Study Of Stability,Bifurcation And Control Algorithms For Dynamical Systems With Time Delays

The bifurcation is a very general fettle that plays a significant role in the understanding and mastery of the periodic solution and the dynamic characteristics of the system.Among them,Hopf bifurcation is a dynamic bifurcation phenomenon,which is the key index to measure the performance of nonlinear dynamic system.At present,Hopf bifurcation not only occupies an important position in control engineering and theoretical research,but also has been widely used in practical models in many disciplines such as finance,medical devices,biochemistry,and even astronomy,and has a profound impact on scientists.And the way of thinking and research of scholars.As we all know,time delay is common in dynamic systems.Many studies have found that due to the appearance of time delay factors,it will destroy the stability of the system and cause the system to undergo unfavorable behaviors such as bifurcation and oscillation.However,proper delay may also help to improve system stability.Hence,the study of dynamic systems with delays has important theoretical and practical importance.Based on previous research results,this paper will focus on the Hopf bifurcation theory of three kinds of dynamic systems with time delay,including the nearest-neighbor coupled neural network,the malicious virus propagation network under the information physical fusion system,and the weak kernel neural network under the proportional differential feedback control(PD control).The specific work is as follows:1.Based on the consideration of the complexity of the neural network in practical applications,this paper breaks through the limitation of only studying the connection of adjacent nodes in the past.And we further considers that there is still connection relationship between non adjacent nodes,studies a nearest neighbor coupled neural network with 6 neurons and 12 discrete delays.The dynamical behavior of the system with its variation is explored using the leakage time lag as the bifurcation parameter,and sufficient terms are derived for the Hopf bifurcation of the system due to the leakage delay.2.Considering that the spread of malicious virus in the information physical fusion system has time delay.This paper explores a malicious virus propagation network with latency,isolation and other delays,studies the propagation mechanism of malicious virus with multiple delays in the Information Physics fusion system.And analyzes the stability and Hopf bifurcation of the dynamic system,Thus,it provides a theoretical basis for eliminating or delaying the adverse bifurcation behavior caused by the spread of malicious virus.3.Aiming at a single neuron network model with discrete and distributed time delays,a proportional-differential feedback controller is proposed.By exploring the stability of the characteristic root of the controlled system,the stability switching of the system with the change of the time delay and the sufficient term of Hopf bifurcation is found.Lastly,the ratio of the critical point of the Hopf bifurcation to the controller variables are obtained.By adjusting the controller variables,it can be seen that the bifurcation activity of the controlled system can be efficiently optimised.