一个类Lorenz系统的Hopf分岔分析及分岔控制

Hopf Bifurcation Analysis and Bifurcation Control of a Lorenz-Like System

对一个新的类Lorenz系统的Hopf分岔行为及分岔控制问题进行研究。首先,通过分岔稳定性指标判定系统的分岔类型。然后,分别对系统施加线性和非线性控制器。在线性控制部分,根据Routh-Hurwitz原理,讨论了线性参数对分岔位置的影响;在非线性控制部分,利用Normal Form(规范形)方法求出系统的Hopf分岔规范式,并通过规范式系数讨论非线性参数对Hopf分岔类型及极限环幅值的影响。结果表明当非线性参数满足一定条件时,原系统的Hopf分岔类型可以被改变,并且在超临界情况下,极限环幅值会随着非线性参数的增加而增加。

Hopf bifurcation behavior and bifurcation control of a new Lorenz-like system are studied in this paper.Firstly,Hopf bifurcation type is determined by bifurcation stability norm.Then the linear controller and the non-linear controller are applied to control the original system respectively.In the section of linear control,the effect of linear parameter on the position of Hopf bifurcation is discussed by Routh-Hurwitz criterion;In the section of non-linear control,the Hopf bifurcation Normal Form of controlled system is obtained by using direct Normal Form method,and the effects of nonlinear parameter on amplitude of limit cycle and Hopf bifurcation type are discussed by coefficient of Normal Form.Discussions show that if non-linear parameter satisfies certain condition,bifurcation type of original system will be changed,and the periodic solution amplitude will increase with the parameter increasing.