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The bifurcation phenomenon generated in neurons is considered to be the cause of many neurological diseases, thus, lots of investigators have been dedicated to bifurcation control, but most of them only focus on the integer-order neuronal models and control methods. Recently, increasing number of research works have shown that the differential equations of fractional-order are more appropriate for description of the electrical properties of certain nerve cell membranes. In this paper, we employ a fractional-order washout filter to realize bifurcation control for fractional-order Morris–Lecar neuronal model. Computer simulation verifies the effectiveness and improved performance of the proposed method than the integer-order washout filter. In addition, the analysis of the fractional-order controller’s impacts on the model is also presented.

In this paper, based on the idea of nonlinear observer, lag synchronization of chaotic fractional system with commensurate and incommensurate order is studied by the stability theorem of linear fractional-order systems. The theoretical analysis of fractional-order systems in this paper is a systematic method. This technique is applied to achieve the lag synchronization of fractional-order Rössler's system, verified by numerical simulation.

In this paper, a chaotic synchronization scheme is proposed to achieve the generalized synchronization between two different fractional-order chaotic systems. Based on the stability theory of fractional-order systems and the pole placement technique, a controller is designed and theoretical proof is given. Two groups of examples are shown to verify the effectiveness of the proposed scheme, the first one is to realize the generalized synchronization between the fractional-order Chen system and the fractional-order Rössler system, the second one is between the fractional-order Lü system and the fractional-order hyperchaotic Lorenz system. The corresponding numerical simulations verify the effectiveness of the proposed scheme.

This paper mainly concerns with the finite-time synchronization of delayed fractional-order quaternion-valued memristor-based neural networks (FQVMNNs). First, the FQVMNNs are studied by separating the system into four real-valued parts owing to the noncommutativity of quaternion multiplication. Then, two state feedback control schemes, which include linear part and discontinuous part, are designed to guarantee that the synchronization of the studied networks can be achieved in finite time. Meanwhile, in terms of the stability theorem of delayed fractional-order systems, Razumikhin technique and comparison principle, some novel criteria are derived to confirm the synchronization of the studied models. Furthermore, two methods are used to obtain the estimation bounds of settling time. Finally, the feasiblity of the synchronization methods in quaternion domain is validated by the numerical examples.

In this paper, the memristor-based fractional-order neural networks with time delay are analyzed. Based on the theories of set-value maps, differential inclusions and Filippov’s solution, some sufficient conditions for asymptotic stability of this neural network model are obtained when the external inputs are constants. Besides, uniform stability condition is derived when the external inputs are time-varying, and its attractive interval is estimated. Finally, numerical examples are given to verify our results.

Compared to the integer-order systems, the system characteristics of the fractional system are closer to the system characteristics of the real engineering system, the study found beyond that, strictly speaking, various physical phenomena in nature are nonlinear. The problem of parameter estimation problem of fractional-order nonlinear systems can be transformed into the problem of parameter optimization problem by constructing an appropriate fitness function. This paper proposes a hybrid improvement algorithm based on whale optimization algorithm (WOA) to solve this problem and verify it both in Lorenz system and Lu system. The simulation result shows that the hybrid improved algorithm is superior to genetic algorithm (GA), particle swarm optimization (PSO), grasshopper optimization algorithm (GOA) and WOA in convergence speed and accuracy.

A novel fractional-order adaptive non-singular terminal sliding mode control (FONTSMC) method is investigated for the synchronization of two nonlinear fractional-order chaotic systems in the presence of external disturbance. The proposed controller consists of a fractional-order non-singular terminal sliding mode surface and an adaptive gain adjusted with sliding surface. Based on Lyapunov stability theory and stability theorem for fractional-order dynamic systems, the controlled system’s stable synchronization is guaranteed. A dual-channel secure communication system is presented to transmit useful signals based on the proposed synchronization controller. Finally, numerical simulations and comparison with fractional-order PID controller, fractional-order PD sliding mode controller and adaptive terminal sliding mode controller are given to demonstrate the effectiveness and the robustness of the proposed FONTSMC control. The application of the proposed synchronization method is studied in the dual-channel secure communication.

This paper addresses the finite-time synchronization problem for fractional-order memristor-based neural networks (FMNNs) with discontinuous activations, in which multiple delays are considered. Firstly, on the basis of set-valued mapping as well as differential inclusions theory, the synchronization issue of drive-response systems is considered as the stabilization of the error system. Then, the state feedback controllers, which contain both discontinuous part and time-delayed part, are designed to analyze the finite-time synchronization of the concerned network model. Making use of the stability theorem of fractional-order systems with multiple time delays, some fractional derivative inequalities and comparison theorem, several sufficient criteria are established for confirming that the synchronization error of the concerned system can reach zero within a limited time. Additionally, the settling time can be optimized by adjusting controller parameter. Finally, the effectiveness of synchronization strategies is validated through the simulation results.

In this paper, a fractional-order hyperchaotic system based on dual memristors is represented to analyze the nonlinear dynamic behaviors via replacing two coupled resistors with dual memristors. It is worthy to note that the fractional-order hyperchaotic system has two zero eigenvalues and three nonzero eigenvalues, therefore the equilibrium plane can be separated into diverse areas which indicates the hyperchaotic system is stable or chaotic. The simulation results illustrate that the initial states have a significant impact on the dynamic behaviors, which can be mirrored by the phase portraits, the bifurcation diagrams, the power spectrum and the time-domain waveform. In particular, the memristor initial boosting phenomenon is investigated in the proposed hyperchaotic system, which implies the memristor initial states determine the attractor offset boosting behaviors under various initial controllers. Clearly, it differs from the variable offset boosting behavior totally due to its multi-dimension and nonlinearity. Finally, numerical results under different initial controllers are exhibited to demonstrate the memristor initial boosting phenomenon. Moreover, a hardware circuit based on PSPICE software is fabricated and its experimental simulations is given to verify the dynamic behaviors effectively.

The dynamic behaviors for fractional-order Cohen–Grossberg neural networks with time-varying delays (FCGNND) are studied in this paper. By introducing the Mittag-Leffler (ML) function, based on properties of fractional calculus, the differential mean-value theorem and Arzela–Ascoli theorem, we give some sufficient theorems to determine the boundedness, global Mittag-Leffler stability (GMLS) and global asymptotical $\omega$-periodicity (GAP) for FCGNND. Finally, a numerical example is given to verify the effectiveness of the theorems.

A proposal of a fractional ($1+\alpha$)-order low-pass filter is presented in this paper. The proposed filter operates in the current-mode and it is designed using Multi-Output Current Followers (MO-CFs), Dual-Output Current Follower (DO-CF), Dual-Output Adjustable Current Amplifier (DO-ACA) and Adjustable Current Amplifiers (ACAs) as active elements within the presented topology of the filter. The filter possesses ability to electronically control its order and also the pole frequency by changing the current gain of current amplifiers (ACAs) already present in the structure. Three different values of the order and pole frequency of the proposed low-pass filter were tested as an example. Design of the proposed filter is supported by simulation and experimental results. Simulations of the circuit are carried out in PSPICE simulator with behavioral models of used active elements. The experimental laboratory measurements are performed with the help of available devices forming equivalent circuits. Simulations and experimental results of the electronical control of the order and pole frequency are compared in this contribution.

In this paper, a systematic design is proposed to generate multi-scroll attractors with hyperchaotic behavior using fractional-order systems, in which switched SC-CNN is triggered with error function. Sprott Systems Case H is reconstructed with fractional-order switched SC-CNN system. Herein, the goal is to increase the complexity of chaotic signals, hence providing safer and reliable communication by generating multi-scroll attractors with hyperchaotic behavior using fractional-order systems. Theoretical analysis of the proposed system’s dynamical behaviors is scrutinized, while numerical investigations are carried out with equilibrium points, Lyapunov exponent, bifurcation diagrams, Poincaré mapping and 0/1 test methods. Numerical results are validated through simulations and on an FPAA platform.

A fractional low-pass filter operating in a low-frequency range is necessary for the filtering of biomedical signals. Thus, we propose a fractional low-pass filter of order ($1+\alpha$) which is implemented using a current follower transconductance amplifier (CFTA). The presented structure is compact. It comprises of five CFTAs along with three grounded capacitors and two resistors. Additionally, this filter structure can be electronically tuned for its order and frequency variation, and these tunings are independent of each other. This electronic tuning is established through the bias current of the active component used. The layout of the proposed filter was designed in Cadence Virtuoso, covering 7920$\mu$m² of chip area. It is operating at ±900mV with a power consumption of 6.8mW. In the simulation results, both pre-layout and post-layout results are included, which indicates that the design is appropriate for fabrication. To check robustness, PVT analysis, Monte-Carlo analysis and THD are also performed. The proposed circuit has also been tested through experiment and its results are also presented. The proposed filter is used to implement a Leaky-Integrate-and-Fire neuron model.

This paper aims to investigate the phenomenon of Diffusionless Lorenz system with fractional-order. We discuss the stability of equilibrium points of the fractional-order system theoretically, and analyze the chaotic behaviors and typical bifurcations numerically. We find rich dynamics in fractional-order Diffusionless Lorenz system with appropriate fractional order and system parameters. Besides, the control problem of fractional-order Diffusionless Lorenz system is examined using feedback control technique, and simulation results show the effectiveness of the method.

On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.

In this paper, the problems of stability and Hopf bifurcation in a class of fractional-order complex-valued single neuron model with time delay are addressed. With the help of the stability theory of fractional-order differential equations and Laplace transforms, several new sufficient conditions, which ensure the stability of the system are derived. Taking the time delay as the bifurcation parameter, Hopf bifurcation is investigated and the critical value of the time delay for the occurrence of Hopf bifurcation is determined. Finally, two representative numerical examples are given to show the effectiveness of the theoretical results.

This paper addresses an important issue in numerical integration of dynamical systems, integer- or fractional-order, with discontinuous vector fields. It is shown that these systems cannot be solved using numerical methods designed for ODEs with continuous functions on the right-hand side, therefore we have to resort to special schemes and procedures in numerical integrations such as continuous approximations of the right-hand sides of the ODEs.

In this paper, we announce a novel 4D chaotic system which belongs to the self-excited attractor and hidden attractor family depending on the parameter values. Lyapunov exponents, bifurcation diagram and bicoherence plot of the CAMO (Camouflage) chaotic system are investigated. Also, fractional-order model of the proposed CAMO system (FOCAMO) is derived and analyzed. FOCAMO chaotic system is then implemented in Field Programmable Gate Array (FPGA) using Adomian decomposition method. Also, power efficiency analysis for various fractional-orders is investigated. The paper helps build a better understanding of chaotic systems with self-excited or hidden attractors.

In this paper, we investigate the dynamical characteristics of four-variable fractional-order Hindmarsh–Rose neuronal model under electromagnetic radiation. The numerical results show that the improved model exhibits more complex dynamical behavior with more bifurcation parameters. Meanwhile, based on the fractional-order Lyapunov stability theory, we propose two adaptive control methods with a single controller to realize chaotic synchronization between two coupled neurons. Finally, numerical simulations show the feasibility and effectiveness of the presented method.

Based on some essential concepts of fractional calculus and the theorem related to the fractional extension of Lyapunov direct method, we present in this paper a synchronization scheme of fractional-order Lur’e systems. A quadratic Lyapunov function is chosen to derive the synchronization criterion. The derived criterion is a suffcient condition for the asymptotic stability of the error system, formulated in the form of linear matrix inequality (LMI). The controller gain can be achieved by solving the LMI. The proposed scheme is illustrated for fractional-order Chua’s circuits and fractional-order four-cell CNN. Numerical results, which agree well with the proposed theorem, are given to show the effectiveness of this scheme.