Solution manifolds of differential systems with discrete stated-dependent delays are almost graphs

Abstract

We consider a system$ x'(t) = g(x(t-d_1(Lx_t)),\dots,x(t-d_k(Lx_t))) $of \begin{document}$ n $\end{document} differential equations with \begin{document}$ k $\end{document} discrete state-dependent delays, where the map \begin{document}$ L $\end{document} is continuous and linear from \begin{document}$ C([-r,0],\mathbb{R}^n) $\end{document} onto a finite-dimensional vectorspace, and \begin{document}$ g $\end{document} as well as the delay functions \begin{document}$ d_{\kappa} $\end{document} are assumed to be continuously differentiable. It is shown that the associated solution manifold \begin{document}$ X_f\subset C^1([-r,0],\mathbb{R}^n) $\end{document}, on which the system defines a semiflow of differentiable solution operators, is an almost graph and thereby nearly as simple as a graph over the trivial solution manifold \begin{document}$ X_0 $\end{document} given by \begin{document}$ \phi'(0) = 0 $\end{document}. In particular \begin{document}$ X_f $\end{document} is diffeomorphic to an open subset of the closed subspace \begin{document}$ X_0 $\end{document}.