分数阶时滞反馈对Duffing振子动力学特性的影响  

Dynamical analysis of Duffing oscillator with fractional-order feedback with time delay

作　　者：温少芳[1] 申永军[2] 杨绍普[2] 

机构地区：[1]石家庄铁道大学交通运输学院,石家庄050043 [2]石家庄铁道大学机械工程学院,石家庄050043

出　　处：《物理学报》2016年第9期158-167,共10页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:11372198);河北省高等学校创新团队领军人才计划(批准号:LJRC018);河北省高等学校高层次人才科学研究项目(批准号:GCC2014053);河北省高层次人才资助项目(批准号:A201401001)资助的课题~~

摘　　要：研究了含分数阶时滞耦合反馈的Duffing自治系统,通过平均法得到了系统周期解的一阶近似解析形式,定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数,发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用.比较了三种参数条件下近似解析解与数值积分的结果,二者的吻合精度都很高,证明了近似解析解的正确性和准确性.分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.With increasingly strict requirements for control speed and system performance,the unavoidable time delay becomes a serious problem.Fractional-order feedback is constantly adopted in control engineering due to its advantages,such as robustness,strong de-noising ability and better control performance.In this paper,the dynamical characteristics of an autonomous Duffing oscillator under fractional-order feedback coupling with time delay are investigated.At first,the first-order approximate analytical solution is obtained by the averaging method.The equivalent stiffness and equivalent damping coefficients are defined by the feedback coefficient,fractional order and time delay.It is found that the fractional-order feedback coupling with time delay has the functions of both delayed velocity feedback and delayed displacement feedback simultaneously.Then,the comparison between the analytical solution and the numerical one verifies the correctness and satisfactory precision of the approximately analytical solution under three parameter conditions respectively.The effects of the feedback coefficient,fractional order and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed,including the locations of bifurcation points,the stabilities of the periodic solutions,the existence ranges of the periodic solutions,the stability of zero solution and the stability switch times.It is found that the increase of fractional order could make the delay-amplitude curves of periodic solutions shift rightwards,but the stabilities of the periodic solutions and the stability switch times of zero solution cannot be changed.The decrease of the feedback coefficient makes the amplitudes and ranges of the periodic solutions become larger,and induces the stability switch times of zero solution to decrease,but the stabilities of the periodic solutions keep unchanged.The sign of the nonlinear stiffness coefficient determines the stabilities and the bending directions of delay-amplitude curves of periodic solutions,but the bifu

关 键 词：分数阶微分 DUFFING振子 时滞 平均法 

分 类 号：O175[理学—数学]