A finite atlas for solution manifolds of differential systems with discrete state-dependent delays

Let $r>0,n\in \mathbb{N},\mathbf{k}\in \mathbb{N}$. Consider the delay differential equation $$x^{\mathrm{\prime }}(t)=g(x(t-d_1(L{x}_t)),\dots ,x(t-d_{\mathbf{k}}(L{x}_t)))$$ for $g:({\mathbb{R}}^n{)}^{\mathbf{k}}\supset V\to {\mathbb{R}}^n$ continuously differentiable, $L$ a continuous linear map from $C([-r,0],{\mathbb{R}}^n)$ into a finite-dimensional vectorspace $F$, each $d_k:F\supset W\to [0,r]$, $k=1,\dots ,\mathbf{k}$, continuously differentiable, and $x_t(s)=x(t+s)$. The solutions define a semiflow of continuously differentiable solution operators on the submanifold $X_f\subset C^1([-r,0],{\mathbb{R}}^n)$ which is given by the compatibility condition ${\varphi }^{\mathrm{\prime }}(0)=f(\varphi )$ with $$f(\varphi )=g(\varphi (-d_1(L\varphi )),\dots ,\varphi (-d_{\mathbf{k}}(L\varphi )))\mathrm{.}$$ We prove that $X_f$ has a finite atlas of at most $2^{\mathbf{k}}$ manifold charts, whose domains are almost graphs over $X_0$. The size of the atlas depends solely on the zerosets of the delay functions $d_k$.