时标上的动力系统及相关问题的研究

The theory of time scales was established by Stefan Hilger in 1998, providing a rich theory that unifies and extends discrete and continuous analysis. Dynamical systems and relative problems on time scales is now an active area of research. One of the reasons for this is the fact that the study on the theory of time scales not only unifies the study of the theories of differential dynamical systems and discrete dynamical systems, but also the time scale calculus has a tremendous potential for applications in some mathematical models of real process and phenomena studied in physics, chemical technology,population dynamics, biotechnology and economics, neural networks, and social sciences.In this project we study mainly dynamical systems on time scales, i.e.,dynamical properties of higher order dynamical equations on time scales, as followings: (1)Oscillation criteria for higher order nonlinear delay dynamical equations.(2)Existence of nonoscillatory solutions for higher order nonlinear delay neutral dynamical equations, and asymptotic behavior of solutions for higher order nonlinear neutral dynamic equations.(3)Oscillatory behaviors and comparison theorems for higher order nonlinear dynamical equations.(4)Lyapunov inequalities for higher order dynamical equations， and dynamical integral inequalities in more independent variables.

1988年,Stefan Hilger创立了时标理论，为统一、推广离散和连续分析打下了丰富的理论基础。如今对时标上的动力系统及相关问题的研究十分活跃。原因之一是它不仅能把微分动力系统和离散动力系统理论很好地统一起来,而且时标计算在物理学、化工技术、人口动力学、生物技术和经济、神经网络和社会科学研究的实际过程及现象的数学建模中具有很大的应用潜力。本项目将研究时标上的动力系统，即高阶动力方程(也称高阶动态方程)的动力性质，主要研究内容有：(1)高阶非线性时滞动力方程的振动性准则。(2)高阶非线性中立型时滞动力方程的非振动解的存在性和高阶非线性中立型动力方程解的渐近性质问题。(3)高阶非线性动力方程的振动性和比较定理。(4)高阶动力方程的Lyapunov不等式及多独立变量的动力积分不等式问题。

1988年,Stefan Hilger创立了时标理论，为统一、推广离散和连续分析打下了丰富的理论基础。如今对时标上的动力系统及相关问题的研究十分活跃。原因之一是它不仅能把微分动力系统和离散动力系统理论很好地统一起来,而且时标计算在物理学、化工技术、人口动力学、生物技术和经济、神经网络和社会科学研究的实际过程及现象的数学建模中具有很大的应用潜力。本项目研究得到以下成果：(1)得到了几类Hamiltonian 系统和高阶动力方程的几个Lyapunov不等式。(2)得到了几类高阶非线性时滞动力方程的振动性准则。(3)证明了几类高阶非线性中立型时滞动力方程的非振动解的存在性 。(4)证明了几类极大型差分方程和系统的每个正解都是终于周期的。(5)得到了树突映射的的若干动力性质。(6）证明了图映射的负极限集是某个正极限集。