Fractional-order Neural Networks Research Articles

Finite-time synchronization of fractional-order memristive neural networks via feedback and periodically intermittent control

This paper discusses the finite-time synchronization (FTS) of fractional-order memristive neural networks (FMNNs) with time-varying delays. Firstly, based on Gronwall–Bellman inequality and the fractional-order derivative of the power function, two novel propositions on finite-time fractional functional differential inequality are built. Secondly, the feedback controller and intermittent controller designed in this paper are all delay-independent controllers, which can work even when the prior state cannot be measured or the specific time delay function is unknown. Thirdly, in addition to the traditional Lyapunov function with absolute value form, a more general and flexile Lyapunov function based on p-norm form is constructed to analyze the criteria of FTS. Meanwhile, an improved estimation of settling time of fractional system is given explicitly, which is more accurate and general than the existing results. Eventually, the validity of theoretical analysis is confirmed by numerical examples.

Stability analysis of fractional order memristor synapse-coupled hopfield neural network with ring structure.

A memristor is a nonlinear two-terminal electrical element that incorporates memory features and nanoscale properties, enabling us to design very high-density artificial neural networks. To enhance the memory property, we should use mathematical frameworks like fractional calculus, which is capable of doing so. Here, we first present a fractional-order memristor synapse-coupling Hopfield neural network on two neurons and then extend the model to a neural network with a ring structure that consists of n sub-network neurons, increasing the synchronization in the network. Necessary and sufficient conditions for the stability of equilibrium points are investigated, highlighting the dependency of the stability on the fractional-order value and the number of neurons. Numerical simulations and bifurcation analysis, along with Lyapunov exponents, are given in the two-neuron case that substantiates the theoretical findings, suggesting possible routes towards chaos when the fractional order of the system increases. In the n-neuron case also, it is revealed that the stability depends on the structure and number of sub-networks.

Dynamical Bifurcations in a Fractional-Order Neural Network with Nonidentical Communication Delays

Dynamical Bifurcations in a Fractional-Order Neural Network with Nonidentical Communication Delays

Equilibrium point, exponential stability and synchronization of numerical fractional-order shunting inhibitory cellular neural networks with piecewise feature

In recent years, exponential Euler difference methods for first-order semi-linear differential equations have been developed rapidly, and various exponential Euler difference methods have become a very important research topic. For fractional-order shunting inhibitory cellular neural networks with time lags, this article first establishes a new difference method named Mittag-Leffler Euler difference, which includes exponential Euler difference. Second, the existence of unique global bounded solution and equilibrium point, exponential stability and synchronization of the derived difference model are investigated. Compared with the classical fractional-order Euler difference method, the Mittag-Leffler discrete shunting inhibitory cellular neural networks can better describe and maintain the dynamic properties of the corresponding continuous-time models. More importantly, it opens up a new way to study discrete-time fractional-order systems and establishes a set of theory and methods to study Mittag-Leffler discrete neural networks.

Local Stabilization of Delayed Fractional-Order Neural Networks Subject to Actuator Saturation

This paper investigates the local stabilization problem of delayed fractional-order neural networks (FNNs) under the influence of actuator saturation. First, the sector condition and dead-zone nonlinear function are specially introduced to characterize the features of the saturation phenomenon. Then, based on the fractional-order Lyapunov method and the estimation technique of the Mittag–Leffler function, an LMIs-based criterion is derived to guarantee the local stability of closed-loop delayed FNNs subject to actuator saturation. Furthermore, two corresponding convex optimization schemes are proposed to minimize the actuator costs and expand the region of admissible initial values, respectively. At last, two simulation examples are developed to demonstrate the feasibility and effectiveness of the derived results.

An improved criterion on finite-time stability for fractional-order fuzzy cellular neural networks involving leakage and discrete delays

This paper concentrates on the finite-time stability (FTS) for fractional-order fuzzy cellular neural networks involving leakage and discrete delays. We propose a method to develop a new FTS criterion which is a supplement to the research on stability for fractional-order systems. The proof mainly depends on a new Henry–Gronwall integral inequality (HGII) with two discrete delays. This inequality generalizes some known ones which have played a significant role in the analysis on fractional-order systems. Finally, we demonstrate the correctness and advantages of the obtained criterion by supplying some examples.

Extended analysis on the global Mittag-Leffler synchronization problem for fractional-order octonion-valued BAM neural networks

In this paper, a new case of neural networks called fractional-order octonion-valued bidirectional associative memory neural networks (FOOVBAMNNs) is established. First, the higher dimensional models are formulated for FOOVBAMNNs with general activation functions and the special linear threshold ones, respectively. On one hand, employing Cayley–Dichson construction in octonion multiplication which is essentially neither commutative nor associative, the system of FOOVBAMNNs is divided into four fractional-order complex-valued ones. On the other hand, Caputo fractional derivative’s character and BAM’s interactive feature are also properly dealt with. Second, the general criteria are obtained by the new design of LKFs, the application of the related inequalities and the construction of the linear feedback controllers for the global Mittag-Leffler synchronization problem of FOOVBAMNNs. Finally, we present two numerical examples to show the realizability and progress of the derived results.

Recent Advances and Applications of Fractional-Order Neural Networks

Recent Advances and Applications of Fractional-Order Neural Networks

Time-dependent and Caputo derivative order-dependent quasi-uniform synchronization on fuzzy neural networks with proportional and distributed delays

This paper focuses on studying the quasi-uniform (Q-U) synchronization on fractional-order fuzzy neural networks (FOFNNs) with the proportional and distributed delays. The impacts of the Caputo derivative order, coefficients of network system and control gain constants on the synchronization performance are taken into account. Two novel time-dependent and Caputo derivative order-dependent algebraic criteria on the Q-U synchronization of FOFNNs are established by using Cauchy–Schwartz inequality, Minkowski inequality, Cp inequality and Hölder inequality, which reveal the internal influences of the time and fractional derivative order on Q-U synchronization with the derivative order interval (0,2). The simulation examples further confirm the effectiveness and practicability of the algebraic criteria in terms of the MATLAB toolbox.

Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks

In the present study, we deal with the stability and the onset of Hopf bifurcation of two type delayed BAM neural networks (integer-order case and fractional-order case). By virtue of the characteristic equation of the integer-order delayed BAM neural networks and regarding time delay as critical parameter, a novel delay-independent condition ensuring the stability and the onset of Hopf bifurcation for the involved integer-order delayed BAM neural networks is built. Taking advantage of Laplace transform, stability theory and Hopf bifurcation knowledge of fractional-order differential equations, a novel delay-independent criterion to maintain the stability and the appearance of Hopf bifurcation for the addressed fractional-order BAM neural networks is established. The investigation indicates the important role of time delay in controlling the stability and Hopf bifurcation of the both type delayed BAM neural networks. By adjusting the value of time delay, we can effectively amplify the stability region and postpone the time of onset of Hopf bifurcation for the fractional-order BAM neural networks. Matlab simulation results are clearly presented to sustain the correctness of analytical results. The derived fruits of this study provide an important theoretical basis in regulating networks.

Global Mittag-Leffler synchronization of discrete-time fractional-order neural networks with time delays

In this article, the problem of the global Mittag-Leffler synchronization is proposed for a sort of discrete-time fractional-order neural networks (DFNNs) with delays. In the first place, a flesh power law inequality pertaining to fractional difference is constructed by means of integration by parts, Young inequality, and some properties about fractional-order difference. In addition, based on aforesaid inequalities, Lyapunov function theory and properties of nabla Mittag-Leffler function as well as inequality techniques, some plentiful criteria are formed to achieve the global Mittag-Leffler synchronization for the delayed DFNNs via devising novel adaptive controller and delay feedback controller. In the end, numerical modeling is given to demonstrate effectiveness of theoretical verdicts.

Fixed-time synchronization of fractional-order complex-valued neural networks with time-varying delay via sliding mode control

Taking into account fractional-order complex-valued neural networks with time-varying delay, the issue of achieving fixed-time synchronization is discussed in this paper. By utilizing the properties of fractional calculus and fractional-order comparison principle, an improved lemma is proposed to derive the fixed-time synchronization conditions. On the basis of sliding model control and Lyapunov stability theorem, an effective sliding mode surface is constructed, which only uses the synchronization error information of FOCVNNs and is composed of fractional and integer integral terms. Further, a suitable sliding model control is constructed, which makes synchronization error converges to zero in a fixed-time. Beyond that, several sufficient conditions are posed to guarantee fixed-time synchronization of the fractional-order complex-valued neural networks and the upper bound of synchronization settling time is estimated. Finally, two numerical simulations are given to demonstrate the effectiveness of the presented theoretical results.

Application of Nonlinear Fractional Differential Equations in Computer Artificial Intelligence Algorithms

Abstract In order to study the application of nonlinear fractional differential equations in computer artificial intelligence algorithms. First, the concept, properties and commonly used neural network models of artificial neural network are introduced, the domestic and foreign status quo of the application of fractional calculus theory to neural network technology is described. Then, the definition, properties and numerical calculation methods of fractional calculus theory are introduced in detail. Then, based on the analysis of artificial intelligence neural network algorithm, the theory of fractional differentiation is introduced, construct BP neural network based on fractional order theory. The Sigmoid function is used as the node function of the neural network, and the actual data is used as the sample set, train a fractional-order network. Finally, by training the network, summarize the change of the two parameters a and p in the function, the impact on the training of the entire network, and make a simple comparison with the fractional order neural network based on the sigmoid function. Experiments show that a variable-order iterative learning algorithm is proposed and applied to the training of neural networks, the results show the feasibility of this algorithm and its advantages in convergence speed and convergence accuracy.

The Passivity of Uncertain Fractional-Order Neural Networks with Time-Varying Delays

In this paper, we study the passive problem of uncertain fractional-order neural networks (UFONNs) with time-varying delays. First, we give a sufficient condition for the asymptotic stability of UFONNs with bounded time-varying delays by using the fractional-order Razumikhin theorem. Secondly, according to the above stability criteria and some properties of fractional-order calculus, a delay-dependent condition that can guarantee the passivity of UFONNs with time-varying delays is given in the form of a linear matrix inequality (LMI) that can be reasonably solved in polynomial time using the LMI Control Toolbox. These conditions are not only delay-dependent but also order-dependent, and less conservative than some existing work. Finally, the rationality of the research results is proved by simulation.

Global Dissipativity Analysis and Stability Analysis for Fractional-Order Quaternion-Valued Neural Networks With Time Delays

This article studies dissipativity analysis of fractional-order quaternion-valued neural networks (FOQVNNs) with time delays. Two specific activation functions are considered along with common bounded and activation functions of Lipschitz-kind. Since quaternion multiplication is not commutative, we must divide the model, which is evaluated by quaternion, into four elements that are real-valued elements. On the basis of the construction of novel Lyapunov functional, and applying fractional-calculus theory, new criteria for the test of the global dissipativity and exponential stability of FOQVNNs model are established. FOQVNNs have also been suggested to provide global dissipativity and exponential stability, whereas nonlinear complex activation functions are constrained by the usage of linear matrix inequality methods, which utilize quaternion matrices and positive quaternion definite matrices. Finally, the effectiveness and superiority of the proposed approach is validated through numerical examples.

On the Finite-Time Boundedness and Finite-Time Stability of Caputo-Type Fractional Order Neural Networks with Time Delay and Uncertain Terms

This study investigates the problem of finite-time boundedness of a class of neural networks of Caputo fractional order with time delay and uncertain terms. New sufficient conditions are established by constructing suitable Lyapunov functionals to ensure that the addressed fractional-order uncertain neural networks are finite-time stable. Criteria for finite-time boundedness of the considered fractional-order uncertain models are also achieved. The obtained results are based on a newly developed property of Caputo fractional derivatives, properties of Mittag–Leffler functions and Laplace transforms. In addition, examples are developed to manifest the usefulness of our theoretical results.

Dynamic Analysis and Audio Encryption Application in IoT of a Multi-Scroll Fractional-Order Memristive Hopfield Neural Network

Fractional-order chaotic systems are widely used in the field of encryption because of its initial value sensitivity and historical memory. In this paper, the fractional-order definition of Caputo is introduced based on a nonideal flux-controlled memristive Hopfield neural network model, when changing the parameters of the fractional-order memristive Hopfield neural network (FMHNN) can generate a different amount of multi-scroll attractors. Some dynamical behaviors are investigated by numerical simulation, especially analyzed coexistence and bifurcation under different orders and different coupling strengths. The results show that the chaotic system of FMHNN has abundant dynamic behaviors. In addition, a chaotic audio encryption scheme under a Message Queueing Telemetry Transport (MQTT) protocol is proposed and implemented by Raspberry Pi; the audio encryption system based on FMHNN has a broad future in intelligent home and other IoT applications.

NOVEL RESULTS ON BIFURCATIONS FOR A FRACTIONAL-ORDER NEURAL NETWORK WITH NEUTRAL DELAYS

In comparison with numerous available results with regard to bifurcations on account of fractional-order neural networks (FONNs) with the retarded delays, this paper is concerned with the stability and bifurcation of FONN with neutral delays. Delay-induced bifurcation results are firstly, procured and the bifurcation diagrams with respect to different delays are nicely plotted. Then it reveals that both time delay and fractional order have important effects in affecting the stability and bifurcation of the developed FONN. It further detects that FONN involving neutral delays can postpone the onset of bifurcations of the developed FONN compared with FONN with the retarded delays. It hints that neutral delays have a vital impact on the dynamics of FONNs in comparison with FONN with the retarded delays, which has outstanding performance in maintaining stability of FONNs. Eventually, numerical examples are addressed to check the availability of the developed results. The obtained criterion can be evaluated as an important contribution to the stability issue of the considered neural networks (NNs).

Asymptotic and Finite-Time Synchronization of Fractional-Order Memristor-Based Inertial Neural Networks with Time-Varying Delay

This paper emphasized on studying the asymptotic synchronization and finite synchronization of fractional-order memristor-based inertial neural networks with time-varying latency. The fractional-order memristor-based inertial neural network model is offered as a more general and flexible alternative to the integer-order inertial neural network. By utilizing the properties of fractional calculus, two lemmas on asymptotic stability and finite-time stability are provided. Based on the two lemmas and the constructed Lyapunov functionals, some updated and valid criteria have been developed to achieve asymptotic and finite-time synchronization of the addressed systems. Finally, the effectiveness of the proposed method is demonstrated by a number of examples and simulations.

Bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks under signed digraph via adaptive strategy

In this work, an adaptive controller, formulated as linear feedback controls plus nonlinear parts, is synthesized to achieve bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks (FMNNs), in which follower FMNNs are linearly coupled under a signed digraph. The salient features of this research lie in two aspects: (1) the assumption on time-varying delays is very weak, since it neither requires boundedness of delays nor restricts the differentiation of time delay functions; (2) the adaptive controller contains no delay term, and it is feasible for both bounded and unbounded activation functions. As the preparatory work for stability analysis of the controlled synchronization error system, the ready-made results on delayed incommensurate fractional-order linear positive systems, especially stability condition and comparison principle, are perfected by relaxing premises of time-varying delays. Besides, a group of differential inclusion inequalities are established as a powerful aid in scaling the bipartite synchronization error system. More relevantly, an algebraic synchronization criterion, formulated in terms of coupling strength, inner coupling matrix, Laplacian matrix and FMNN system parameters, is proved with the benefit of vector Lyapunov function and positive system theory. With the controller utilized, linear feedback controls can be exerted merely on partial neurons of some selected FMNNs, which is exemplified by numerical simulations.