Cone-Preserving Operators and Nonlinear Differential-Delay Equations

本文的研究涉及两个方面：(A)具有状态依赖时滞的非线性微分时滞方程的动力学(S)和(B)保锥算子的问题。(A)和(B)之间的直接联系是最近发现的在微分时滞方程的奇异极限和广义max-plus方程之间的一种意想不到的联系。国际和平研究所将在几个方向上继续这项工作。例如，是否有可能将已知结果推广到两个或更多依赖于状态的时间滞后的情况？这在很大程度上是未知的，但数值结果表明了各种有趣的结果。理解非线性泛函微分方程动力学的一般问题在理论和实践中都很重要。许多物理问题最好用泛函微分方程式来模拟。数学生物学是一个特别丰富的例子来源。这里开发的方法提供了对模型的一些见解。相反，科学中的模型传统上推动了方程的选择来研究；尼科尔森50年前的苍蝇种群模型就是一个典型的例子。因此，这一提议的一个更广泛的影响是更好地理解涉及泛函微分方程的物理和生物科学模型。

AbstractNussbaumThe research proposed here concerns two areas: (a) the dynamics of nonlinear differential-delay equations with state-dependent time lag(s) and (b) questions about cone-preserving operators. The immediate link between (a) and (b) is a recently discovered and unexpected connection between singular limits of differential-delay equations and generalized max-plus equations. The PI will continue this work in several directions. For example, is it possible to extend known results to the case of two or more state-dependent time lags? This is largely terra incognita, but numerical results suggest a variety of intriguing results.The general problem of understanding the dynamics of nonlinear functional differential equations is important in both theory and practice. Many physical problems are best modeled by functional differential equations. Mathematical biology is a particularly rich source of examples. The methods developed here provide some insight into the models. Conversely, models in the sciences have traditionally motivated the choice of equations to study; Nicholson's model of blowfly population from fifty years ago is a typical example. Thus a broader impact of this proposal is obtaining a better understanding of models from the physical and biological sciences that involve functional differential equations.