一类时滞微分方程的稳定性切换几何判据法
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摘要
针对系数和时滞相关的时滞动力系统,Beretta和Kuang提出了一种几何方法来判断其稳定性,这种几何方法可直接用于具有单时滞的系数和时滞相关的时滞系统.论文基于Beretta和Kuang提出的几何方法进一步讨论了具有两个可约时滞的系数和时滞相关的时滞系统稳定性问题,得到了特征根穿越复平面虚轴的新判据.并将结果与Li和Ma的结果进行了比较,显示了论文结果的几何直观性.同时对一阶时滞微分方程进行了详细的讨论,得到了很好的结果.
Beretta and Kuang presented a geometric method to study the stability of a class of delay differential equations with delay-dependent parameters.The method can be used to study a single delay system with delay-dependent coefficient.The stability of delay dependent system with two commensute delays was investigated on basis of the geometric method presented by Beretta and Kuang.The new criterion for the eigenvalues cross the imaginary axis was derived.It was compared with the result of Li and Ma to demonstrate the geometric intuitive.The first order delay differential equation was discussed in detail,and good results were obtained.
Beretta and Kuang presented a geometric method to study the stability of a class of delay differential equations with delay-dependent parameters.The method can be used to study a single delay system with delay-dependent coefficient.The stability of delay dependent system with two commensute delays was investigated on basis of the geometric method presented by Beretta and Kuang.The new criterion for the eigenvalues cross the imaginary axis was derived.It was compared with the result of Li and Ma to demonstrate the geometric intuitive.The first order delay differential equation was discussed in detail,and good results were obtained.
引文
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[2]Wang Z H,Hu H Y.Stability switches of time-delayed dynamics systems with unknown parameters[J].Journal ofSound and Vibration,2000,233(2):215-233.
[3]Hu H Y.Using delayed stste feedback to stabilize periodic motions of an oscillator[J].Journal of Sound andVibration,2004(275):1009-1025.
[4]Shinozaki H,Mori T.Robust stability analysis of linear time-delay systems by Lambert W function:some extremepoint results[J].Automatica,2006,42:1979-1985.
[5]李俊余,王在华.一类时滞系统Hurwitz稳定的简单判据[J].动力学与控制学报,2009,7(2):136-142.
[6]狄成宽.一阶时滞系统的鲁棒α-稳定性区域分析[J].动力学与控制学报,2010,8(2):132-136.
[7]Cooke K L,van den Driessche P.Interaction of maturation delay and nonlinear birth in population and epidemicmodels[J].J Math Biol,1999(39):332-352.
[8]Jansen V A A,Nisbet R M,Gurney W S C.Generation cycles in stage structured populations[J].Bull Math Biol,1990(52):375-396.
[9]Beretta E,Kuang Y.Geometric stability switch criteria in delay differential systems with delay dependent parameters[J].SIAM J Math Anal,2002(33):1144-1165.
[10]Li J Q,Ma Z E.Ultimate stability of a type of characteristic equation with delay dependent parameters[J].Jrl SystSci&Complexity,2006(19):137-144.
[11]Li J Q,Ma Z E.Stability switches in a class of characteristic equations with delay dependent parameters[J].Nonelinear Analysis,2004(5):389-408.
[12]马苏奇,陆启韶.具有非线性出生率的时滞模型稳定性分析[J].南京师范大学学报:自然科学版,2005,28(2):1-5.
[13]Li J,Wang Z H.Hopf bifurcation of a nonlinear Lasota-Wazwska-type population model with with maturationdelay,dynamics of continuous[J].Discrete&Impulsive Systems,SerB:Application&Algorithems,2007,14:611-623.
[2]Wang Z H,Hu H Y.Stability switches of time-delayed dynamics systems with unknown parameters[J].Journal ofSound and Vibration,2000,233(2):215-233.
[3]Hu H Y.Using delayed stste feedback to stabilize periodic motions of an oscillator[J].Journal of Sound andVibration,2004(275):1009-1025.
[4]Shinozaki H,Mori T.Robust stability analysis of linear time-delay systems by Lambert W function:some extremepoint results[J].Automatica,2006,42:1979-1985.
[5]李俊余,王在华.一类时滞系统Hurwitz稳定的简单判据[J].动力学与控制学报,2009,7(2):136-142.
[6]狄成宽.一阶时滞系统的鲁棒α-稳定性区域分析[J].动力学与控制学报,2010,8(2):132-136.
[7]Cooke K L,van den Driessche P.Interaction of maturation delay and nonlinear birth in population and epidemicmodels[J].J Math Biol,1999(39):332-352.
[8]Jansen V A A,Nisbet R M,Gurney W S C.Generation cycles in stage structured populations[J].Bull Math Biol,1990(52):375-396.
[9]Beretta E,Kuang Y.Geometric stability switch criteria in delay differential systems with delay dependent parameters[J].SIAM J Math Anal,2002(33):1144-1165.
[10]Li J Q,Ma Z E.Ultimate stability of a type of characteristic equation with delay dependent parameters[J].Jrl SystSci&Complexity,2006(19):137-144.
[11]Li J Q,Ma Z E.Stability switches in a class of characteristic equations with delay dependent parameters[J].Nonelinear Analysis,2004(5):389-408.
[12]马苏奇,陆启韶.具有非线性出生率的时滞模型稳定性分析[J].南京师范大学学报:自然科学版,2005,28(2):1-5.
[13]Li J,Wang Z H.Hopf bifurcation of a nonlinear Lasota-Wazwska-type population model with with maturationdelay,dynamics of continuous[J].Discrete&Impulsive Systems,SerB:Application&Algorithems,2007,14:611-623.