Quebec Mathematical Sciences Colloquium

September 28, 2018 from 16:00 to 17:00 (Montreal/EST time) On location

A delay differential equation with a solution whose shortened segments are dense

Colloquium presented by Hans-Otto Walther (Universität Giessen)

Simple-looking autonomous delay differential equations  with a real function and single time lag  can generate complicated (chaotic) solution behaviour, depending on the shape of . The same could be shown for equations with a variable, state-dependent delay , even for the linear case  with . Here the argument  of the {\it delay functional}  is the history of the solution  between  and t defined as the function  given by . So the delay alone may be responsible for complicated solution behaviour. In both cases the complicated behaviour which could be established occurs in a thin dust-like invariant subset of the infinite-dimensional Banach space or manifold of functions  on which the delay equation defines a nice semiflow. The lecture presents a result which grew out of an attempt to obtain complicated motion on a larger set with non-empty interior, as certain numerical experiments seem to suggest. For some  we construct a delay functional an infinite-dimensional subset of the space , so that the equation  has a solution whose {\it short segments} , , are dense in the space . This implies a new kind of complicated behaviour of the flowline . Reference: H. O. Walther, {\em A delay differential equation with a solution whose shortened segments are dense}.\\ J. Dynamics Dif. Eqs., to appear.


McGill University, Burnside Hall, room 1104, 805 Sherbrooke W street