Fractional-order Neural Networks Research Articles

Control of the Bifurcation Behaviors of Delayed Fractional-Order Neural Networks with Cooperation–Competition Topology

In the real world, the competition and cooperation relationship exists in numerous systems. For instance, the competition–cooperation structure of a biological neural network is determined by the excitatory and inhibitory effects of neurons. The dynamic behaviors of a neural network model with a competition–cooperation structure are studied in this article, focusing particularly on the bifurcation and control problems. By selecting time delay as the parameter, a new sufficient condition for Hopf bifurcation is given and the impact of the fractional order on bifurcation behavior is determined for the network. Furthermore, a time-delay feedback controller is introduced to manage Hopf bifurcation behaviors, and, meanwhile, the stability domain is expanded. Our findings indicate that both fractional order and time delay play a crucial role in controlling the stability and Hopf bifurcation of the given model. Lastly, the accuracy of our theoretical results is verified through several numerical simulations, and the impact of control parameters on the bifurcation behavior of the network model is discussed in detail.

Exponential Quasi-Synchronization of Fractional-Order Fuzzy Cellular Neural Networks via Impulsive Control

This paper investigates the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks with parameters mismatch via impulsive control. Firstly, under the framework of the generalized Caputo fractional-order derivative, a new fractional-order impulsive differential inequality is established. Secondly, based on this fractional-order impulsive differential inequality, a general criterion for the quasi-synchronization of fractional-order systems is obtained. Then, specific to the fractional-order fuzzy cellular neural network model in this paper, the criteria and error estimation of the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks can be obtained. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

Does a Fractional-Order Recurrent Neural Network Improve the Identification of Chaotic Dynamics?

This paper presents a quantitative study of the effects of using arbitrary-order operators in Neural Networks. It is based on a Recurrent Wavelet First-Order Neural Network (RWFONN), which can accurately identify several chaotic systems (measured by the mean square error and the coefficient of determination, also known as R-Squared, r2) under a fixed parameter scheme in the neural algorithm. Using fractional operators, we analyze whether the identification capabilities of the RWFONN are improved, and whether it can identify signals from fractional-order chaotic systems. The results presented in this paper show that using a fractional-order Neural Network does not bring significant advantages in the identification process, compared to an integer-order RWFONN. Nevertheless, the neural algorithm (modeled with an integer-order derivative) proved capable of identifying fractional-order dynamical systems, whose behavior ranges from periodic and multi-stable to chaotic oscillations. That is, the performances of the Neural Network model with an integer-order derivative and the fractional-order network are practically identical, making the use of fractional-order RWFONN-type networks meaningless. The results deepen the work previously published by the authors, and contribute to developing structures based on robust and generic neural algorithms to identify more than one chaotic oscillator without retraining the Neural Network.

Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse

Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful for enhancing the processing capacity and storage capabilities of neural networks. When integrated into systems like Hopfield neural networks, memristors enable the study of complex dynamic behaviors, such as chaos and multistability. Moreover, fractional calculus is significant for their ability to model memory effects, enabling more accurate simulations of complex systems. Fractional-order Hopfield networks, in particular, exhibit chaotic and multistable behaviors not found in integer-order models. By combining memristors with fractional-order Hopfield neural networks, these systems offer the possibility of investigating different dynamic phenomena in artificial neural networks. This study investigates the dynamical behavior of a fractional-order Hopfield neural network (HNN) incorporating a memristor with a piecewise segment function in one of its synapses, highlighting the impact of fractional-order derivatives and memristive synapses on the stability, robustness, and dynamic complexity of the system. Using a network of four neurons as a case study, it is demonstrated that the memristive fractional-order HNN exhibits multistability, coexisting chaotic attractors, and coexisting limit cycles. Through spectral entropy analysis, the regions in the initial condition space that display varying degrees of complexity are mapped, highlighting those areas where the chaotic series approach a pseudo-random sequence of numbers. Finally, the proposed fractional-order memristive HNN is implemented on a Field-Programmable Gate Array (FPGA), demonstrating the feasibility of real-time hardware realization.

Fractional derivative of Hermite fractal splines on the fractional-order delayed neural networks synchronization

The purpose of this research is twofold. First, the master–slave synchronization of fractional-order neural networks is explored with time delays using aperiodic intermittent control. Then we present a sufficient condition for master–slave synchronization of delayed fractional-order neural networks via average-width intermittent control technique. A numerical simulation is used to demonstrate the efficacy of the derived results. Second, a novel investigation of the Caputo-fractional derivative of Hermite fractal splines is accomplished. Moreover, its box counting dimension is estimated and related with the Caputo-fractional order. Additionally, we propose an image encryption algorithm utilizing the semi-tensor product (STP). The efficiency of the algorithm is evaluated through the application of statistical measures.

Finite-Time Synchronization Criteria for Caputo Fractional-Order Uncertain Memristive Neural Networks with Fuzzy Operators and Transmission Delay Under Communication Feedback

Unlike existing memristive neural networks or fuzzy neural networks, this article investigates a class of Caputo fractional-order uncertain memristive neural networks (CFUMNNs) with fuzzy operators and transmission delay to realistically model complex environments. Especially, the fuzzy symbol AND and the fuzzy symbol OR as well as nonlinear activation behaviors are all concerned in the generalized master-slave networks. Based on the characteristics of the neural networks being studied, we have designed distinctive information feedback control protocols including three different functional sub-modules. Combining comparative theorems, inequality techniques, and stability theory, novel delay-independent conditions can be derived to ensure the finite-time synchronization (FTS) of fuzzy CFUMNNs. Besides, the upper bound of the settling time can be effectively evaluated based on feedback coefficients and control parameters, which makes the achievements of this study more practical for engineering applications such as signal encryption and secure communications. Ultimately, simulation experiments show the feasibility of the derived results.

Optimizing Fractional-Order Convolutional Neural Networks for Groove Classification in Music Using Differential Evolution

This study presents a differential evolution (DE)-based optimization approach for fractional-order convolutional neural networks (FOCNNs) aimed at enhancing the accuracy of groove classification in music. Groove, an essential element in music perception, is typically influenced by rhythmic patterns and acoustic features. While FOCNNs offer a promising method for capturing these subtleties through fractional-order derivatives, they face challenges in efficiently converging to optimal parameters. To address this, DE is applied to optimize the initial weights and biases of FOCNNs, leveraging its robustness and ability to explore a broad solution space. The proposed DE-FOCNN was evaluated on the Janata dataset, which includes pre-rated music tracks. Comparative experiments across various fractional-order values demonstrated that DE-FOCNN achieved superior performance in terms of higher test accuracy and reduced overfitting compared to a standard FOCNN. Specifically, DE-FOCNN showed optimal performance at fractional-order values such as v = 1.4. Further experiments demonstrated that DE-FOCNN achieved higher accuracy and lower variance compared to other popular evolutionary algorithms. This research primarily contributes to the optimization of FOCNNs by introducing a novel DE-based approach for the automated analysis and classification of musical grooves. The DE-FOCNN framework holds promise for addressing other related engineering challenges.

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.

Adaptive Strategies and its Application in the Mittag-Leffler Synchronization of Delayed Fractional-Order Complex-Valued Reaction-Diffusion Neural Networks

Adaptive Strategies and its Application in the Mittag-Leffler Synchronization of Delayed Fractional-Order Complex-Valued Reaction-Diffusion Neural Networks

New Event-Triggered Synchronization Criteria for Fractional-Order Complex-Valued Neural Networks with Additive Time-Varying Delays

This paper explores the synchronization control issue for a class of fractional-order Complex-valued Neural Networks (FOCVNNs) with additive time-varying delays (TVDs) utilizing a sampled-data-based event-triggered mechanism (SDBETM). First, an innovative free-matrix-based fractional-order integral inequality (FMBFOII) and an improved fractional-order complex-valued integral inequality (FOCVII) are proposed, which are less conservative than the existing classical fractional-order integral inequality (FOII). Secondly, an SDBETM is inducted to conserve network resources. In addition, a novel Lyapunov–Krasovskii functional (LKF) enriched with additional information regarding the fractional-order derivative, additive TVDs, and triggering instants is constructed. Then, through the integration of the innovative FOCVII, LKF, SDBETM, and other analytical methodologies, we deduce two criteria in the form of linear matrix inequalities (LMIs) to ensure the synchronization of the master–slave FOCVNNs. Finally, numerical simulations are illustrated to confirm the validity of the proposed results.

Asymptotic Synchronization for Caputo Fractional-Order Time-Delayed Cellar Neural Networks with Multiple Fuzzy Operators and Partial Uncertainties via Mixed Impulsive Feedback Control

To construct Caputo fractional-order time-delayed cellar neural networks (FOTDCNNs) that characterize real environments, this article introduces partial uncertainties, fuzzy operators, and nonlinear activation functions into the network models. Specifically, both the fuzzy AND operator and the fuzzy OR operator are contemplated in the master–slave systems. In response to the properties of the considered cellar neural networks (NNs), this article designs a new class of mixed control protocols that utilize both the error feedback information of systems and the sampling information of impulse moments to achieve network synchronization tasks. This approach overcomes the interference of time delays and uncertainties on network stability. By integrating the fractional-order comparison principle, fractional-order stability theory, and hybrid control schemes, readily verifiable asymptotic synchronization conditions for the studied fuzzy cellar NNs are established, and the range of system parameters is determined. Unlike previous results, the impulse gain spectrum considered in this study is no longer confined to a local interval (−2,0) and can be extended to almost the entire real number domain. This spectrum extension relaxes the synchronization conditions, ensuring a broader applicability of the proposed control schemes.

A Novel Fractional Model and Its Application in Network Security Situation Assessment

The evaluation process of the Fractional Order Model is as follows. To address the commonly observed issue of low accuracy in traditional situational assessment methods, a novel evaluation algorithm model, the fractional-order BP neural network optimized by the chaotic sparrow search algorithm (TESA-FBP), is proposed. The fractional-order BP neural network, by incorporating fractional calculus, demonstrates enhanced dynamic response characteristics and historical dependency, showing exceptional potential for handling complex nonlinear problems, particularly in the field of network security situational awareness. However, the performance of this network is highly dependent on the precise selection of network parameters, including the fractional order and initial values of the weights. Traditional optimization methods often suffer from slow convergence, a tendency to be trapped in local optima, and insufficient optimization accuracy, which significantly limits the practical effectiveness of the fractional-order BP neural network. By introducing cubic chaotic mapping to generate an initial population with high randomness and global coverage capability, the exploration ability of the sparrow search algorithm in the search space is effectively enhanced, reducing the risk of falling into local optima. Additionally, the Estimation of Distribution Algorithm (EDA) constructs a probabilistic model to guide the population toward the globally optimal region, further improving the efficiency and accuracy of the search process. The organic combination of these three approaches not only leverages their respective strengths, but also significantly improves the training performance of the fractional-order BP neural network in complex environments, enhancing its generalization ability and stability. Ultimately, in the network security situational awareness system, this integration markedly enhances the prediction accuracy and response speed.

Finite-time contractive stability for fractional-order nonlinear systems with delayed impulses: Applications to neural networks

This paper deals with finite-time stability (FTS) and finite-time contractive stability (FTCS) criteria of fractional-order nonlinear systems (FONSs), where the state of the systems consists of a finite number of delayed impulsive instants. To achieve these results, we have extended the theories on FTS and FTCS criteria from integer-order NSs to fractional-order systems. Employing the fractional-order Lyapunov stability theory, the sufficient conditions of the above-mentioned stability criteria are derived for a class of FONSs with state-dependent delayed impulses. Then, to ensure the applicability of the proposed results, we have analyzed the desired performances for several neural network (NN) models, such as non-autonomous NNs (NANNs), Cohen–Grossberg NNs (CGNNs), switched NNs (SNNs) and bidirectional associative memory NNs (BAMNNs). Finally, four representative numerical examples are illustrated to demonstrate the assurance of the obtained stability conditions of the respective state-dependent impulsive fractional-order NNs (FONNs).

Quasi-synchronization of fractional-order complex-value BAM neural networks with time delays and discontinuous activations

Quasi-synchronization of fractional-order complex-value BAM neural networks with time delays and discontinuous activations

Further finite-time stability analysis of neural networks with proportional delay

Different from fixed delays between neurons, proportional delay is a time-varying unbounded delay. In this paper, the issue of finite-time stability of fractional-order neural networks with proportional delay is investigated. To address the proportional delay terms in the system, a new Gronwall-type inequality with delay is proposed. Based on this inequality, order-related criterion for finite-time stability of fractional-order neural networks with proportional delay is obtained. Finally, some numerical experiments demonstrate the effectiveness and superiority of the results.

Finite-time quasi-projective synchronization of fractional-order reaction-diffusion delayed neural networks

This paper investigates the finite-time quasi-projective synchronization (FTQPS) issue of fractional-order reaction-diffusion neural networks (FORDNNs). To the best of our knowledge, this paper introduces the concept of FTQPS for the first time. First, an integral-type Lyapunov function is constructed relying on the characterization of the reaction-diffusion term and some inequality methods. Subsequently, the nonlinear feedback control strategy is designed to achieve the FTQPS goal and some sufficient conditions are obtained to guarantee FTQPS of FORDNNs. Further, the system's synchronization speed is measured by estimating the settling time. It should be noted that the above control strategy is also applicable to conventional integer-order reaction-diffusion neural networks with time delays. Finally, a numerical example is used to illustrate the validity of the theoretical analysis presented.

Finite-time stability of fractional-order quaternion-valued memristive neural networks with time delay

This study addresses the finite-time stability (FTS) problem of fractional-order quaternion-valued memristive neural networks (FQMNNs) with time delay. Firstly, by leveraging set-valued mapping, differential inclusion, and contracting mapping principle, the sufficient condition is provided to assure the existence and uniqueness of the equilibrium point. Secondly, given the fractional order differential inequality and Gronwall inequality, a finite-time stability (FTS) criterion is obtained for the delayed FQMNNs with the fractional order between 0 and 1. Additionally, a sufficient condition is proposed to guarantee the FTS of FQMNNs with the fractional order between 1 and 2. Compared with existing results, this study not only extends the fractional order between 0 and 2, but also incorporates quaternions, which significantly enhances the FTS theory. Finally, two numerical examples demonstrate the practicality of the obtained results.

Finite-time cluster synchronization of multi-weighted fractional-order coupled neural networks with and without impulsive effects

In this paper, finite-time cluster synchronization (FTCS) of multi-weighted fractional-order neural networks is studied. Firstly, a FTCS criterion of the considered neural networks is obtained by designing a new delayed state feedback controller. Secondly, a FTCS criterion for the considered neural networks with mixed impulsive effects is given by constructing a new piecewise controller, where both synchronizing and desynchronizing impulses are taken into account. It should be noted that it is the first time that finite-time cluster synchronization of multi-weighted neural networks has been investigated. Finally, numerical simulations are given to show the validity of the theoretical results.

Designing sampled data controller for time-delayed fractional-order neural networks via a new functional approach

Designing sampled data controller for time-delayed fractional-order neural networks via a new functional approach

Adaptive Synchronization of Fractional-Order Uncertain Complex-Valued Competitive Neural Networks under the Non-Decomposition Method

This paper is devoted to the study of adaptive synchronization for fractional-order uncertain complex-valued competitive neural networks (FOUCVCNNs) using the non-decomposition method. Firstly, a new network model named FOUCVCNNs is proposed, which is not separated into two real-valued subsystems in order to keep its intrinsic speciality. In addition, a novel adaptive controller is designed to reduce the cost of control. Meanwhile, with the help of fractional Lyapunov theory, 1-norm analysis framework and inequality techniques, several effective synchronization criteria for FOUCVCNNs are obtained by constructing an appropriate Lyapunov function. Finally, the reliability of the results obtained is tested through numerical simulation.