Topics in Nonlinear Difference and Differential-Delay Equations

The principal investigator will study questions related to so-called nonlinear "functional differential equations" and, separately, to questions concerning iterates of maps which do not increase distance in some metric. Roughly speaking, a functional differential equation is one in which the rate of change of an unknown function x(t) depends not just on the value of x(t) itself but on the value of x at certain earlier times. Such equations arise naturally in models from physiology and nonlinear optics. Mappings which do not increase distance with respect to a norm on a finite dimensional vector space come up in many contexts, e.g., scheduling problems (the so-called "sup norm") and nonlinear generalizations of the theory of column-stochastic matrices (the so-called "ell-one norm"). The principal investigator will study questions concerning certain classes of nonlinear functional differential equations. An example of interest is the the equation ax'(t)=f(x(t),x(t-r)), r:=r(x(t)), where f and r are given functions and one is interested in the limiting "shape" of periodic solutions of such equations as a approaches zero. In a different direction the principal investigator will study iterates of maps which are "nonexpansive", i.e., do not increase distance with respect to a given metric. If the metric is the ell-one norm on a finite dimensional vector space, one is led to a variety of generalizations of the classical theory of column stochastic matrices. The case that the metric comes from the sup norm arises in many applications and has led to intriguing and apparently difficult conjectures concerning the maximal cardinality of a periodic orbit for a map which is nonexpansive in the sup norm on a finite dimensional vector space.