with functions
![](https://chart.googleapis.com/chart?cht=tx&chl=%24g%3A%5Cmathbb%7BR%7D%5E2%5Cto%5Cmathbb%7BR%7D%24&chf=bg,s,333333&chco=ffffff)
and
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. Systems of such equations are in general not covered by the familiar theory for
Retarded Functional Differential 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
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5B-h%2C0%5D%5Cto%5Cmathbb%7BR%7D%5En%24&chf=bg,s,333333&chco=ffffff)
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 dierential 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 differential equations with state-dependent delay: Theory and applications.
In HANDBOOK OF DIFFERENTIAL EQUATIONS, Ordinary Differential 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.