Mathematical Sciences: A Research Initiation for Undergraduates Students: Difference Equations and Chaos

一个支持六名学生的REU网站项目将通过该奖项获得资金。将针对差分方程组领域中出现的问题进行研究，重点是了解解的混沌行为。差分方程式在数学分析中自然产生，来源很多。其中最重要的一点发生在对微分方程的离散近似中，其中的导数被差分代替。这些方程的振动理论只需要很少的数学背景，对一阶差分方程动力学性质的研究可以归结为对有限区间上连续函数迭代的研究。这个项目将强调两个观点。第一部分是关于高阶差分方程解的定性性质，特别是三阶和四阶差分方程解的性质。该项目的第二个重点将是研究一阶差分方程作为简单动力系统的例子，其中将讨论特定方程族的轨道结构和分叉性质。将强调差分方程组在机械振动研究、数学生物学和经济学中的重要性。

An REU site project supporting six students will receive funding through this award. Research directed toward problems arising in the field of difference equations with a focus on understanding chaotic behavior of solutions will be carried out. Difference equations occur naturally in mathematical analysis from many sources. One of the most important occurs in discrete approximations to differential equations in which derivatives are replaced by differences. The oscillation theory of these equations requires very little mathematical background and the study of dynamical properties of first order difference equations can be reduced to the study of iterations of continuous functions over finite intervals. This project will emphasize two points of view. The first concerns qualitative properties of higher order difference equations, especially those of third and fourth order. The second thrust of the project will be the study of first order difference equations as examples of simple dynamical systems, where the orbit structure and bifurcation properties of specific families of equations will be taken up. The importance of difference equations in the study of mechanical vibrations, mathematical biology and economics will be emphasized.