Fractional order maps are a hot research topic; many new mathematical models are suitable for developing new applications in different areas of science and engineering. In this paper, a new class of a 2D fractional hyperchaotic map is introduced using the Caputo-like difference operator. The hyperchaotic map has no equilibrium and lines of equilibrium points, depending on the values of the system parameters. All of the chaotic attractors generated by the proposed fractional map are hidden. The system dynamics are analyzed via bifurcation diagrams, Lyapunov exponents, and phase portraits for different values of the fractional order. The results show that the fractional map has rich dynamical behavior, including hidden homogeneous multistability and offset boosting. The paper also illustrates a novel theorem, which assures that two hyperchaotic fractional discrete systems achieve synchronized dynamics using very simple linear control laws. Finally, the chaotic dynamics of the proposed system are stabilized at the origin via a suitable controller.

This paper deals with chaos control in an arch microelectromechanical system (MEMS) from the fractional calculus perspective. There is a growing need for effective controllers in various technological fields, and it is important to consider disruptions, uncertainties, and control input limitations when designing a practical controller. To address this problem, we propose a novel disturbance-observer-based terminal sliding mode control technique for stabilizing and controlling chaos in a fractional-order arch MEMS resonator. The design of this technique takes into account uncertainty, disturbances, and control input saturation in the fractional-order system. The proposed control technique is practical for real-world applications because it includes control input saturation. The equation for a fractional-order arch MEMS resonator is presented, and its nonlinear vibration and chaotic behavior are studied. The design process for the proposed control technique is then described. The Lyapunov stability theorem is used to prove the finite-time convergence of the proposed controller and disturbance observer. The proposed controller is applied to the arch MEMS resonator, and numerical simulations are used to demonstrate its effectiveness and robustness for uncertain nonlinear systems. The results of these simulations clearly show the effectiveness of the proposed control technique.

In the past few years, the world has suffered from an untreated infectious epidemic disease (COVID-19), caused by the so-called coronavirus, which was regarded as one of the most dangerous and viral infections. From this point of view, the major objective of this intended paper is to propose a new mathematical model for the coronavirus pandemic (COVID-19) outbreak by operating the Caputo fractional-order derivative operator instead of the traditional operator. The behavior of the positive solution of COVID-19 with the initial condition will be investigated, and some new studies on the spread of infection from one individual to another will be discussed as well. This would surely deduce some important conclusions in preventing major outbreaks of such disease. The dynamics of the fractional-order COVID-19 mathematical model will be shown graphically using the fractional Euler Method. The results will be compared with some other concluded results obtained by exploring the conventional model and then shedding light on understanding its trends. The symmetrical aspects of the proposed dynamical model are analyzed, such as the disease-free equilibrium point and the endemic equilibrium point coupled with their stabilities. Through performing some numerical comparisons, it will be proved that the results generated from using the fractional-order model are significantly closer to some real data than those of the integer-order model. This would undoubtedly clarify the role of fractional calculus in facing epidemiological hazards.

A hydraulic generator regulating system with electrical, mechanical, and hydraulic constitution is a complex nonlinear system, which is analyzed in this research. In the present study, the dynamical behavior of this system is investigated. Afterward, the input/output feedback linearization theory is exerted to derive the controllable model of the system. Then, the chaotic behavior of the system is controlled using a robust controller that uses a Chebyshev neural network as a disturbance observer in combination with a non-singular robust terminal sliding mode control method. Moreover, the convergence of the system response to the desired output in the presence of uncertainty and unexpected disturbances is demonstrated through the Lyapunov stability theorem. Finally, the effectiveness and appropriate performance of the proposed control scheme in terms of robustness against uncertainty and unexpected disturbances are demonstrated through numerical simulations.

<abstract><p>In this research, we investigate the direct interaction acquisition method to create a stochastic computational formula of cholera infection evolution via the fractional calculus theory. Susceptible people, infected individuals, medicated individuals, and restored individuals are all included in the framework. Besides that, we transformed the mathematical approach into a stochastic model since it neglected the randomization mechanism and external influences. The descriptive behaviours of systems are then investigated, including the global positivity of the solution, ergodicity and stationary distribution are carried out. Furthermore, the stochastic reproductive number for the system is determined while for the case $ \mathbb{R}_{0}^{s} &gt; 1, $ some sufficient condition for the existence of stationary distribution is obtained. To test the complexity of the proposed scheme, various fractional derivative operators such as power law, exponential decay law and the generalized Mittag-Leffler kernel were used. We included a stochastic factor in every case and employed linear growth and Lipschitz criteria to illustrate the existence and uniqueness of solutions. So every case was numerically investigated, utilizing the newest numerical technique. According to simulation data, the main significant aspects of eradicating cholera infection from society are reduced interaction incidence, improved therapeutic rate, and hygiene facilities.</p></abstract>