Fractional-order Neural Networks Research Articles

Projective Multi-Synchronization of Fractional-order Complex-valued Coupled Multi-stable Neural Networks with Impulsive Control

In this paper, we study a projective multi-synchronization problem for fractional-order complex-valued coupled multi-stable neural networks (FCVCMNNs) with time-delays. Using a complex decomposition approach, FCVCMNNs are divided into their real and imaginary components. Our method uses certain conditions for each subnetwork to achieve the multiple local equilibrium points or stable periodic orbits that are exponentially stable, which, when combined with the Lyapunov functions method, result in FCVCMNNs that are projectively multi-synchronized. The FCVCMNNs, on the other hand, are examined directly through the use of the Lyapunov functional method and linear matrix inequality (LMI). Various new sufficient conditions in the form of complex-valued LMIs are presented for the projective multi-synchronization of the considered FCVCMNNs. As a final step, we provide two numerical simulations to verify the effectiveness of the main results derived in this paper.

New approach to finite-time stability for fractional-order BAM neural networks with discrete and distributed delays

• A novel generalized FODGI with two different orders is developed, which includes the one in [39, Theorem 3.2] as a special case and can be used to analyze the qualitative properties of various FODSs with two different orders. • As a straightforward generalization, a novel generalized FODGI with multiple different orders is also established, which includes the one in [40, Theorem 1.4] as a special case. • Two novel FTS criteria of FOBAMNNs with the orders 0 < ℘ < 1 and 1 < ℘< 2 are established by the obtained FODGI with two different orders. In this approach, the term w ( t ) and delay term w ( t − τ ) can be analyzed in a unified framework. This is to say, the delay term w ( t − τ ) needs not to be transformed to the term w ( t ). • Comparisons with the existing contributions: a) The obtained results in this article is less conservative than the ones in the existing works [30, 31]. b) There are delay terms in the obtained novel FODGI with multiple different orders. Therefore, the proposed inequality in this article is more general. The finite-time stability (FTS) for fractional-order bidirectional associative memory neural networks (FOBAMNNs) with discrete and distributed delays is investigated in this article. Firstly, a novel fractional-order delayed Gronwall inequality (FODGI) with two different orders is developed. As a straightforward generalization, a novel generalized FODGI with multiple different orders is also established, which can be applied to investigate the stability of fractional-order delayed systems (FODSs) with multiple different orders. Secondly, based on the former inequality, two novel FTS criteria of FOBAMNNs with the orders 0 < ℘ < 1 and 1 < ℘ < 2 are established. Finally, three examples are exhibited to illustrate the effectiveness and the less conservativeness of the proposed methods.

Adaptive Event-Triggered Synchronization of Uncertain Fractional Order Neural Networks with Double Deception Attacks and Time-Varying Delay.

This paper investigates the problem of adaptive event-triggered synchronization for uncertain FNNs subject to double deception attacks and time-varying delay. During network transmission, a practical deception attack phenomenon in FNNs should be considered; that is, we investigated the situation in which the attack occurs via both communication channels, from S-C and from C-A simultaneously, rather than considering only one, as in many papers; and the double attacks are described by high-level Markov processes rather than simple random variables. To further reduce network load, an advanced AETS with an adaptive threshold coefficient was first used in FNNs to deal with deception attacks. Moreover, given the engineering background, uncertain parameters and time-varying delay were also considered, and a feedback control scheme was adopted. Based on the above, a unique closed-loop synchronization error system was constructed. Sufficient conditions that guarantee the stability of the closed-loop system are ensured by the Lyapunov-Krasovskii functional method. Finally, a numerical example is presented to verify the effectiveness of the proposed method.

Sliding Mode Matrix-Projective Synchronization for Fractional-Order Neural Networks

This work generalizes the projection scaling factor to a general constant matrix and proposes the matrix-projection synchronization (MPS) for fractional-order neural networks (FNNs) based on sliding mode control firstly. This kind of scaling factor is far more complex than the constant scaling factor, and it is highly variable and difficult to predict in the process of realizing the synchronization for the driving and response systems, which can ensure high security and strong confidentiality. Then, the fractional-order integral sliding surface and sliding mode controller for FNNs are designed. Furthermore, the criterion for realizing MPS is proved, and the reachability and stability of the synchronization error system are analyzed, so that the global MPS is realized for FNNs. Finally, a numerical application is given to demonstrate the feasibility of theory analysis. MPS is more general, so it is reduced to antisynchronization, complete synchronization, projective synchronization (PS), and modified PS when selecting different projective matrices. This work will enrich the synchronization theory of FNNs and provide a feasible method to study the MPS of other fractional-order dynamical models.

Pinning multisynchronization of delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations

This paper concerns the multisynchronization issue for delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations. First, the coexistence of multiple equilibrium states for isolated subnetwork is analyzed. By means of state-space decomposition, fractional-order Halanay inequality and Caputo derivative properties, the novel algebraic sufficient conditions are derived to ensure that the addressed networks with arbitrary activation functions have multiple locally stable almost periodic orbits or equilibrium points. Then, based on the obtained multistability results, a pinning control strategy is designed to realize the multisynchronization of the N coupled networks. By the aid of graph theory, depth first search method and pinning control law, some sufficient conditions are formulated such that the considered neural networks can possess multiple synchronization manifolds. Finally, the multistability and multisynchronization performance of the considered neural networks with different activation functions are illustrated by numerical examples.

Mittag-Leffler stability and asymptotic [formula omitted]-periodicity of fractional-order inertial neural networks with time-delays

In this paper, the stability for a class fractional-order inertial neural networks with time-delay are investigated. Moreover, some sufficient conditions for the Mittag-Leffler stability and the asymptotical ω-periodicity are obtained, by the appropriate transformation, using the property of the Riemann-Liouville fractional integral and derivative. In the end, results of the theoretical derivation are verified by virtue of two numerical simulation examples.

DYNAMICAL ANALYSIS OF A SELF-CONNECTION FRACTIONAL-ORDER NEURAL NETWORK

This paper researches the problem of bifurcation for a ring fractional-order neural network (FONN) with self-connection delay and communication delay. Self-connection delay is firstly regarded as a bifurcation parameter to examine the bifurcations of the developed FONN, and the critical values of bifurcations with respect to self-connection delay are derived. Second, communication delay is further taken as a bifurcation parameter to investigate the bifurcations of the designed FONN, and the stability zones and bifurcation points are determined. It reveals that FONN exhibits excellent stability performance when electing a lesser value of them, and the stability performance is devastated and Hopf bifurcation arises upon taking a larger one. Eventually, the efficiency of the developed theory is appraised on account of numerical verifications.

Synchronization of fractional-order spatiotemporal complex-valued neural networks in finite-time interval and its application

This paper focuses on the synchronization of fractional-order complex-valued neural networks (FOCVNNs) with reaction–diffusion terms in finite-time interval. Different from the existing complex-valued neural networks (CVNNs), the reaction–diffusion phenomena and fractional derivative are first considered into the system, meanwhile, the parameter switching (the system parameters will switch with the state) is considered, which makes the presented model more comprehensive. By choosing an appropriate Lyapunov function, the driver and response systems achieve Mittag-Leffler synchronization under a suitable controller. In addition, based on the fractional calculus theorem and the basic inequality methods, a criterion of synchronization for the error system in finite-time interval is derived and the upper bound of the corresponding finite synchronization time can be obtained. Finally, two examples are provided, one is a numerical example to explain the effectiveness of the main results, and the other shows that the results of this paper can be applied to image encryption for any size with high-security coefficient.

Pseudo-State Estimation for Fractional Order Neural Networks

Pseudo-State Estimation for Fractional Order Neural Networks

Finite-time stability of fractional-order fuzzy cellular neural networks with time delays

The finite-time stability of fractional-order fuzzy cellular neural networks with time delays is investigated. One of the difficulties in stability analysis of fractional-order delayed systems lies in how to handle the delay terms in the systems. In view of this difficulty, firstly, a new fractional-order Gronwall inequality with time delay is developed; this can be widely used to study finite-time stability of a variety of fractional-order systems with time delays. Secondly, a new criterion for the finite-time stability of fractional-order fuzzy cellular neural networks with time delays is derived in terms of this inequality; it is less conservative than the one using the existing fractional-order Gronwall inequality with time delays. Finally, two examples are given to expound the effectiveness and less conservativeness of the proposed results.

Finite-time stabilization of fractional-order fuzzy quaternion-valued BAM neural networks via direct quaternion approach

This paper investigates fractional-order fuzzy quaternion-valued BAM neural networks (FOFQBAMNNs) without decomposition. By virtue of a novel contraction mapping, the existence and uniqueness of the equilibrium point is yielded. Furthermore, according to some basic knowledge on fractional calculus, inequality techniques of fuzzy logic and reduction to absurdity, some criteria are yielded to guarantee finite-time stabilization of FOFQBAMNNs via original quaternion-valued controllers, and the settling times of corresponding finite-time stabilization are derived. Finally, the feasibility of our obtained theoretical results is illustrated by some numerical simulations.

Global synchronization of uncertain fractional-order BAM neural networks with time delay via improved fractional-order integral inequality

This paper is concerned with the problem of global synchronization for a class of uncertain fractional-order bidirectional associative memory (FOBAM) neural networks (NNs) with constant time delay. Some linear matrix inequality (LMI) based conditions have been established for ensuring the stability behavior of the governing augmented system derived from the considered fractional-order master system and the constructed slave system. In such LMI conditions, for utilizing the information on the time delay terms and the order of the fractional derivative, some fractional-order integral inequalities have been developed. Next, by defining a new Lyapunov–Krasovskii functional (LKF) and based on the improved fractional-order inequalities, the global synchronization criteria have been established in terms of a solvable set of LMIs for the considered class of uncertain FOBAM NNs without uncertain matrices. Further, the obtained results have been extended to the case of uncertain FOBAM NNs. Finally, a numerical example is presented to illustrate the feasibility and effectiveness of the proposed theoretical results.

Exponential stability for nonlinear fractional order sampled-data control systems with its applications

This paper investigates some topics about fractional order nonlinear systems with sampled-data. First, according to comparison principle and Laplacian transform method, sufficient conditions are derived to guarantee that the fractional order sampled-data control systems are globally and exponentially stable. Then, based on the stability results above and some properties of fractional order integral and derivative, the sampled-data controller is designed for the fractional order neural networks. Furthermore, the synchronization criteria of fractional order dynamical networks with sampled-data communications are obtained based on matrix technique and above analysis methods. Finally, three numerical examples are provided to illustrate the effectiveness of the derived results.

Exact coexistence and locally asymptotic stability of multiple equilibria for fractional-order delayed Hopfield neural networks with Gaussian activation function

This paper explores the multistability issue for fractional-order Hopfield neural networks with Gaussian activation function and multiple time delays. First, several sufficient criteria are presented for ensuring the exact coexistence of 3n equilibria, based on the geometric characteristics of Gaussian function, the fixed point theorem and the contraction mapping principle. Then, different from the existing methods used in the multistability analysis of fractional-order neural networks without time delays, it is shown that 2n of 3n total equilibria are locally asymptotically stable, by applying the theory of fractional-order linear delayed system and constructing suitable Lyapunov function. The obtained results improve and extend some existing multistability works for classical integer-order neural networks and fractional-order neural networks without time delays. Finally, an illustrative example with comprehensive computer simulations is given to demonstrate the theoretical results.

New results on finite-time stability of fractional-order neural networks with time-varying delay

New results on finite-time stability of fractional-order neural networks with time-varying delay

Hidden coexisting firings in fractional-order hyperchaotic memristor-coupled HR neural network with two heterogeneous neurons and its applications.

The firing patterns of each bursting neuron are different because of the heterogeneity, which may be derived from the different parameters or external drives of the same kind of neurons, or even neurons with different functions. In this paper, the different electromagnetic effects produced by two fractional-order memristive (FOM) Hindmarsh-Rose (HR) neuron models are selected for characterizing different firing patterns of heterogeneous neurons. Meanwhile, a fractional-order memristor-coupled heterogeneous memristive HR neural network is constructed via coupling these two heterogeneous FOM HR neuron models, which has not been reported in the adjacent neuron models with memristor coupling. With the study of initial-depending bifurcation behaviors of the system, it is found that the system exhibits abundant hidden firing patterns, such as periods with different topologies, quasiperiodic firings, chaos with different topologies, and even hyperchaotic firings. Particularly, the hidden hyperchaotic firings are perfectly detected by two-dimensional Lyapunov stability graphs in the two-parameter space. Meanwhile, the hidden coexisting firing patterns of the system are excited from two scattered attraction domains, which can be confirmed from the local attraction basins. Furthermore, the color image encryption based on the system and the DNA approach owns great keyspace and a good encryption effect. Finally, the digital implementations based on Advanced RISC Machine are in good coincidence with numerical simulations.

Novel methods to global Mittag-Leffler stability of delayed fractional-order quaternion-valued neural networks

In this paper, a type of fractional-order quaternion-valued neural networks (FOQVNNs) with leakage and time-varying delays is established to simulate real-world situations, and the global Mittag-Leffler stability of the system is investigated by using the non-decomposition method. First, to avoid decomposing the system into two complex-valued systems or four real-valued systems, a new sign function for quaternion numbers is introduced based on the ones for real and complex numbers. And two novel lemmas for quaternion-valued sign function and Caputo fractional derivative are established in quaternion domain, which are used to investigate the stability of FOQVNNs. Second, a concise and flexible quaternion-valued state feedback controller is directly designed and a novel 1-norm Lyapunov function composed of the absolute values of real and imaginary parts is established. Then, based on the designed quaternion-valued state feedback controller and the proposed lemmas, some sufficient conditions are given to ensure the global Mittag-Leffler stability of the system. Finally, a numerical simulation is given to verify the theoretical results.

Bifurcations due to different delays of high-order fractional neural networks

This paper expounds the bifurcations of two-delayed fractional-order neural networks (FONNs) with multiple neurons. Leakage delay or communication delay is viewed as a bifurcation parameter, stability zones and bifurcation conditions with respect to them are commendably established, respectively. It declares that both leakage delay and communication delay immensely influence the stability and bifurcation of the developed FONNs. The explored FONNs illustrate superior stability performance if selecting a lesser leakage delay or communication delay, and Hopf bifurcation generates once they overstep their critical values. The verification of the feasibility of the developed analytic results is implemented via numerical experiments.

Globallyβ-Mittag-Leffler stability andβ-Mittag-Leffler convergence in Lagrange sense for impulsive fractional-order complex-valued neural networks

This paper explores the globally β-Mittag-Leffler stability in Lagrange sense for the fractional-order complex-valued neural network (FOCVNN) with impulsive effects. By Lyapunov method and matrix inequalities, some novel sufficient conditions are obtained to guarantee the globally β-Mittag-Leffler stability in Lagrange sense for two class of complex-valued (CV) activation functions. The convergent rate is also given, which is controlled by the parameters of the addressed system. The existence and uniqueness of the solution for this system do not require consideration. To show the validity and usefulness of the results, two numerical stimulations are provided.

Global asymptotic stability and S-asymptotic [formula omitted]-periodicity of impulsive non-autonomous fractional-order neural networks

This paper is concerned with the globally S-asymptotic ω-periodicity of impulsive non-autonomous fractional-order (FO) neural networks (NNs). The definition of the piecewise S-asymptotic ω-periodicity of the considered system is given and a new Banach space is constructed. Based on the Banach mapping principle, several novel criteria for the existence and uniqueness of piecewise S-asymptotically ω-periodical solutions are obtained. Then by the FO differential and integral inequalities, we discuss the globally asymptotic stability and globally asymptotic periodicity of the probed FONN. Our methods and results are new. Finally, a numerical example is given to verify the validity of our findings.