A FINITE ATLAS FOR SOLUTION MANIFOLDS OF DIFFERENTIAL SYSTEMS WITH DISCRETE STATE-DEPENDENT DELAYS

Let r > 0,n is an element of N, k is an element of N. Consider the delay differential equation x'(t) = g(x(t - d(1)(Lx(t))),..., x(t - d(k)(Lx(t)))) for g : (R-n)(k) superset of V -> R-n continuously differentiable, L a continuous linear map from C([-r, 0], R-n) into a finite-dimensional vector-space F, each d(k ): F superset of D W -> [0, r], k = 1,... ,k, continuously differentiable, and x(t) (s) = x(t + s). The solutions define a semi-flow of continuously differentiable solution operators on the sub-manifold X-f subset of C-1 ([- r, 0], R-n) which is given by the compatibility condition phi'(0) = f(phi) with f(phi) = f(phi(-d(1)(L phi)), ...,phi(-d(k)(L-phi))). We prove that X-f has a finite atlas of at most 2(k) manifold charts, whose domains are almost graphs over X-0. The size of the atlas depends solely on the zero-sets of the delay functions d(k).