Bifurcation Control And Application Of Fractional-order Systems With Time Delay

The application of fractional calculus in physics and engineering is a hotspot in recent years.It is more accurate to introduce the fractional derivative into the model to describe some phenomena,and the uncertainty of the order makes the fractional-order system have infinite memory.The information interaction between different neurons must have different time delays,which will affect the dynamic behavior of the network.The concept of fractional order is introduced in PD controller.In addition to proportional gain and differential gain parameters,there is also an adjustable order parameter,which has more advantages in regulating system.Therefore,it is meaningful to study and control the dynamical behavior of fractional-order networks with multiple delays.On the basis of reviewing the predecessors of advances,this paper explores the dynamic behavior of fractional-order neural networks with multiple time delays,and explores the bifurcation control problem of small-world networks.The specific work is as follows:1.The dynamic behavior of fractional-order single-neuron networks with leakage is explored.Adopting leakage delay as bifurcation parameter,the corresponding characteristic equation was analyzed,and the sufficient conditions for Hopf bifurcation were obtained.In addition,the influence of order on system stability domain is obtained.2.This paper is concerned with the stability and Hopf bifurcation of fractional-order neural networks with discrete and distributed delays.By introducing two virtual neurons to the original network,a new four-neuron network only involving discrete delays is formed.It is the sum of discrete delays that is adopted as the bifurcation parameter,which demonstrates the existence of Hopf bifurcation.It is found that the critical value of bifurcation can be effectively manipulated by choosing appropriate system parameters and order.Furthermore,we describe the relationships between the parameters and the onset of bifurcation.3.A neoteric fractional-order Proportional-Derivative(PD)feedback controller is proposed to address the problem of bifurcation control for the small-world network model with discrete delay.We first determine the stability condition of the original system,then add the controller to adjust the occurrence of bifurcation,and finally obtain the sufficient condition for bifurcation.By regulating the controller parameters,the dynamic behavior for the controlled system can be effectively optimized.Finally,the relationships between the onset of the Hopf bifurcation and the controller parameters are obtained.