Topics in Nonlinear Difference and Differential-Delay Equations

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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.