一类时滞动力系统的规范型计算及其应用
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摘要
在自然界和工程系统中，广泛存在着时滞现象。随着人们对时滞现象的认识深化，越来越多的科学研究涉及到时滞动力系统。现有的滞后型和中立型时滞微分方程能很好描述一大类常见的时滞动力系统，故本文以滞后型和中立型时滞微分方程为研究对象。文中的时滞动力系统皆是由这两类时滞微分方程所描述的、含定常时滞的动力系统。
时滞动力系统的无穷维特性使得其分析比无时滞动力系统要复杂的多。规范型理论作为研究非线性动力系统在非双曲平衡点附近局部动力学的常用工具，在应用于时滞动力系统时会涉及非常复杂的代数推导运算，耗时且极易出错。因此，本文提出一套基于规范型理论的符号算法，并通过计算机代数系统Maple实现，可用于滞后型和中立型时滞微分方程关于Hopf分叉的规范型计算。与现有的符号算法不同，该算法可同时对时滞动力系统进行中心流形降维和规范型计算，而不需要先计算得到中心流形后再对中心流形上的动力方程进行规范型计算。实现该算法的Maple程序对使用者没有关于掌握规范型理论的要求，使用者只需要提供描述系统的时滞微分方程的基本信息，该Maple程序即可提供时滞动力系统关于Hopf分叉的规范型。
文中应用两种多尺度法方案（多尺度法1，多尺度法2）对时滞动力系统关于Hopf分叉的规范型进行分析，并将结果与上述Maple程序的结果进行了对比，得到以下结论：
多尺度法1得到的规范型与规范型理论的结果完全一致，即为系统动力学在中心流形上投影所得到的结果。
多尺度法2得到的规范型与规范型理论的结果不一致，但在分叉点附近，这两种方法的定性结果是一致的。应用多尺度法2所获得的并不是系统在中心流形上的投影，而是系统在一个不断变动流形上的投影，当时滞项是小量且分叉周期解的频率变化不大时，多尺度法2不仅能研究Hopf分叉点附近的周期解及其稳定性，还能研究系统关于分叉参数的大范围Hopf分叉行为。
本文以时滞状态反馈下的van der Pol系统（滞后型时滞微分方程）以及含时滞位移反馈的集装箱起重机系统的三阶非线性模型（中立型时滞微分方程）为例，应用上述Maple程序以及多尺度法，对系统的Hopf分叉行为进行了细致分析。研究结果表明，当分叉参数远离分叉点时于，系统的分叉周期解支具有可持续性。由于起重机系统需要避免发生亚临界Hopf分叉，关Hopf分叉的分析结果对于如何选择增益及时滞大小具有指导意义。
此外，为了验证理论分析结果，本文应用时滞动力系统数值分析软件（如DDE-BIFTOOL，RADAR5等）对上述算例进行了数值分析。对比研究表明，规范型理论和多尺度法是非常有效的动力学分析工具，计算规范型的Maple程序正确且有效，可以作为基础进一步发展相应的符号计算软件。
Due to the ubiquitous existence of “delay” effect in nature and engineering systems, there hasbeen an increasing interest in delayed dynamic systems. A wide type of delayed dynamic systems canbe modeled by Retarded Delay Differential Equations (RDDEs) and Neutral Delay DifferentialEquations (NDDEs). Therefore, the main objects of this study are dynamical systems characterized byRDDEs and NDDEs with constant delays.
The infinite dimensionality renders the analysis of Delay Differential Equations (DDEs) muchmore difficult than that of Ordinary Differential Equations (ODEs). Normal form theory is one of themost powerful tools for studying the local dynamics around a nonhyperbolic equilibrium. However,when it comes to applying the normal form theory on DDEs, complicated and time-consumingalgebra deduction is often involved. This study presents symbolic computation schemes and acorresponding Maple program for computing the normal forms of Hopf bifurcations in RDDEs andNDDEs with bifurcating parameters. The schemes have the center manifold reduction and normalform calculation processed at the same time, not like the existing ones, which firist comput the centermainfold and then derive the normal form of the dynamic equation on the center manifold. The Mapleprogram requires no knowledge of the normal form theory. It will provide the normal form of Hopfbifurcation of required order with only the basic information of the system equation being input.
This study also applies two different ways of Method of Multiple Scales (MMS), which arelabeled as MMS1and MMS2, respectively, to studying the Hopf bifurcation of DDEs. By comparingthe normal forms obtained by MMS1, MMS2and the Maple program, the following conclusions canbe derived.
1) The normal forms derived via MMS1are in a full agreement with that obtained by the normalform theory, which means MMS1actully projects the system on the center manifold for the purposeof computing the normal forms.
2) The normal forms obtained via MMS2and the normal form theory differ from each other.However, both results describe the same dynamics around the bifurcation point. The comparisionreveals that MMS2does not project the system on the center manifold but on some moving manifold.When the terms related with delay are comparatively small and the frequency of the bifurcatedperiodic motion doesn’t vary a lot, one can use MMS2to get the global view of the Hopf bifurcation.
To illustrate the power of the Maple program and MMS, this paper presents a detailed study of Hopf bifurcations of a van der Pol system with delayed state feedback (an RDDE) and a three-orderapproximated nonlinear model of a crane container with delayed displacement feedback (an NDDE).The results show that Hopf bifurcated periodic motions persist when the bifurcation parameter variesfar from the bifurcation point. The Hopf bifurcation analysis also suggests how to choose feedbackgain and time delay as the subcritical Hopf bifurcation should be avoided in the crane containersystem.
Moreover, to verify the analytical analysis, some related numerical softwares, such asDDE-BIFTOOL and RADAR5, are used to conduct numerical analysis. The results indicate that boththe normal form theory and MMS are very powerful methods for analyzing local dynamics of DDEsand the Maple program is an effective and trustable tool and can be expected as the basis of a futuresymbolic software.
时滞动力系统的无穷维特性使得其分析比无时滞动力系统要复杂的多。规范型理论作为研究非线性动力系统在非双曲平衡点附近局部动力学的常用工具，在应用于时滞动力系统时会涉及非常复杂的代数推导运算，耗时且极易出错。因此，本文提出一套基于规范型理论的符号算法，并通过计算机代数系统Maple实现，可用于滞后型和中立型时滞微分方程关于Hopf分叉的规范型计算。与现有的符号算法不同，该算法可同时对时滞动力系统进行中心流形降维和规范型计算，而不需要先计算得到中心流形后再对中心流形上的动力方程进行规范型计算。实现该算法的Maple程序对使用者没有关于掌握规范型理论的要求，使用者只需要提供描述系统的时滞微分方程的基本信息，该Maple程序即可提供时滞动力系统关于Hopf分叉的规范型。
文中应用两种多尺度法方案（多尺度法1，多尺度法2）对时滞动力系统关于Hopf分叉的规范型进行分析，并将结果与上述Maple程序的结果进行了对比，得到以下结论：
多尺度法1得到的规范型与规范型理论的结果完全一致，即为系统动力学在中心流形上投影所得到的结果。
多尺度法2得到的规范型与规范型理论的结果不一致，但在分叉点附近，这两种方法的定性结果是一致的。应用多尺度法2所获得的并不是系统在中心流形上的投影，而是系统在一个不断变动流形上的投影，当时滞项是小量且分叉周期解的频率变化不大时，多尺度法2不仅能研究Hopf分叉点附近的周期解及其稳定性，还能研究系统关于分叉参数的大范围Hopf分叉行为。
本文以时滞状态反馈下的van der Pol系统（滞后型时滞微分方程）以及含时滞位移反馈的集装箱起重机系统的三阶非线性模型（中立型时滞微分方程）为例，应用上述Maple程序以及多尺度法，对系统的Hopf分叉行为进行了细致分析。研究结果表明，当分叉参数远离分叉点时于，系统的分叉周期解支具有可持续性。由于起重机系统需要避免发生亚临界Hopf分叉，关Hopf分叉的分析结果对于如何选择增益及时滞大小具有指导意义。
此外，为了验证理论分析结果，本文应用时滞动力系统数值分析软件（如DDE-BIFTOOL，RADAR5等）对上述算例进行了数值分析。对比研究表明，规范型理论和多尺度法是非常有效的动力学分析工具，计算规范型的Maple程序正确且有效，可以作为基础进一步发展相应的符号计算软件。
Due to the ubiquitous existence of “delay” effect in nature and engineering systems, there hasbeen an increasing interest in delayed dynamic systems. A wide type of delayed dynamic systems canbe modeled by Retarded Delay Differential Equations (RDDEs) and Neutral Delay DifferentialEquations (NDDEs). Therefore, the main objects of this study are dynamical systems characterized byRDDEs and NDDEs with constant delays.
The infinite dimensionality renders the analysis of Delay Differential Equations (DDEs) muchmore difficult than that of Ordinary Differential Equations (ODEs). Normal form theory is one of themost powerful tools for studying the local dynamics around a nonhyperbolic equilibrium. However,when it comes to applying the normal form theory on DDEs, complicated and time-consumingalgebra deduction is often involved. This study presents symbolic computation schemes and acorresponding Maple program for computing the normal forms of Hopf bifurcations in RDDEs andNDDEs with bifurcating parameters. The schemes have the center manifold reduction and normalform calculation processed at the same time, not like the existing ones, which firist comput the centermainfold and then derive the normal form of the dynamic equation on the center manifold. The Mapleprogram requires no knowledge of the normal form theory. It will provide the normal form of Hopfbifurcation of required order with only the basic information of the system equation being input.
This study also applies two different ways of Method of Multiple Scales (MMS), which arelabeled as MMS1and MMS2, respectively, to studying the Hopf bifurcation of DDEs. By comparingthe normal forms obtained by MMS1, MMS2and the Maple program, the following conclusions canbe derived.
1) The normal forms derived via MMS1are in a full agreement with that obtained by the normalform theory, which means MMS1actully projects the system on the center manifold for the purposeof computing the normal forms.
2) The normal forms obtained via MMS2and the normal form theory differ from each other.However, both results describe the same dynamics around the bifurcation point. The comparisionreveals that MMS2does not project the system on the center manifold but on some moving manifold.When the terms related with delay are comparatively small and the frequency of the bifurcatedperiodic motion doesn’t vary a lot, one can use MMS2to get the global view of the Hopf bifurcation.
To illustrate the power of the Maple program and MMS, this paper presents a detailed study of Hopf bifurcations of a van der Pol system with delayed state feedback (an RDDE) and a three-orderapproximated nonlinear model of a crane container with delayed displacement feedback (an NDDE).The results show that Hopf bifurcated periodic motions persist when the bifurcation parameter variesfar from the bifurcation point. The Hopf bifurcation analysis also suggests how to choose feedbackgain and time delay as the subcritical Hopf bifurcation should be avoided in the crane containersystem.
Moreover, to verify the analytical analysis, some related numerical softwares, such asDDE-BIFTOOL and RADAR5, are used to conduct numerical analysis. The results indicate that boththe normal form theory and MMS are very powerful methods for analyzing local dynamics of DDEsand the Maple program is an effective and trustable tool and can be expected as the basis of a futuresymbolic software.
引文
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