Multistability of delayed fractional-order competitive neural networks

Mittag-Leffler stability and application of delayed fractional-order competitive neural networks

In the article, the Mittag-Leffler stability and application of delayed fractional-order competitive neural networks (FOCNNs) are developed. By virtue of the operator pair, the conditions of the coexistence of equilibrium points (EPs) are discussed and analyzed for delayed FOCNNs, in which the derived conditions of coexistence improve the existing results. In particular, these conditions are simplified in FOCNNs with stepped activations. Furthermore, the Mittag-Leffler stability of delayed FOCNNs is established by using the principle of comparison, which enriches the methodologies of fractional-order neural networks. The results on the obtained stability can be used to design the horizontal line detection of images, which improves the practicability of image detection results. Two simulations are displayed to validate the superiority of the obtained results.

Anti-periodic motion and mean-square exponential convergence of nonlocal discrete-time stochastic competitive lattice neural networks with fuzzy logic

This paper firstly establishes the discrete-time lattice networks for nonlocal stochastic competitive neural networks with reaction diffusions and fuzzy logic by employing a mix techniques of finite difference to space variables and Mittag-Leffler time Euler difference to time variable. The proposed networks consider both the effects of spatial diffusion and fuzzy logic, whereas most of the existing literatures focus only on discrete-time networks without spatial diffusion. Firstly, the existence of a unique ω-anti-periodic in distribution to the networks is addressed by employing Banach contractive mapping principle and the theory of stochastic calculus. Secondly, global exponential convergence in mean-square sense to the networks is discussed on the basis of constant variation formulas for sequences. Finally, an illustrative example is used to show the feasible of the works in the current paper with the help of MATLAB Toolbox. The work in this paper is pioneering in this regard and it has created a certain research foundations for future studies in this area.

Coexistence and local stability of multiple equilibrium points for fractional-order state-dependent switched competitive neural networks with time-varying delays

This paper investigates the coexistence and local stability of multiple equilibrium points for a class of competitive neural networks with sigmoidal activation functions and time-varying delays, in which fractional-order derivative and state-dependent switching are involved at the same time. Some novel criteria are established to ensure that such n-neuron neural networks can have [Formula: see text] total equilibrium points and [Formula: see text] locally stable equilibrium points with m₁+m₂=n, based on the fixed-point theorem, the definition of equilibrium point in the sense of Filippov, the theory of fractional-order differential equation and Lyapunov function method. The investigation implies that the competitive neural networks with switching can possess greater storage capacity than the ones without switching. Moreover, the obtained results include the multistability results of both fractional-order switched Hopfield neural networks and integer-order switched Hopfield neural networks as special cases, thus generalizing and improving some existing works. Finally, four numerical examples are presented to substantiate the effectiveness of the theoretical analysis.

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.

Fin-TS and Fix-TS on fractional quaternion delayed neural networks with uncertainty via establishing a new Caputo derivative inequality approach

This paper investigates the finite time synchronization (Fin-TS) and fixed time synchronization (Fix-TS) issues on Caputo quaternion delayed neural networks (QDNNs) with uncertainty. A new Caputo fractional differential inequality is constructed, then Fix-TS settling time of the positive definite function is estimated, which is very convenient to derive Fix-TS condition to Caputo QDNNs. By designing the appropriate self feedback and adaptive controllers, the algebraic discriminant conditions to achieve Fin-TS and Fix-TS on Caputo QDNNs are proposed based on quaternion direct method, Lyapunov stability theory, extended Cauchy Schwartz inequality, Jensen inequality. Finally, the correctness and validity of the presented results under the different orders are verified by two numerical examples.

Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays

This study on the local stability of quaternion-valued neural networks is of great significance to the application of associative memory and pattern recognition. In the research, we study local Lagrange exponential stability of quaternion-valued neural networks with time delays. By separating the quaternion-valued neural networks into a real part and three imaginary parts, separating the quaternion field into 34n subregions, and using the intermediate value theorem, sufficient conditions are proposed to ensure quaternion-valued neural networks have 34n equilibrium points. According to the Halanay inequality, the conditions for the existence of 24n local Lagrange exponentially stable equilibria of quaternion-valued neural networks are established. The obtained stability results improve and extend the existing ones. Under the same conditions, quaternion-valued neural networks have more stable equilibrium points than complex-valued neural networks and real-valued neural networks. The validity of the theoretical results were verified by an example.

Fixed-time synchronization of discontinuous competitive neural networks with time-varying delays

In this article, the fixed-time (FXT) synchronization of discontinuous competitive neural networks (CNNs) involving time-varying delays is investigated. Firstly, two kinds of discontinuous FXT control schemes are proposed and two forms of Lyapunov function are constructed based on p-norm and 1-norm to discuss the FXT synchronization of CNNs. By means of nonsmooth analysis and some inequality techniques, some simple criteria are obtained to achieve FXT synchronization and the upper bound of the settling time with less conservativeness is provided. Furthermore, the effect of time scale on FXT synchronization of CNNs is considered. Lastly, some numerical results for an example are provided to demonstrate the derived theoretical results.

Dynamic Analysis and Bifurcation Study on Fractional-Order Tri-Neuron Neural Networks Incorporating Delays

In this manuscript, we principally probe into a class of fractional-order tri-neuron neural networks incorporating delays. Making use of fixed point theorem, we prove the existence and uniqueness of solution to the fractional-order tri-neuron neural networks incorporating delays. By virtue of a suitable function, we prove the uniformly boundedness of the solution to the fractionalorder tri-neuron neural networks incorporating delays. With the aid of the stability theory and bifurcation knowledge of fractional-order differential equation, a new delay-independent condition to guarantee the stability and creation of Hopf bifurcation of the fractional-order tri-neuron neuralnetworks incorporating delays is established. Taking advantage of the mixed controller that contains state feedback and parameter perturbation, the stability region and the time of onset of Hopf bifurcation of the fractional-order trineuron neural networks incorporating delays are successfully controlled. Software simulation plots are displayed to illustrate the established key results. The obtained conclusions in this article have important theoretical significance in designing and controlling neural networks.

Optimization of fractional-order chaotic cellular neural networks by metaheuristics

Artificial neural networks have demonstrated to be very useful in solving problems in artificial intelligence. However, in most cases, ANNs are considered integer-order models, limiting the possible applications in recent engineering problems. In addition, when dealing with fractional-order neural networks, almost any work shows cases when varying the fractional order. In this manner, we introduce the optimization of a fractional-order neural network by applying metaheuristics, namely: differential evolution (DE) and accelerated particle swarm optimization (APSO) algorithms. The case study is a chaotic cellular neural network (CNN), for which the main goal is generating fractional orders of the neurons whose Kaplan-Yorke dimension is being maximized. We propose a method based on Fourier transform to evaluate if the generated time series is chaotic or not. The solutions that do not have chaotic behavior are not passed to the time series analysis (TISEAN) software, thus saving execution time. We show the best solutions provided by DE and APSO of the attractors of the fractional-order chaotic CNNs.

Chaos Control for a Fractional-Order Jerk System via Time Delay Feedback Controller and Mixed Controller

In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PDσ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system.