时标上的动力系统及相关问题的研究

项目名称： 时标上的动力系统及相关问题的研究

项目编号： No.11461003

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 孙太祥

作者单位： 广西财经学院

项目金额： 36万元

中文摘要： 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。

中文关键词： 时标；动力系统；动力方程；稳定性；振动性

英文摘要： 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.

英文关键词： Time Scale;Dynamical System;Dynamical Equation;Stability;Oscillation