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

Abstract

We show that for a system $$x^{\prime }(t)=g(x(t-d_1(L{x}_t)),\dots ,x(t-d_k(L{x}_t)))$$ 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 $d_{\kappa }$ are assumed to be continuously differentiable.

16 pages