Min Xiao

Bifurcations in a delayed fractional complex-valued neural network

Complex-valued neural networks (CVNNs) with integer-order have attracted much attention, and which have been well discussed. Fractional complex-valued neural networks (FCVNNs) are more suitable to describe the dynamical properties of neural networks, but have rarely been studied. It is the first time that the stability and bifurcation of a class of delayed FCVNN is investigated in this paper. The activation function can be expressed by separating into its real and imaginary parts. By using time delay as the bifurcation parameter, the dynamical behaviors that including local asymptotical stability and Hopf bifurcation are discussed, the conditions of emergence of bifurcation are obtained. Furthermore, it reveals that the onset of the bifurcation point can be delayed as the order increases. Finally, an illustrative example is provided to verify the correctness of the obtained theoretical results.

Hopf Bifurcation of an

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.

(n+1)

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.

-Neuron Bidirectional Associative Memory Neural Network Model With Delays

This paper investigates an issue of bifurcation control for a novel incommensurate fractional-order predator-prey system with time delay. Firstly, the associated characteristic equation is analyzed by taking time delay as the bifurcation parameter, and the conditions of creation for Hopf bifurcation are established. It is demonstrated that time delay can heavily effect the dynamics of the proposed system and each order has a major influence on the creation of bifurcation simultaneously. Then, a linear delayed feedback controller is introduced to successfully control the Hopf bifurcation for such system. It is shown that the control effort is markedly influenced by feedback gain. It is further found that the onset of the bifurcation can be delayed as feedback gain decreases. Finally, two illustrative examples are exploited to verify the validity of the obtained newly results.

Undamped Oscillations Generated by Hopf Bifurcations in Fractional-Order Recurrent Neural Networks With Caputo Derivative

This paper proposes to use a state feedback method to control the Hopf bifurcation for a novel congestion control model, i.e. the exponential random early detection (RED) algorithm with a single link and a single source. The gain parameter of the congestion control model is chosen as the bifurcation parameter. The analysis shows that in the absence of the state feedback controller, the model loses stability via the Hopf bifurcation early, and can maintain a stationary sending rate only in a certain domain of the gain parameter. When applying the state feedback controller to the model, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the model possesses a stable sending rate in a larger parameter range. Furthermore, explicit formulae to determine the properties of the Hopf bifurcation are obtained. Numerical simulations are given to justify the validity of the state feedback controller in bifurcation control.

Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders

Abstract By only designing the internal coupling, quasi synchronization of heterogeneous complex networks coupled by N nonidentical Duffing-type oscillators without any external controller is investigated in this paper. To achieve quasi synchronization, the average of states of all nodes is designed as the virtual target. Heterogeneous complex networks with two kinds of nonlinear node dynamics are analyzed firstly. Some sufficient conditions on quasi synchronization are obtained without designing any external controller. Quasi synchronization means that the states of all nonidentical nodes will keep a bounded error with the virtual target. Then the heterogeneous complex network with impulsive coupling which means the network only has coupling at some discrete impulsive instants, is further discussed. Some sufficient conditions on heterogeneous complex network with impulsive coupling are derived. Based on these results, heterogeneous complex network can still reach quasi synchronization even if its nodes are only coupled at discrete impulsive instants. Finally, two examples are provided to verify the theoretical results.

Bifurcation control of a congestion control model via state feedback

In this technical note, fractional-order congestion control systems are introduced for the first time. In comparison with the conventional integer-order dual congestion control algorithms, the fractional control algorithms are more accurate and versatile. Bifurcation theory in fractional-order differential equations is still an outstanding problem. Sufficient conditions for the occurrence of Hopf bifurcations are extended from integer-order dynamical systems to fractional-order cases. Then, these conditions are used to establish the existence of Hopf bifurcations for the delayed fractional-order model of dual congestion control algorithms proposed in this note. Finally, the onsets of bifurcations are identified, where Hopf bifurcations occur and a family of oscillations bifurcate from the equilibrium. Illustrative examples are also provided to demonstrate the theoretical results.

Synchronization of coupled heterogeneous complex networks

This paper mainly studies the stability and Hopf bifurcation criteria of hub-based genetic regulatory networks with multiple delays and bidirectional couplings. The hub-structured network is an important motif in complex networks, which provides a new view angle on structure to describe the regulation mechanism between genes (including both mRNAs and proteins). It is well known that hubs play a leading role in characterizing the network dynamical behaviors. However, the dynamics of hubs or star-coupled systems is not well understood. In this paper, we first examine the existence of the positive equilibria in this type of genetic networks. By analyzing the associated characteristic equation, we present sufficient conditions of biochemical parameters for delay-independent local stability in hub-coupled genetic regulatory networks. Then we investigate their Hopf bifurcation when such networks lose their stability. Specific conditions for delay-dependent stability and Hopf bifurcations in genetic networks with hub structure are derived. It is found that the dynamics of hub-structured genetic regulatory networks has no direct relationship with single time delay or individual connection, but instead depends on the sum of all delays among all genes and the product of the connection strengths between all genes. Finally, some simulation examples are provided to substantiate our analysis.

Stability and Bifurcation of Delayed Fractional-Order Dual Congestion Control Algorithms

In this paper, we propose a delayed fractional-order congestion control model which is more accurate than the original integer-order model when depicting the dual congestion control algorithms. The presence of fractional orders requires the use of suitable criteria which usually make the analytical work so harder. Based on the stability theorems on delayed fractionalorder differential equations, we study the issue of the stability and bifurcations for such a model by choosing the communication delay as the bifurcation parameter. By analyzing the associated characteristic equation, some explicit conditions for the local stability of the equilibrium are given for the delayed fractionalorder model of congestion control algorithms. Moreover, the Hopf bifurcation conditions for general delayed fractional-order systems are proposed. The existence of Hopf bifurcations at the equilibrium is established. The critical values of the delay are identified, where the Hopf bifurcations occur and a family of oscillations bifurcate from the equilibrium. Same as the delay, the fractional order normally plays an important role in the dynamics of delayed fractional-order systems. It is found that the critical value of Hopf bifurcations is crucially dependent on the fractional order. Finally, numerical simulations are carried out to illustrate the main results.

Stability and Bifurcation Analysis of Arbitrarily High-Dimensional Genetic Regulatory Networks With Hub Structure and Bidirectional Coupling

Abstract Bifurcation and control of fractional-order systems are still an outstanding problem. In this paper, a novel delayed fractional-order model of small-world networks is introduced and several topics related to the dynamics and control of such a network are investigated, such as the stability, Hopf bifurcations, and bifurcation control. The nonlinear interactive strength is chosen as the bifurcation parameter to analyze the impact of the interactive strength parameter on the dynamics of the fractional-order small-world network model. Firstly, the stability domain of the equilibrium is completely characterized with respect to network parameters, delays and orders, and some explicit conditions for the existence of Hopf bifurcations are established for the delayed fractional-order model. Then, a fractional-order Proportional-Derivative (PD) feedback controller is first put forward to successfully control the Hopf bifurcation which inherently happens due to the change of the interactive parameter. It is demonstrated that the onset of Hopf bifurcations can be delayed or advanced via the proposed fractional-order PD controller by setting proper control parameters. Meanwhile, the conditions of the stability and Hopf bifurcations are obtained for the controlled fractional-order small-world network model. Finally, illustrative examples are provided to justify the validity of the control strategy in controlling the Hopf bifurcation generated from the delayed fractional-order small-world network model.