds for T-type operators and is violated for L-type operators. This property is very important for the application of GFC in physics and other sciences. The presence or violation of the additivity property can be associated with qualitative differences in the behavior of physical processes and systems. In this paper, we define L-type line GF integrals and L-type line GF gradients. For these L-type operators, the gradient theorem is proved in this paper. In general, the L-type line GF integral over a simple line is not equal to the sum of the L-type line GF integrals over lines that make up the entire line. In this work, it is shown that there exist two cases when the additivity property holds for the L-type line GF integrals. In the first case, the L-type line GF integral along the line is equal to the sum of the L-type line GF integrals along parts of this line only if the processes, which are described by these lines, are independent. Processes are called independent if the history of changes in the subsequent process does not depend on the history of the previous process. In the second case, we prove the additivity property holds for the L-type line GF integrals, if the conditions of the GF gradient theorems are satisfied. Full article

In this work, novel Ostrowski-type inequalities for dissimilar function classes and generalized fractional integrals (FITs) are presented. We provide a useful identity for differentiable functions under FITs, which results in special expressions for functions whose derivatives have convex absolute values. A new condition for bounded variation functions is examined, as well as expansions to bounded and Lipschitzian derivatives. Our comprehension is improved by comparison with current findings, and recommendations for future study areas are given. Full article

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. Full article

Reactive power dispatch (RPD) in electric power systems, integrated with renewable energy sources, is gaining popularity among power engineers because of its vital importance in the planning, designing, and operation of advanced power systems. The goal of RPD is to upgrade the power system performance by minimizing the transmission line losses, enhancing voltage profiles, and reducing the total operating costs by tuning the decision variables such as transformer tap setting, generator’s terminal voltages, and capacitor size. But the complex, non-linear, and dynamic characteristics of the power networks, as well as the presence of power demand uncertainties and non-stationary behavior of wind generation, pose a challenging problem that cannot be solved efficiently with traditional numerical techniques. In this study, a new fractional computing strategy, namely, fractional hybrid particle swarm optimization (FHPSO), is proposed to handle RPD issues in electric networks integrated with wind power plants (WPPs) while incorporating the power demand uncertainties. To improve the convergence characteristics of the Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA), the proposed FHPSO incorporates the concepts of Shannon entropy inside the mathematical model of traditional PSOGSA. Extensive experimentation validates FHPSO effectiveness by computing the best value of objective functions, namely, voltage deviation index and line loss minimization in standard power systems. The proposed FHPSO shows an improvement in percentage of 61.62%, 85.44%, 86.51%, 93.15%, 84.37%, 67.31%, 61.64%, 61.13%, 8.44%, and 1.899%, respectively, over ALC_PSO, FAHLCPSO, OGSA, ABC, SGA, CKHA, NGBWCA, KHA, PSOGSA, and FPSOGSA in case of traditional optimal reactive power dispatch(ORPD) for IEEE 30 bus system. Furthermore, the stability, robustness, and precision of the designed FHPSO are determined using statistical interpretations such as cumulative distribution function graphs, quantile-quantile plots, boxplot illustrations, and histograms. Full article

This paper investigates the problem of observer-based control for a class of nonlinear systems described by the Caputo–Hadamard fractional-order derivative. Given the growing interest in fractional-order systems for their ability to capture complex dynamics, ensuring their practical stability remains a significant challenge. We propose a novel concept of practical stability tailored to nonlinear Hadamard fractional-order systems, which guarantees that the system solutions converge to a small ball containing the origin, thereby enhancing their robustness against perturbations. Furthermore, we introduce a practical observer design that extends the classical observer framework to fractional-order systems under an enhanced One-Sided Lipschitz (OSL) condition. This extended OSL condition ensures the convergence of the proposed practical observer, even in the presence of significant nonlinearities and disturbances. Notably, the novelty of our approach lies in the extension of both the practical observer and the stability criteria, which are innovative even in the integer-order case. Theoretical results are substantiated through numerical examples, demonstrating the feasibility of the proposed method in real-world control applications. Our contributions pave the way for the development of robust observers in fractional-order systems, with potential applications across various engineering domains. Full article

We utilize Lyapunov exponents to quantitatively assess the hyperchaos and categorize the limit sets of complex dynamical systems. While there are numerous methods for computing Lyapunov exponents in integer-order systems, these methods are not suitable for fractional-order systems because of the nonlocal characteristics of fractional-order derivatives. This paper introduces innovative eight-dimensional chaotic systems that investigate fractional-order dynamics. These systems exploit the memory effect inherent in the Grünwald–Letnikov (G-L) derivative. This approach enhances the system’s applicability and compatibility with traditional integer-order systems. An 8D Chen’s fractional-order system is utilized to showcase the effectiveness of the presented methodology for hyperchaotic systems. The simulation results demonstrate that the proposed algorithm outperforms existing algorithms in both accuracy and precision. Moreover, the study utilizes the 0–1 Test for Chaos, Kolmogorov–Sinai (KS) entropy, the Kaplan–Yorke dimension, and the Perron Effect to analyze the proposed eight-dimensional fractional-order system. These additional metrics offer a thorough insight into the system’s chaotic behavior and stability characteristics. Full article