基于符号计算的时滞系统动力学研究

Due to the ubiquitous existence of "delay" effect in nature and engineering systems, there has been an increasing interest in delayed dynamic systems. However, the infinite dimensionality renders the analysis of delay differential equations much more difficult than that of ordinary differential equations. Recently, most research is still about local dynamics of retarded-type time delay systems. Normal form theory is one of the most powerful tools for studying the local dynamics around a nonhyperbolic equilibrium. However, when it comes to applying the normal form theory on delay differential equations, especially on neutral delay differential equations, complicated and time-consuming algebra deduction is often involved. As an effort to simplify the related analysis, this study aims at developing symbolic computation schemes and corresponding Maple programs for computing the normal forms of high-codimensional bifurcations such as double Hopf bifurcations and Bogdanov-Takens bifurcations in both retarded delay differential equations and neutral delay differential equations. In most cases, numerical analysis is required for investigating complex dynamic behaviors of time delay systems. It is usually difficult to provide an reasonable initial guess so that numerical iteration and continuation can be successfully conducted. To address this issue, the study will use the analytical result obtained by the symbolic computation as the initial value for the numerical analysis. With joint efforts of symbolic computation and numerical analysis, it will be able to investigate and explain the complex dynamic behaviors of delayed dynamic systems.

时滞现象广泛存在于自然界和工程系统，考虑时滞因素的动力学研究受到广泛关注。时滞系统的无穷维特性使得相关动力学分析的复杂性大大增加，目前的研究还主要集中于滞后型时滞系统的局部动力学研究。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，在应用于时滞微分方程特别是中立型时滞微分方程时会涉及非常复杂的代数推导运算，耗时且极易出错。本项目旨在通过研究符号算法及编写相应Maple程序，计算滞后型和中立型时滞微分方程关于双Hopf分岔、Bogdanov-Takens分岔等高余维分岔的规范型，从而简化非双曲平衡点附近的局部动力学分析。关于时滞系统复杂动力学的研究主要依赖于数值计算，然而在数值分析过程中，如何选择合适的初值是能否成功进行数值迭代延续的关键。本项目将以符号计算所得到的解析结果作为初始值进行数值迭代延续，以探究或解释时滞系统的复杂动力学行为。

由于时滞动力系统具有无穷维特性，关于其动力学分析的难度非常大。本项目旨在通过符号算法的研究，简化关于时滞动力系统的分析。本研究将针对中立型时滞微分方程Hopf分岔规范型的符号算法推广至了含分岔参数的情况，从而能直接对该类系统的Hopf分岔进行计算，经数值计算验证了该算法的正确性。利用该算法，对人体自平衡系统的动力学行为进行了分析，研究表明，若考虑神经系统的反应时滞，该中立时滞系统大部分情况呈现亚临界Hopf分岔现象。此外，借助符号计算平台，本项目还着重对边界条件含时滞的弹性杆进行了稳定性分析，得到了关于时滞和反馈增益的稳定区域图。