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

Abstract

Let r > 0, n ∈ N,k ∈ N. Consider the delay differential equation x(t) = g(x(t− d1(Lxt)), . . . , x(t− dk(Lxt))) for g : (R) ⊃ V → R continuously differentiable, L a continuous linear map from C([−r, 0],R) into a finite-dimensional vectorspace F , each dk : F ⊃ W → [0, r], k = 1, . . . ,k, continuously differentiable, and xt(s) = x(t + s). The solutions define a semiflow of continuously differentiable solution operators on the submanifold Xf ⊂ C([−r, 0],R) which is given by the compatibility condition φ′(0) = f(φ) with f(φ) = g(φ(−d1(Lφ)), . . . , φ(−dk(Lφ))). We prove that Xf has a finite atlas of at most 2 k manifold charts, whose domains are almost graphs over X0. The size of the atlas depends solely on the zerosets of the delay functions dk.