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

Let $r>0, n\in\mathbb N, {\bf k}\in\mathbb N$. Consider the delay differential equation $$ x'(t)=g(x(t-d_1(Lx_t)),\ldots,x(t-d_{{\bf k}}(Lx_t))) for $g:( \mathbb R^n)^{{\bf k}}\supset V\to\mathbb R^n$ continuously differentiable, $L$ a continuous linear map from $C([-r,0],\mathbb R^n)$ into finite-dimensional vector-space $F$, each $d_k:F\supset W\to[0,r]$, $k=1,\ldots,{\bf k}$, and $x_t(s)=x(t+s)$. The solutions define semi-flow of differentiable solution operators on sub-manifold $X_f\subset C^1([-r,0],\mathbb which is given by compatibility condition $\phi'(0)=f(\phi)$ with $$f (\phi)=g(\phi(-d_1(L\phi)),\ldots,\phi(-d_{{\bf k}}(L\phi))). We prove that $X_f$ has finite atlas at most $2^{{\bf k}}$ manifold charts, whose domains are almost graphs over $X_0$. size depends solely zero-sets functions $d_k$.