Stability Analysis of Fractional-Order Neural Networks with Time Delay

Stability and synchronization of fractional-order reaction-diffusion inertial time-delayed neural networks with parameters perturbation

This study is centered around the dynamic behaviors observed in a class of fractional-order generalized reaction-diffusion inertial neural networks (FGRDINNs) with time delays. These networks are characterized by differential equations involving two distinct fractional derivatives of the state. The global uniform stability of FGRDINNs with time delays is explored utilizing Lyapunov comparison principles. Furthermore, global synchronization conditions for FGRDINNs with time delays are derived through the Lyapunov direct method, with consideration given to various feedback control strategies and parameter perturbations. The effectiveness of the theoretical findings is demonstrated through three numerical examples, and the impact of controller parameters on the error system is further investigated.

A Comprehensive Review of Continuous-/Discontinuous-Time Fractional-Order Multidimensional Neural Networks

The dynamical study of continuous-/discontinuous-time fractional-order neural networks (FONNs) has been thoroughly explored, and several publications have been made available. This study is designed to give an exhaustive review of the dynamical studies of multidimensional FONNs in continuous/discontinuous time, including Hopfield NNs (HNNs), Cohen-Grossberg NNs, and bidirectional associative memory NNs, and similar models are considered in real ( [Formula: see text]), complex ( [Formula: see text]), quaternion ( [Formula: see text]), and octonion ( [Formula: see text]) fields. Since, in practice, delays are unavoidable, theoretical findings from multidimensional FONNs with various types of delays are thoroughly evaluated. Some required and adequate stability and synchronization requirements are also mentioned for fractional-order NNs without delays.

Stability of Delay Hopfield Neural Networks with Generalized Riemann-Liouville Type Fractional Derivative

The general delay Hopfield neural network is studied. We consider the case of time-varying delay, continuously distributed delays, time-varying coefficients, and a special type of a Riemann-Liouville fractional derivative (GRLFD) with an exponential kernel. The kernels of the fractional integral and the fractional derivative in this paper are Sonine kernels and satisfy the first and the second fundamental theorems in calculus. The presence of delays and GRLFD in the model require a special type of initial condition. The applied GRLFD also requires a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. An inequality for Lyapunov type of convex functions with the applied GRLFD is proved. It is combined with the Razumikhin method to study stability properties of the equilibrium of the model. As a partial case we apply quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included.

Synchronization and adaptive control for coupled fractional-order reaction-diffusion neural networks with multiple couplings

In this paper, two kinds of coupled fractional-order reaction-diffusion neural networks (CFORDNNs) with multiple state couplings or spatial diffusion couplings are proposed. By resorting to the Laplace transform and the properties of Mittag-Leffler functions, sufficient synchronization conditions are derived for the concerned network models. In addition, to guarantee the synchronization of these two networks, several appropriate adaptive control schemes are also developed. Ultimately, the validity of the devised adaptive strategies are verified by adopting some numerical examples with simulation results.

Asymptotic stability and synchronization of fractional order Hopfield neural networks with unbounded delay

In this paper, the asymptotic stability and synchronization of fractional order Hopfield neural networks (FOHNNs) with time‐varying delay are investigated. First, we propose an extended fractional Halanay inequality, which plays an essential role in the proof of the main results. Using the Banach fixed point theorem, the existence and uniqueness of the equilibrium point of the system is proved. Through the extended Halanay inequality and a useful fractional derivative inequality, the sufficient conditions of the asymptotic stability and synchronization are given for FOHNNs. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results.

Bifurcations of a Fractional-Order Four-Neuron Recurrent Neural Network with Multiple Delays

This paper investigates the bifurcation issue of fractional-order four-neuron recurrent neural network with multiple delays. First, the stability and Hopf bifurcation of the system are studied by analyzing the associated characteristic equations. It is shown that the dynamics of delayed fractional-order neural networks not only depend heavily on the communication delay but also significantly affects the applications with different delays. Second, we numerically demonstrate the effect of the order on the Hopf bifurcation. Two numerical examples illustrate the validity of the theoretical results at the end.

Modeling Analysis on Coupling Mechanisms of Mountain–Basin Human–Land Systems: Take Yuxi City as an Example

The result of a human–land relationship in geographical environment systems is a human–land coupling system, which is a comprehensive process of interaction and infiltration between human economic and social systems and the natural ecosystem. Based on the recognition that the human–land system is a nonlinear system coupled by multiple factors, a time delay fractional order dynamics model with a Holling-II-type transformation rate was constructed, the stability analysis of the system was carried out, the transformation times of different land classes were clarified, and the coupled dynamics model parameters of mountainous areas and basin areas were obtained by using the land-use change survey data and socio-economic statistical data in Yuxi City, respectively: the transformation parameter of the production and living land to the unused land in mountainous areas and basin areas (aM, 0.0486 and aB, 0.0126); the transformation parameter of unused land to production and living land in mountainous areas and basin areas (bM 0.0062 and bB, 0.0139); the transformation parameter of unused land to the forest and grass land in mountainous areas and basin areas (sM, 0.0051 and sB, 0.0028); the land area required to maintain the individual unit in mountainous areas and basin areas (hM, 0.0335 and hB, 0.0165); the average reclamation capacity in mountainous areas and basin areas (dM, 0.03 and dB, 0.05); the inherent growth rate of populations in mountainous areas and basin areas (rM, 0.0563 and rB, 0.151). Through analyzing the coupling mechanisms of human–land systems, the countermeasures for the difference between mountainous areas and basin areas in the future development are put forward. The mountainous area should reduce the conversion of forest and grass land to production and living land by reducing the average reclamation or development capacity, reducing the excessive interference of human beings on unused land, and speeding up its natural recovery and succession to forest and grass land. In addition to reducing the average reclamation or development capacity in basin areas, the reclamation or development rate of the idle land and degraded land should be increased, and the conversion of idle land and degraded land into productive and living land should be encouraged by certain scientific and technological means.

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.

A Novel Fractional-Order Chaotic Phase Synchronization Model for Visual Selection and Shifting

Visual information processing is one of the fields of cognitive informatics. In this paper, a two-layer fractional-order chaotic network, which can simulate the mechanism of visual selection and shifting, is established. Unlike other object selection models, the proposed model introduces control units to select object. The first chaotic network layer of the model is used to implement image segmentation. A control layer is added as the second layer, consisting of a central neuron, which controls object selection and shifting. To implement visual selection and shifting, a strategy is proposed that can achieve different subnets corresponding to the objects in the first layer synchronizing with the central neuron at different time. The central unit acting as the central nervous system synchronizes with different subnets (hybrid systems), implementing the mechanism of visual selection and shifting in the human system. The proposed model corresponds better with the human visual system than the typical model of visual information encoding and transmission and provides new possibilities for further analysis of the mechanisms of the human cognitive system. The reasonability of the proposed model is verified by experiments using artificial and natural images.

Mittag-Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers

In this paper, we propose a fractional-order neural network system with time-varying delays and reaction-diffusion terms. We first develop a new Mittag-Leffler synchronization strategy for the controlled nodes via impulsive controllers. Using the fractional Lyapunov method sufficient conditions are given. We also study the global Mittag-Leffler synchronization of two identical fractional impulsive reaction-diffusion neural networks using linear controllers, which was an open problem even for integer-order models. Since the Mittag-Leffler stability notion is a generalization of the exponential stability concept for fractional-order systems, our results extend and improve the exponential impulsive control theory of neural network system with time-varying delays and reaction-diffusion terms to the fractional-order case. The fractional-order derivatives allow us to model the long-term memory in the neural networks, and thus the present research provides with a conceptually straightforward mathematical representation of rather complex processes. Illustrative examples are presented to show the validity of the obtained results. We show that by means of appropriate impulsive controllers we can realize the stability goal and to control the qualitative behavior of the states. An image encryption scheme is extended using fractional derivatives.

Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay

As we know, the notion of dissipativity is an important dynamical property of neural networks. Thus, the analysis of dissipativity of neural networks with time delay is becoming more and more important in the research field. In this paper, the authors establish a class of fractional-order complex-valued neural networks (FCVNNs) with time delay, and intensively study the problem of dissipativity, as well as global asymptotic stability of the considered FCVNNs with time delay. Based on the fractional Halanay inequality and suitable Lyapunov functions, some new sufficient conditions are obtained that guarantee the dissipativity of FCVNNs with time delay. Moreover, some sufficient conditions are derived in order to ensure the global asymptotic stability of the addressed FCVNNs with time delay. Finally, two numerical simulations are posed to ensure that the attention of our main results are valuable.