In this paper, a fractional-order recurrent neural network is proposed and several topics related to the dynamics of such a network are investigated, such as the stability, Hopf bifurcations, and undamped oscillations. The stability domain of the trivial steady state is completely characterized with respect to network parameters and orders of the commensurate-order neural network. Based on the stability analysis, the critical values of the fractional order are identified, where Hopf bifurcations occur and a family of oscillations bifurcate from the trivial steady state. Then, the parametric range of undamped oscillations is also estimated and the frequency and amplitude of oscillations are determined analytically and numerically for such commensurate-order networks. Meanwhile, it is shown that the incommensurate-order neural network can also exhibit a Hopf bifurcation as the network parameter passes through a critical value which can be determined exactly. The frequency and amplitude of bifurcated oscillations are determined.

This paper studies the stability and Hopf bifurcation in a class of high-dimension neural network involving the discrete and distributed delays under a new framework. By introducing some virtual neurons to the original system, the impact of distributed delay can be described in a simplified way via an equivalent new model. This paper extends the existing works on neural networks to high-dimension cases, which is much closer to complex and real neural networks. Here, we first analyze the Hopf bifurcation in this special class of high dimensional model with weak delay kernel from two aspects: one is induced by the time delay, the other is induced by a rate parameter, to reveal the roles of discrete and distributed delays on stability and bifurcation. Sufficient conditions for keeping the original system to be stable, and undergoing the Hopf bifurcation are obtained. Besides, this new framework can also apply to deal with the case of the strong delay kernel and corresponding analysis for different dynamical behaviors is provided. Finally, the simulation results are presented to justify the validity of our theoretical analysis.

Recent studies on Hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two, three, four, five, or six neurons. It is well known that neural networks are complex and large-scale nonlinear dynamical systems, so the dynamics of the delayed neural networks are very rich and complicated. Although discussing the dynamics of networks with a few neurons may help us to understand large-scale networks, there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large-scale networks. In this paper, a general delayed bidirectional associative memory neural network model with n +1 neurons is considered. By analyzing the associated characteristic equation, the local stability of the trivial steady state is examined, and then the existence of the Hopf bifurcation at the trivial steady state is established. By applying the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution. Furthermore, the paper highlights situations where the Hopf bifurcations are particularly critical, in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections. It is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons. Numerical simulations are carried out to illustrate the main results.

This paper investigates bipartite consensus of multi-agent systems under unknown false data injection attacks. Attacks are launched in the channels from the controller to the actuator. To compensate the impact of attacks, an extended observer including the absolute output information is firstly constructed, and then an observer-based event-triggered controller is designed to save resources. Moreover, an observer-based fully asynchronous event-triggered controller is proposed to further reduce the communication cost. Sufficient criteria are deduced to guarantee bipartite consensus and Zeno behavior is further ruled out. Finally, numerical examples are given to illustrate the effectiveness of the derived results.

Bifurcation control remains largely unresolved for fractional-order dynamical systems. This article addresses the optimal control issue of bifurcation for a delayed complex networks model with Caputo derivative, where the time delay is selected as the variable parameter. We first devise a fractional-order proportional-integral-derivative (PID) feedback synthesis for regulation of the Hopf bifurcation embedded in a delayed small-world network model with Caputo derivative. Dynamic stability criterion and Hopf bifurcation condition are obtained by carrying out the eigenvalue analysis of the controlled network. The stability range of the parameters of the PID control is evaluated completely for the small-world network. We can optimize the dynamics of stability and bifurcation of small-world networks by manipulating the gain parameters. Finally, we implement some simulations to show the performance of the presented PID scheme. The numerical simulations verify the advantage of the fractional PID controller in bifurcation control.