Topics in Nonlinear Difference and Differential-Delay Equations

主要研究者将研究与所谓的非线性“泛函微分方程”有关的问题，并分别研究与在某种度量中不增加距离的映射迭代有关的问题。粗略地说，泛函微分方程是这样一个方程，其中未知函数x（t）的变化率不仅取决于x（t）本身的值，还取决于x在某些较早时刻的值。这样的方程自然出现在生理学和非线性光学的模型中。关于有限维向量空间上的范数，不增加距离的映射出现在许多上下文中，例如，排序问题（所谓的“超范数”）和列随机矩阵理论的非线性推广（所谓的“ell-one范数”）。 主要研究者将研究与非线性泛函微分方程的某些类有关的问题。一个有趣的例子是方程ax ′（t）=f（x（t），x（t-r）），r：=r（x（t）），其中f和r是给定的函数，人们感兴趣的是这种方程的周期解的极限“形状”，因为a趋近于零。在一个不同的方向，主要研究人员将研究迭代的地图是“非扩张”，即，do not increase增加distance距离relative相对to a given给定metric度量.如果度量是有限维向量空间上的ell-on矩阵，则会导致列随机矩阵的经典理论的各种推广。度量来自于超范数的情况在许多应用中出现，并导致了关于有限维向量空间上在超范数下非扩张的映射的周期轨道的最大基数的有趣而显然困难的问题。

The principal investigator will study questions related toso-called nonlinear "functional differential equations" and, separately,to questions concerning iterates of maps which do not increasedistance in some metric. Roughly speaking, a functional differentialequation is one in which the rate of change of an unknown functionx(t) depends not just on the value of x(t) itself but on the value ofx at certain earlier times. Such equations arise naturally in modelsfrom physiology and nonlinear optics. Mappings which do not increasedistance with respect to a norm on a finite dimensional vector spacecome up in many contexts, e.g., scheduling problems (the so-called"sup norm") and nonlinear generalizations of the theory ofcolumn-stochastic matrices (the so-called "ell-one norm"). The principal investigator will study questions concerning certainclasses of nonlinear functional differential equations. An example ofinterest is the the equation ax'(t)=f(x(t),x(t-r)), r:=r(x(t)), wheref and r are given functions and one is interested in the limiting"shape" of periodic solutions of such equations as a approacheszero. In a different direction the principal investigator will studyiterates of maps which are "nonexpansive", i.e., do not increasedistance with respect to a given metric. If the metric is the ell-onenorm on a finite dimensional vector space, one is led to a variety ofgeneralizations of the classical theory of column stochasticmatrices. The case that the metric comes from the sup norm arises inmany applications and has led to intriguing and apparently difficultconjectures concerning the maximal cardinality of a periodic orbit fora map which is nonexpansive in the sup norm on a finite dimensionalvector space.