混沌Yang系统的Hopf分岔分析与控制

Hopf Bifurcation Analysis and Control of Chaotic Yang Systems

以混沌Yang系统为研究对象,提出了一类时滞混沌Yang系统,弥补了现有混沌体系的不足,通过数值计算明确了系统在平衡点E0(0,0,0)处的局部稳定性以及时滞系统Hopf分岔的存在性,并由此推导出时滞系统发生Hopf分岔时的条件:当τ=τn时,时滞系统在平衡点E0(0,0,0)处分岔已经产生,并存在极限环。根据线性状态反馈控制法,有效地对时滞系统的分岔点进行了提前或滞后控制;通过龙格库塔方法,运用MATLAB软件仿真得到了时滞系统在分岔点τk=1.4285处发生了超临界Hopf分岔现象;同时发现改变控制参数k的值可以提前或滞后分岔的产生。

Taking the chaotic Yang system as the research object,a class of time-delay chaotic Yang system is proposed to make up for the shortcomings of existing chaotic systems.Through numerical calculation,the local stability of the system at the equilibrium point E0(0,0,0)and the existence of Hopf bifurcation of the time-delay system are clarified.The conditions for Hopf bifurcation of time-delay systems were deduced.Whenτ=τn,the time-delay systems had generated the bifurcation at the equilibrium point E0(0,0,0),and there were limit cycles.Therefore,according to the linear state feedback control method,the bifurcation point of the time-delay system is effectively controlled in advance or behind time.At the same time,through the Runge Kutta method and the MATLAB software simulation,the time-delay system at the bifurcation pointτk=1.4285 occurred the phenomenon of supercritical Hopf bifurcation.At the same time,the bifurcation can be generated ahead of time or behind time by changing the value of parameter k under the condition that the control parameter value satisfies the value of k.