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The Master Stability Function (MSF) is a crucial tool for understanding the synchronization behavior of coupled nonlinear oscillators. In recent years, the field of fractional calculus and its applications in nonlinear dynamics gained significant attention, leading to an expanded focus on the network dynamics of such systems. This study aims to explore this emerging area by deriving the MSF for coupled fractional-order nonlinear oscillators and investigating their relationship with coupling strength and fractional order. To provide a comprehensive comparison, we utilize the well-known nonlinear oscillators to study the differences and similarities between integer-order and fractional-order MSFs. Our analysis reveals that, similar to integer-order systems, fractional-order coupled nonlinear oscillators exhibit MSFs that can be characterized by the presence of negative values within a finite interval of the normalized coupling parameter. This negative region is crucial as it indicates stable synchronization. Furthermore, we categorize the fractional-order MSFs using the same classifications applied to integer-order MSFs. This classification helps in systematically understanding and comparing the synchronization properties of both types of systems. Our findings are supported by extensive numerical simulations, which demonstrate that the majority of fractional-order coupled oscillators exhibit higher classes of MSF. This higher classification suggests that fractional-order systems have a superior ability to achieve synchronization compared to their integer-order counterparts. In summary, this study underscores the significance of fractional calculus in enhancing our understanding of synchronization in complex systems. The derived MSF for fractional-order nonlinear oscillators provides valuable insights into their dynamic behavior and opens new avenues for research in various scientific and engineering disciplines.

This research focuses on studying the influence of the Hall current on the propagation and reflection of elastic waves in a non-local isotropic rotating solid. The dispersion relation is derived to determine the speed of propagation, revealing the presence of three coupled quasi-waves within the solid: coupled qP-wave, qT-wave and qSV-wave. The rotational motion and the Hall current introduce anisotropic characteristics to the medium, leading to the emergence of quasi-type waves. The rotation disrupts the isotropic nature of the solid, transforming it into an anisotropic medium. Consequently, the purely longitudinal and transverse waves are converted into quasi-longitudinal and quasi-transverse waves. The speed of the propagating waves is dependent on specific elastic parameters. By employing free boundary conditions, the mathematical calculation and graphical representation of wave amplitude ratios are determined. The influence of rotational frequency, non-locality, fractional order and Hall current parameters on the computed results is investigated. The conservation of energy validates the accuracy of the obtained results. Furthermore, it is observed that the previously reported results in the literature can be obtained as a special case when rotation and the Hall current are absent.

In this work, a Liouville–Caputo fractional order (FO) derivative for the mathematical system based on the accelerating universe in the modified gravity (AUMG), i.e. FO-AUMG is proposed to get more accurate solutions. The nonlinear dynamics of the FO-AUMG is classified into five dynamics. The performances of the designed nonlinear FO-AUMG are numerically stimulated with the stochastic procedures of Levenberg–Marquardt backpropagated (LMB) scheme-based neural networks. The statics for FO-AUMS is used for the nonlinear FO-AUMG as 72%, 16% and 12% for training, authorization, and testing. Twenty neurons in hidden layers have been used to approximate the solution of the nonlinear FO-AUMS. The comparison of three different cases of the nonlinear FO-AUMS is performed with dataset generated by Adams method. To validate the uniformity, legitimacy, precision, and competence of LMB-based adaptive neural networks, the outcomes of the state transitions parameters, regression, correlation, error-histogram plots have been exploited.

Chaos synchronization of the Duffing, Lorenz and Rössler systems with fractional orders are studied theoretically and numerically. Three methods are applied in this paper: combination of active-passive decomposition (APD) and one-way coupling methods, Pecora–Carroll method, bidirectional coupling method. The sufficient conditions of achieving synchronization between two identical fractional systems are derived by using the Laplace transform theory. Numerical simulations demonstrate the effectiveness of the proposed synchronization schemes for these fractional systems.

In this paper, lag full state hybrid projective synchronization (LFSHPS) in fractional-order chaotic systems is first studied. We show that LFSHPS does exist in fractional-order chaotic systems. Based on active control theory, synchronization schemes for LFSHPS of the fractional-order chaotic systems are given. Numerical simulations are provided to illustrate and verify the effectiveness of the proposed methods.

Based on the stability theory of fractional order systems, an effective but theoretically rigorous nonlinear control method is proposed to synchronize the fractional order chaotic systems. Using this method, chaos synchronization between two identical fractional order unified systems is studied. Simulation results are shown to illustrate the effectiveness of this method.

In the paper, generalized chaotic synchronization of a class of fractional order systems is studied. Based on the stability theory of linear fractional order systems, a generalized synchronization scheme is presented, and theoretical analysis is provided to verify its feasibility. The proposed method can realize generalized synchronization not only of fractional order systems with same dimension, but also of systems with different dimensions. Besides, the function relation of generalized synchronization can be linear or nonlinear. Numerical simulations show the effectiveness of the scheme.

A new fractional order chaotic n-scroll modified Chua circuit is introduced. It can generate n-scroll with a total order less than three. The equilibrium points are classified into two types according to the characteristics of the eigenvalues of the Jacobian matrix at the equilibrium points. To overcome some disadvantages of the traditional fractional order controller, a new fractional order control method is developed for stabilizing the system to any expected equilibrium point. Numerical examples are provided to verify the effectiveness of the proposed scheme.

In this paper, based on the fact that the inductors and capacitors are of fractional order in nature, the four-order mathematical model of the fractional order quadratic Boost converter in type I and type II discontinuous conduction mode (DCM) — DCM is established by using fractional calculus theory. Direct current (DC) analysis is conducted by using the DC equivalent model and the transfer functions are derived from the corresponding alternating current (AC) equivalent model. The DCM–DCM regions of type I and type II are obtained and the relations between the regions and the orders are found. The influence of the orders on the performance of the quadratic Boost converter in DCM–DCM is verified by numerical and circuit simulations. Simulation results demonstrate the correctness of the fractional order model and the efficiency of the proposed theoretical analysis.

In this paper, the memristor-based fractional-order neural networks (MFNN) with delay and with two types of stabilizing control are described in detail. Based on the Lyapunov direct method, the theories of set-value maps, differential inclusions and comparison principle, some sufficient conditions and assumptions for global stabilization of this neural network model are established. Finally, two numerical examples are presented to demonstrate the effectiveness and practicability of the obtained results.

This paper investigates the dynamic evolution characteristics of the hydropower station by introducing the fractional order damping forces. A careful analysis of the dynamic characteristics of the generator shaft system is carried out under different values of fractional order. It turns out the vibration state of the axis coordinates has a certain evolution law with the increase of the fractional order. Significantly, the obtained law exists in the horizontal evolution and vertical evolution of the dynamical behaviors. Meanwhile, some interesting dynamical phenomena were found in this process. The outcomes of this study enrich the nonlinear dynamic theory from the engineering practice of hydropower stations.

In this paper, the issue of synchronization and anti-synchronization for fractional-delayed memristor-based chaotic system is studied by using active control strategy. Firstly, some explicit conditions are proposed to guarantee the synchronization and anti-synchronization of the proposed system. Secondly, the influence of order and time delay on the synchronization (anti-synchronization) is discussed. It reveals that synchronization (anti-synchronization) is faster as the order increases or the time delay decreases. Finally, some numerical simulations are presented to verify the validity of our theoretical analysis.

In order to depict the effect of driver’s memory on car-following behavior, a new kind of car-following model is proposed by using fractional order differential equation in this paper. Its dynamic equation is defined by Caputo fractional order derivative. And the order of derivative is the measurement of driver’s memory. In addition, discrete formulas of the position and velocity of the new model are given. The Optimal Velocity (OV) model is taken as an example to introduce how to get the fractional order car-following model from an ordinary model. The simulation results show that the Fractional Order Optimal Velocity (FOOV) model is more stable, and it can avoid unrealistic acceleration values of the OV model in the cases of starting and braking processes. Moreover, magnitudes of the speed and headway fluctuation of the FOOV model with a suitable order are smaller than those of the OV model. This indicates that the memory characteristic of drivers increases the stability of traffic flow.

In this paper, a fractional-order (and an integer-order) chaotic system, obtained from Chua’s circuit by substituting Chua’s diode with two active coupled memristors (MRs) characterized by quadratic nonlinearity, is introduced to probe the memristive coupling effect. Two MRs connected in parallel are coupled by the flux. For the integer-order memristive system, the dynamical characteristics depending on the coupling strength coefficient between MRs without changing the circuit parameters are illustrated theoretically and numerically by using phase portraits, time domain diagram, bifurcation diagram and the Lyapunov diagram. Then based on the Adams–Bashforth–Moulton algorithm, the study of dynamic behavior of the fractional-order memristive system containing the time-delay reveals that appropriately setting the coupling strength between MRs generates more interesting attractors that differ from its integer-order counterpart. Besides, the effects of the order and the time-delay on dynamics are discussed briefly. Finally, the simulation results verify the validity of the theoretical analysis.

The traditional fractional-order particle swarm optimization (FOPSO) algorithm depends on the fractional order $\alpha$, and it is easy to fall into local optimum. To overcome these disadvantages, a novel perspective with PID gains tuning procedure is proposed by combining the time factor with FOPSO, i.e. a new fractional-order particle swarm optimization called TFFV-PSO, which reduces the dependence on the fractional order to enhance the ability of particles to escape from local optimums. According to its influence on the performance of the algorithm, the time factor is varied with population diversity parameters to balance the exploration and exploitation capabilities of the particle swarm, so as to adjust the convergence speed of the algorithm, then it follows that a better convergence performance will be obtained. The improved method is tested on several benchmark functions and applied to tune the PID controller parameters. The experimental results and the comparison with previous other methods show that our proposed TFFV-PSO provides an adequate velocity of convergence and a satisfying accuracy, as well as even better robustness.

In this paper, a variable threshold voltage metal oxide semiconductor (VTMOS) field effect transistor is used to improve an ultra-low voltage, ultra-low power current conveyor transconductance amplifier (CCTA). To achieve the desired result, an analytical subthreshold VTMOS model is used. Designs that utilize the TSMC 0.18 μm technology are verified using PSPICE simulation. The power consumption is simply 0.12 μW at a ± 0.2 V supply voltage. The proposed CCTA is synthesized using fractional-order (FO) universal filters that can simultaneously realise low pass (LP), high pass (HP) and bandpass (BP) responses with independent control of quality factor and pole frequency by transconductances (g_m). Moreover, the circuit has low input and high output impedance which would be an ideal choice for cascading in current-mode circuit. The FO filters are constructed using two FO capacitors of orders α and β (0 < α, β ≤ 1). The FO filters provide improved performance in terms of pole frequency compared with conventional-order filters. The filter has a low power consumption of 0.71 μW at a ± 0.2 V supply voltage. The validity of the proposed filter is verified through PSPICE simulations.

This paper presents design of electronically reconfigurable fractional-order filter that is able to be configured to operate as fractional-order low-pass filter (FLPF) or fractional-order high-pass filter (FHPF). Its slope of attenuation between pass band and stop band, i.e., order of the filter, is electronically adjustable in the range between 1 and 2. Also, pole frequency can be electronically controlled independently with respect to other tuned parameters. Moreover, particular type of approximation can be also controlled electronically. This feature set is available both for FLPF and FHPF-type of response. Presented structure of the filter is based on well-known follow-the-leader feedback (FLF) topology adjusted in our case for utilization with just simple active elements operational transconductance amplifiers (OTAs) and adjustable current amplifiers (ACAs), both providing possibility to control its key parameter electronically. This paper explains how reconfigurable third-order FLF topology is used in order to approximate both FLPF and FHPF in concerned frequency band of interest. Design is supported by PSpice simulations for three particular values of order of the filter (1.25, 1.5, 1.75), for several values of pole frequency and for two particular types of approximation forming the shape of both the magnitude and phase response. Moreover, theoretical presumptions are successfully confirmed by laboratory measurements with prepared prototype based on behavioral modeling.

Many practical systems, such as thermal system, economic system and electric power system, can be more accurately described by the fractional-order system rather than integer-order system. Therefore, it is an important topic to study the fractional-order system and estimate its parameters. The problem of parameter estimation is essentially a multi-dimensional parameter optimization problem. In this paper, according to the average value of position information, an improved Tent mapping and a piecewise mutation probability, a modified particle swarm optimization (MPSO) algorithm is presented to solve the parameter estimation problem. The performance of MPSO is tested with eight benchmark functions, which proves the effectiveness of the algorithm. Based on the double-dispersion Cole model, the proposed MPSO algorithm is used to estimate the parameters for the generated simulated datasets. Experimental results show that the MPSO algorithm for parameters identification of the Cole model is an effective and promising method with high accuracy and good robustness.

The passive synthesis of the fractional-order driving-point immittance functions is of great significance for the modeling and design of fractional-order circuits. This paper studies the synthesis of a class of two-element-order fractional immittance functions based on variable substitution. First, the method of realizing the two-variable immittance functions corresponding to this kind of two-element-order fractional immittance functions is established; second, we perform the inverse substitution of variables to realize the two-element-order fractional immittance network; finally, the synthesis program of this kind of two-element-order fractional immittance functions is proposed, and specific examples are given for simulation verification. This synthesis method overcomes the disadvantages of using a multi-port transformer, and is more conducive to network analysis.

In recent years, there has been expanding research on the applications of fractional calculus to the areas of signal processing, modeling and controls. Analog circuit implementation of chaotic systems is used in studying nonlinear dynamical phenomena, which is also applied in realizing the controller development. In this paper, chain fractance and tree fractance circuits are constructed to realize the fractional-order Chen–Lee system. The results are in good agreement with those obtained from numerical simulation. This study shows that not only is this system related to gyro motion but can also be applied to electronic circuits for secure communication.