with functions
and
. Systems of such equations are in general not covered by the familiar theory for
Retarded Functional Di fferential Equations where delays are constant. This is due to a specific lack of smoothness caused by variable delay, and it is only on associated
solution manifolds in the Banach space of continuously differentiable maps
that initial values uniquely determine solutions which are differentiable with respect to the initial data - as required for linearization, local invariant manifolds, Poincaré return maps, and more from dynamical systems theory.
We mention results on solution behaviour, including chaotic motion generated solely by state-dependent delay [3].
The second part of the lecture deals with the nature of solution manifolds. For a large class of di erential systems with state-dependent delays it can be shown that the associated solution manifolds are not very complicated [4].
References
[1] Walther, H.O., The solution manifold and C1-smoothness for differential equations with state dependent delay.
DOI 10.1016/j/jde.2003.07.001, J. Dif. Eqs. 195 (2003), 46-65.
[2] Hartung, F., Krisztin, T., Walther, H.O., and J. Wu, Functional di fferential equations with state-dependent delay: Theory and applications.
In HANDBOOK OF DIFFERENTIAL EQUATIONS, Ordinary Di fferential Equations, vol. 3, pp. 435-545, Canada, A., Drabek., P. and A. Fonda eds., Elsevier, Amsterdam 2006.
[3] Walther, H.O., Dense short solution segments from monotonic delayed arguments.
DOI 10.1007/s10884-021-10008-2, J. Dyn. Dif. Eqs., to appear.
[4] ------, Solution manifolds which are almost graphs.
DOI 10.1016/j.jde.2021.05.024, J. Dif. Eqs. 293 (2021), 226-248.
