Dynamic Analysis Of High-dimensional Fractionalorder Neural Networks And Design Of Bifurcation Control

Due to the complexity and wide application prospect of neural networks,the neural bifurcation dynamics has always been the research hotspot of bifurcation dynamics.Unfortunately,most existing work on dynamics of neural networks is still limited to the low-dimensional situation,and few work involves the dynamics of high-dimensional neural networks.The main reason is that it is difficult to calculate the determinant of the high-dimensional matrix and acquire the rule of the characteristic equation with the increase of neuron node number through traditional matrix determinant operation,and the complexity of high-order characteristic equations makes dynamics analysis difficult.In addition,due to the in-depth research of bifurcation dynamics of complex networks,scholars began to pay attention to the research of bifurcation control.However,most bifurcation control tools still use traditional simple control strategies for now,such as state feedback control,time-delay feedback control,hybrid control and so on.Nevertheless,these control strategies have their own defects in the application of bifurcation control.Therefore,more bifurcation control tools need to be further developed.The contributionsbf this study are as follows:1)High-dimensional neural network models with chain structure and ring structure are established.The characteristic equations of the systems are summarized by means of flow-graph representation of matrix,flow-graph decomposition and Coates flow-graph formula.The stability conditions and bifurcation thresholds based on the delay of the corresponding systems are analyzed by analyzing the characteristic equations of these two neural networks.The result shows that with the increase of the number of neuron,the bifurcation threshold of the chain-structure neural network decreases gradually and there is a lower bound value,while the bifurcation threshold of the ring structure neural network decreases by oscillation.The bifurcation thresholds of these two kinds of neural networks have a correlation law,which is determined by the structural properties.2)The bifurcation dynamics of high-dimensional BAM neural networks is studied.The characteristic equation of the system is obtained by linearization method and matrix determinant operation.The stability condition of the system and the bifurcation threshold based on the system delay are analyzed by eigenvalue analysis method.The result indicates that the bifurcation threshold of the BAM neural network decreases with the increase of the number of neuron,and the smaller the absolute value of the difference between the numbers of neurons in two layersis is,the smaller the bifurcation threshold is.3)The above three kinds of high-dimensional neural networks are developed into the corresponding fractional-order neural networks by introducing the theory of fractional calculus,and the stability conditions and bifurcation threshold of the corresponding fractional-order systems are analyzed by the bifurcation theory of the fractional order system.The result illustrates that the bifurcation thresholds of these three kinds of fractional-order systems increase with the decrease of system order.4)Combined with the classical Proportional-Integral-Derivative(PID)control strategy and fractional calculus theory,a fractional-order PID control strategy is proposed,and the designed control strategy is applied to the bifurcation control of a fractional-order system with the same order.The effects of the control gains on the stability and bifurcation of the controlled system are illustrated by theoretical analysis and simulation experiments.The result shows that the bifurcation threshold of the controlled system increase with the decrease of system order.5)Based on the above research on the fractional-order PID control strategy,a fractional-order PD control strategy with variable order is designed,and the designed fractional-order PD control strategy is applied to the bifurcation control of an integer-order system to study the effects of the control gains and order of controller on the stability and bifurcation of the controlled system.The simulation result shows that with the increase of the order of the controller,the bifurcation threshold of the controlled system always increases first and then decreases,which means that there is an optimal order to maximize the bifurcation threshold of the controlled system.6)Considering some defects of the traditional time-delay feedback control strategy in the applications of stability control and bifurcation control,the distributed time-delay term is used to replace the discrete time-delay term of the traditional time-delay feedback control strategy to design a time-delay feedback control strategy with distributed characteristics.By analyzing the mathematical properties of the distributed time-delay term,the structural block diagram of the designed control strategy is provided,and the designed control strategy is applied to the time-delay system to analyze the influence of the controller parameters on the stability and bifurcation dynamics of the controlled system.In this study,the bifurcation dynamics of high-dimensional chain structure neural network,high-dimensional ring structure neural network and high-dimensional BAM neural network are analyzed,and the traditional matrix determinant operation is replaced by the flow-graph representation of matrix,flow-graph decomposition and Coates flow-graph formula.The results show that it is easier to obtain the characteristic equation of the neural network model with specific structure through these flow-graph methods.The obtained characteristic equation has the structural properties of the model,and more intuitively shows the variation law of the characteristic equation with the increase of the number of neural nodes.Based on the above research on bifurcation dynamics,this study improves some existing bifurcation control strategies.By combining the fractional calculus theory with the classical PID control strategy,the fractional-order PID bifurcation control strategy and PD bifurcation control strategy with variable order are designed.In addition,a distributed time-delay feedback control strategy is designed to make up for some defects of the traditional time-delay feedback control strategy in the application of stability control and bifurcation control.The results show that the designed time-delay feedback control strategy with distributed characteristics can not only retain the excellent characteristics of the traditional time-delay feedback control strategy,but also effectively avoid some defects of the traditional time-delay feedback control strategy.