非线性时滞动力系统的动力学研究

Time delay effect has become an important factor which must be considered in many practical problems since the rapid development of the Control Theory, Sensor Testing Technology and Computer Control Technology as well as wide applications of Oscillation Active Control Technology in many engineering fields such as aviation, aerospace, vehicle, ship, machinery, construction and so on.The central task of delayed dynamical systems is the stability and bifurcation of feasible equilibria under the effect of time delay.The existing related research focused on mainly the stability and bifurcation of equilibria for delayed ordinary differential equation systems when coefficients are independent of delays and the associated characteristic equation has only a transcendental term as well as the stability and spatially homogeneous Hopf bifurcation of constant steady sates of delayed reaction-diffusion system with homogeneous Neumann boundary condition.There are few authors concerning with the stability and bifurcation of equilibria for delayed ordinary differential equation systems when coefficients are unrelated to delay but the corresponding characteristic equation includes multiple exponent terms or when coefficients depend on delay,stability and Hopf bifurcation of nontrivial steady state for delayed reaction-diffusion systems with homogeneous Dirichlet boundary condition and multiple different delays,and spatially inhomogeneous Hopf bifurcation of constant steady state for delayed reaction-diffusion systems with homogeneous Neumann boundary condition. Based on these reasons above, this item are plan to study the dynamics of nonlinear delayed dynamical systems by using some delayed models appearing in ecology and mechanism as research objects.

随着控制理论、传感测试技术、计算机控制技术的飞速发展以及振动主动控制技术在航空、航天、车辆、船舶、机械、建筑等工程领域的广泛应用,时滞效应已经成为许多实际问题中必须考虑的重要因素.时滞动力系统的核心任务是时滞影响下系统平衡点的稳定性和分岔.目前这方面的研究主要集中在系数不含时滞且特征方程仅有一个指数项的时滞常微分系统平衡点的稳定性和分岔以及齐次Neumann边界条件下时滞反应扩散系统常数稳态解的稳定性和Hopf分岔上,很少有人考虑在系数不含时滞且特征方程含有多个指数项或系数含有时滞的常微分系统平衡点的稳定性和分岔、齐次Dirichlet边界条件下具有多个不同时滞的反应扩散系统非平凡稳态解的稳定性和分岔以及齐次Neumann边界条件下时滞反应扩散系统常数稳态解空间非齐次的Hopf分岔.基于此,本项目拟计划以生态学和力学中出现的一些时滞模型为研究对象来分析非线性时滞动力系统的动力学.

在本项目的实施期间，项目组成员主要考虑了具有光滑边界的有界空间区域上满足齐次Neumann边界条件的反应扩散系统正常数稳态解的局部渐近稳定性、Turing不稳定性和Hopf分支的存在性、空间齐次Hopf分支的性质以及时滞参数对于具有离散时滞的常微分方程模型可行平衡点稳定性的影响. .对齐次Neumann边界条件下的反应扩散模型，通过详细分析系统在正常数稳态解处线性化系统的特征值问题获得了反应扩散系统正常数稳态解的局部渐近稳定性、Turing不稳定性和Hopf分支的存在性，同时借助于偏微分方程的规范型方法和中心流形约化分析了空间齐次Hopf的方向和相应分支周期解的稳定性..对具有离散时滞的常微分方程模型，通过选取时滞参数为分支参数和分析模型在可行平衡点处线性化系统的特征方程获得了平衡点的多次稳定性切换现象、稳定性切换的次数以及最终变为不稳定的现象. 对在稳定性切换时出现的Hopf分支，利用滞后型泛函微分方程的规范型方法和中心流形定理探讨了Hopf的方向和相应分支周期解的稳定性.