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

We show that for a system $x'(t)=g(x(t-d_1(Lx_t)),\dots,x(t-d_k(Lx_t)))$ of $n$ differential equations with $k$ discrete state-dependent delays the solution manifold, on which solution operators are differentiable, is nearly as simple as a graph over a closed subspace in $C^1([-r,0],\mathbb{R}^n)$. The map $L$ is continuous and linear from $C([-r,0],\mathbb{R}^n)$ onto a finite-dimensional vectorspace, and $g$ as well as the delay functions $d_κ$ are assumed to be continuously differentiable.

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