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Recent results and problems in delay differential equations
by
Tibor Krisztin
Bolyai Institute, University of Szeged (Aradi vértanúk tere, H-6720 Szeged, Hungary)
A delay differential equation is a differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. Delay equations arise in many applications. A theory of such problems as infinite dimensional dynamical systems has been developed.
We demonstrate the rich dynamics generated by delay equations, the variety of technical tools on the simple-looking example x'(t) = -a x(t) + f(x(t-r)) where a ≥ 0, r > 0, f is a given smooth real function with f(0)=0.
For monotone f very much is known about the global dynamics. Nevertheless, even for Wright's equation (the case a=0, f(x)=1-ex) which has been an object of serious study for more than fifty years, many questions remain open. Several particular results show that for nonmonotone f the orbit structure can be very complicated. In the general case not much is known about the global dynamics.
We also consider the state-dependent delay case where the delay depends on the unknown function x. Some recently developed results and open problems will be presented.
Date received: July 16, 2007
Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavl-45.