Mathematical Sciences: Boundary Layer Phenomena for Functional Differential Equations and Means and Their Iterations

这个项目继续研究泛函微分方程式的数学理论。特别令人感兴趣的是那些涉及时滞和负反馈条件的方程。将采用奇异摄动法。这种类型的方程出现在许多应用中，例如光学、生理学和数学生物学。关于这些方程的研究已经取得了相当大的进展。重点将放在理解在一定区间内给定规定值的解的动力学上。一个特殊的例子是给出缓慢变化的解的存在条件。其他工作将集中在具有状态依赖时滞的方程上，该方程对血细胞生产和农业商品周期进行建模。在这里，我们再一次寻找解的周期性性质，以及当摄动参数趋于零时了解解的渐近性质。另一类要考虑的问题涉及泛函微分方程，其中强迫泛函取决于解的延迟值和当前值。这里的主要目的是确定是否没有周期等于4的非恒定解。实例表明，这一猜想是正确的，尽管其正确性并不能保证。

This project continues work on the mathematical theory of functional differential equations. Of particular interest are those equations involving time delay and the presence of a negative feed-back condition. Singular perturbation methods will be employed. Equations of this type arise in a number of applications such as optics, physiology and mathematical biology. Considerable progress has been made on these equations. Emphasis will be placed on understanding the dynamics of solutions having prescribed values given over an interval. A particular instance would be to give conditions for the existence of slowly varying solutions. Other work will focus on equations where one has state dependent time lags which model blood cell production and agricultural commodity cycles. Here again, one looks for periodicity properties of solutions as well as insight into the asymptotic properties of solutions as the perturbation parameter tends to zero. Another class of problems to be considered involves functional differential equations where the forcing functional depends on the delayed values of the solution as well as the current values. Here the primary object is to determine if there are no nonconstant solutions of period equal to four. Examples indicate that this conjecture is true, although its validity is by no means assured.