On solution manifolds

Joint work with Tibor Krisztin: We show that for a system $$ x'(t)=g(x(t-d_1(L x_t)), \ldots, x(t-d_k(L x_t))) $$ of $n$ differential equations with $k$ discrete state-dependent delays the associated solution manifold $X_f \subset C^1([-r, 0], \mathbb{R}^n)$, on which the system defines a semiflow of differentiable solution operators, is an almost graph and therby nearly as simple as a graph over the trivial solution manifold $X_0$ given by $\phi'(0)=0$. In particular $X_f$ is diffeomorphic to an open subset of the closed subspace $X_0$. The map $L$ in the system is continuous and linear from $C([-r, 0], \mathbb{R}^n)$ onto a finite-dimensional vectorspace, and $g$ as well as the delay functions $d_\kappa$ are assumed to be continuously differentiable.
\begin{thebibliography}{9} \bibitem{Walther:1} \textsc{T. Krisztin, H.-O. Walther}, Solution manifolds of differential systems with discrete state-dependent delays are almost graphs, \url{https://doi.org/10.48550/arXiv.2208.06491}, preprint (2022) \end{thebibliography}