Fractional Differential Equations

Explicit formula and graphs of few special functions are derived in the paper on the basis of various definitions of various fractional derivatives and various fractional integrals. Their applications are also reviewed in the paper.

/pdf/applications-of-fractional-calculus-1c2m93jo56.pdf

Applications of Fractional Calculus

This paper is mainly concerned with the synchronization problem of reaction–diffusion neural networks (RDNNs) with delays and its direct application in image secure communications. An adaptive control is designed without a sign function in which the controller gain matrix is a function of time. The synchronization criteria are established for an error model derived from master–slave models through solving the set of linear matrix inequalities derived by constructing the suitable novel Lyapunov–Krasovskii functional candidate, Green’s formula, and Wirtinger’s inequality. If the proposed sufficient conditions are satisfied, then the global asymptotic synchronization of the error model is guaranteed. The numerical illustrations are provided to demonstrate the validity of the derived synchronization criteria. In addition, the role of system parameters is picturized through the chaotic nature of RDNNs and those unprecedented solutions is utilized to promote better security of image transactions. As is evident, the enhancement of image encryption algorithm is designed with two levels, namely, image watermarking and diffusion process. The contributions of this paper are discussed as concluding remarks.

Adaptive Synchronization of Reaction–Diffusion Neural Networks and Its Application to Secure Communication

This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop control and adaptive state feedback control is designed to guarantee the global projective lag synchronization of the addressed FOMBNNs model. Secondly, by using a Lyapunov–Krasovskii functional and Barbalet’s lemma, a new brand of sufficient criterion is proposed to ensure the projective lag synchronization of the FOMBNNs model considered. Moreover, as special cases by using a hybrid control scheme, some sufficient conditions are derived to ensure the global projective synchronization, global complete synchronization and global anti-synchronization for the FOMBNNs model considered. Finally, numerical simulations are provided to check the accuracy and validity of our obtained synchronization results.

https://www.mdpi.com/2227-7390/7/8/759/pdf

Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays

In this paper, to investigate the exponential synchronization of stochastic neural networks, a new periodically intermittent discrete observation control (PIDOC) is first proposed. Different from the existing periodically intermittent control, our control in control time is feedback control based on discrete-time state observations (FCDSOs) instead of a continuous-time one. By employing the Lyapunov method, graph theory, and theory of differential inclusions, the exponential synchronization of stochastic neural networks with a discontinuous right-hand side is realized by PIDOC and some sufficient conditions are presented. Especially, when control width tends to control period, PIDOC will be reduced to a general FCDSO and we give some detailed discussions. Then, we provide some corollaries about synchronization in mean square, asymptotical synchronization in mean square, and exponential synchronization of stochastic neural networks under FCDSO. Finally, some numerical simulations are provided to demonstrate our analytical results.

Intermittent Discrete Observation Control for Synchronization of Stochastic Neural Networks

Master-slave exponential synchronization of delayed complex-valued memristor-based neural networks via impulsive control.

This article solves the event-triggered exponential synchronization problem for a class of complex-valued memristive neural networks with time-varying delays. The drive-response complex-valued memristive neural networks are translated into two real-valued memristive neural networks through the method of separating the complex-valued memristive neural networks into real and imaginary parts. In order to reduce the information exchange frequency between the sensor and the controller, a novel event-triggered mechanism with the event-triggering functions is introduced in wireless communication networks. Some sufficient conditions are established to achieve the event-triggered exponential synchronization for drive-response complex-valued memristive neural networks with time-varying delays. In addition, to guarantee that the Zeno behavior cannot occur, a positive lower bound for the interevent times is explicitly derived. Finally, numerical simulations are provided to illustrate the effectiveness and superiority of the obtained theoretical results.

Event-Triggered Exponential Synchronization for Complex-Valued Memristive Neural Networks With Time-Varying Delays

Finite-time synchronization of fractional-order memristive recurrent neural networks with discontinuous activation functions

This study focuses on the exponential stabilisation problem for a general class of memristive neural networks subjected to both stochastic disturbance and time-varying delays under periodically intermittent adaptive control. The stochastic disturbances are described as Brownian motions in the considered networks. An adaptive updated rule and a periodically intermittent adaptive control strategy are designed for the exponential stabilisation of memristive neural networks subjected to both stochastic disturbance and time-varying delays. Then, by adopting the adaptive control technique, differential inclusion theory and set-valued maps, and by building a new Lyapunov–Krasovskii functional, many novel sufficient conditions are proposed to guarantee exponential stabilisation for stochastic memristive neural networks. Different from existing results on stabilisation of stochastic memristive neural networks, the obtained criteria in this study are directly derived according to the parameters of networks. Finally, an example is carried out to demonstrate the validity of the theoretic results.

https://ietresearch.onlinelibrary.wiley.com/doi/epdf/10.1049/iet-cta.2017.0021

Huiyuan Li

Papers

Master-slave exponential synchronization of delayed complex-valued memristor-based neural networks via impulsive control.

Event-Triggered Exponential Synchronization for Complex-Valued Memristive Neural Networks With Time-Varying Delays

Finite-time synchronization of fractional-order memristive recurrent neural networks with discontinuous activation functions

Exponential stabilisation of stochastic memristive neural networks under intermittent adaptive control

Exponential adaptive synchronization of stochastic memristive chaotic recurrent neural networks with time-varying delays☆