Quasi-projective and complete synchronization of fractional-order complex-valued neural networks with time delays

Complete synchronization of discrete-time fractional-order BAM neural networks with leakage and discrete delays

This paper concerns complete synchronization (CS) problem of discrete-time fractional-order BAM neural networks (BAMNNs) with leakage and discrete delays. Firstly, on the basis of Caputo fractional difference theory and nabla l-Laplace transform, two equations about the nabla sum are strictly proved. Secondly, two extended Halanay inequalities that are suitable for discrete-time fractional difference inequations with arbitrary initial time and multiple types of delays are introduced. In addition, through applying Caputo fractional difference theory and combining with inequalities gained from this paper, some sufficient CS criteria of discrete-time fractional-order BAMNNs with leakage and discrete delays are established under adaptive controller. Finally, one numerical simulation is utilized to certify the effectiveness of the obtained theoretical results.

Synchronization Analysis of Discrete-Time Fractional-Order Quaternion-Valued Uncertain Neural Networks

This article studies synchronization issues for a class of discrete-time fractional-order quaternion-valued uncertain neural networks (DFQUNNs) using nonseparation method. First, based on the theory of discrete-time fractional calculus and quaternion properties, two equalities on the nabla Laplace transform and nabla sum are strictly proved, whereafter three Caputo difference inequalities are rigorously demonstrated. Next, based on our established inequalities and equalities, some simple and verifiable quasi-synchronization criteria are derived under the quaternion-valued nonlinear controller, and complete synchronization is achieved using quaternion-valued adaptive controller. Finally, numerical simulations are presented to substantiate the validity of derived results.

Almost periodic quasi-projective synchronization of delayed fractional-order quaternion-valued neural networks

This paper examines the issue of almost periodic quasi-projective synchronization of delayed fractional-order quaternion-valued neural networks. First, using a direct method rather than decomposing the fractional quaternion-valued system into four equivalent fractional real-valued systems, using Banach's fixed point theorem, according to the basic properties of fractional calculus and some inequality methods, we obtain that there is a unique almost periodic solution for this class of neural network with some sufficient conditions. Next, by constructing a suitable Lyapunov functional, using the characteristic of the Mittag-Leffler function and the scaling idea of the inequality, the adequate conditions for the quasi-projective synchronization of the established model are derived, and the upper bound of the systematic error is estimated. Finally, further use Matlab is used to carry out two numerical simulations to prove the results of theoretical analysis.

Adaptive control-based synchronization of discrete-time fractional-order fuzzy neural networks with time-varying delays

This paper is concerned with complete synchronization for discrete-time fractional-order fuzzy neural networks (DFFNNs) with time-varying delays. First, three original equalities and two Caputo σ-difference inequalities are established based on theory of discrete-time fractional Calculus. Next, a novel discrete-time adaptive controller with time-varying delay is designed, by virtue of 1-norm Lyapunov function and newly established lemmas herein as well as inequality techniques and contradiction method, some judgement conditions are derived to guarantee complete synchronization for the explored DFFNNs. Benefitting from discrete-time adaptive control strategy and our analysis method, the conservatism of the derived synchronization criteria is reduced. Ultimately, the effectiveness of our theoretical results and secure communication scheme are demonstrated through two numerical examples.

Quasi-projective and complete synchronization of discrete-time fractional-order delayed neural networks

This paper presents new theoretical results on quasi-projective synchronization (Q-PS) and complete synchronization (CS) of one kind of discrete-time fractional-order delayed neural networks (DFDNNs). At first, three new fractional difference inequalities for exploring the upper bound of quasi-synchronization error and adaptive synchronization are established by dint of Laplace transform and properties of discrete Mittag-Leffler function, which vastly expand a number of available results. Furthermore, two controllers are designed including nonlinear controller and adaptive controller. And on the basis of Lyapunov method, the aforementioned inequalities and properties of fractional-order difference operators, some sufficient synchronization criteria of DFDNNs are derived. Because of the above controllers, synchronization criteria in this paper are less conservative. At last, numerical examples are carried out to illustrate the usefulness of theoretical upshots.

State-based event-triggered consensus strategy for Takagi-Sugeno fuzzy fractional-order multiagent systems with switching topologies

This article addresses the event-triggered consensus problem for Takagi-Sugeno fuzzy fractional-order multiagent systems with switching topologies. First, to effectively avoid the frequent communication among agents, a state-based event-triggered consensus strategy is designed, which uses the local information from neighboring agents at sampling moments. Then, several sufficient conditions, which rely on the fractional derivative number and time delay information, are presented to guarantee the consensus of fractional-order multiagent systems based on Takagi-Sugeno fuzzy model. Moreover, Zeno behaviors are precluded by proving that the interval length of the two consecutive event-triggering moments for each agent is greater than a positive constant. Finally, some numerical examples are presented, which not only demonstrated the rationality of our proposed consensus protocol but also shown that the presented consensus method based on the designed event-triggered control protocol has the advantage for avoiding communication congestion compared to the existing results.

Mittag–Leffler Synchronization of Caputo-Delayed Quaternion BAM Neural Networks via Adaptive and Linear Feedback Control Designs

The Mittag–Leffler synchronization (MLS) issue for Caputo-delayed quaternion bidirectional associative memory neural networks (BAM-NNs) is studied in this paper. Firstly, a novel lemma is proved by the Laplace transform and inverse transform. Then, without decomposing a quaternion system into subsystems, the adaptive controller and the linear controller are designed to realize MLS. According to the proposed lemma, constructing two different Lyapunov functionals and applying the fractional Razumikhin theorem and inequality techniques, the sufficient criteria of MLS on fractional delayed quaternion BAM-NNs are derived. Finally, two numerical examples are given to illustrate the validity and practicability.

Artificial neural networks: a practical review of applications involving fractional calculus

In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.

Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays

This paper investigates a class of fractional-order delayed impulsive gene regulatory networks (GRNs). The proposed model is an extension of some existing integer-order GRNs using fractional derivatives of Caputo type. The existence and uniqueness of an almost periodic state of the model are investigated and new criteria are established by the Lyapunov functions approach. The effects of time-varying delays and impulsive perturbations at fixed times on the almost periodicity are considered. In addition, sufficient conditions for the global Mittag–Leffler stability of the almost periodic solutions are proposed. To justify our findings a numerical example is also presented.

Chaotic dynamics in a novel COVID-19 pandemic model described by commensurate and incommensurate fractional-order derivatives

Mathematical models based on fractional-order differential equations have recently gained interesting insights into epidemiological phenomena, by virtue of their memory effect and nonlocal nature. This paper investigates the nonlinear dynamic behavior of a novel COVID-19 pandemic model described by commensurate and incommensurate fractional-order derivatives. The model is based on the Caputo operator and takes into account the daily new cases, the daily additional severe cases, and the daily deaths. By analyzing the stability of the equilibrium points and by continuously varying the values of the fractional order, the paper shows that the conceived COVID-19 pandemic model exhibits chaotic behaviors. The system dynamics are investigated via bifurcation diagrams, Lyapunov exponents, time series, and phase portraits. A comparison between integer-order and fractional-order COVID-19 pandemic models highlights that the latter is more accurate in predicting the daily new cases. Simulation results, besides to confirming that the novel fractional model well fit the real pandemic data, also indicate that the numbers of new cases, severe cases, and deaths undertake chaotic behaviors without any useful attempt to control the disease.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11071-021-06867-5.

Exponential Stability of Fractional-Order Complex Multi-Links Networks With Aperiodically Intermittent Control

In this article, the exponential stability problem for fractional-order complex multi-links networks with aperiodically intermittent control is considered. Using the graph theory and Lyapunov method, two theorems, including a Lyapunov-type theorem and a coefficient-type theorem, are given to ensure the exponential stability of the underlying networks. The theoretical results show that the exponential convergence rate is dependent on the control gain and the order of fractional derivative. To be specific, the larger control gain, the higher the exponential convergence rate. Meanwhile, when aperiodically intermittent control degenerates into periodically intermittent control, a corollary is also provided to ensure the exponential stability of the underlying networks. Furthermore, to show the practicality of theoretical results, as an application, the exponential stability of fractional-order multi-links competitive neural networks with aperiodically intermittent control is investigated and a stability criterion is established. Finally, the effectiveness and feasibility of the theoretical results are demonstrated through a numerical example.

Finite-Time Projective Synchronization of Caputo Type Fractional Complex-Valued Delayed Neural Networks

This paper focuses on investigating the finite-time projective synchronization of Caputo type fractional-order complex-valued neural networks with time delay (FOCVNNTD). Based on the properties of fractional calculus and various inequality techniques, by constructing suitable the Lyapunov function and designing two new types controllers, i.e., feedback controller and adaptive controller, two sufficient criteria are derived to ensure the projective finite-time synchronization between drive and response systems, and the synchronization time can effectively be estimated. Finally, two numerical examples are presented to verify the effectiveness and feasibility of the proposed results.

Finite-time cluster synchronization in complex-variable networks with fractional-order and nonlinear coupling

This paper is primarily concentrated on finite-time cluster synchronization of fractional-order complex-variable networks with nonlinear coupling by utilizing the non-decomposition method. Firstly, two control strategies are designed which are relevant to complex-valued sign functions. Thereafter, by employing fractional-order stability theory and complex function theory, several criteria are deduced to ensure finite-time cluster synchronization under the framework within a new norm consisting of absolute values for real and imaginary components. Furthermore, the setting time is effectively estimated based on some significant properties of fractional-order Caputo derivation and Mittag-Leffler functions. Lastly, two numerical examples are given to verify the effectiveness of theoretical results.