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

TL;DR

This paper demonstrates that the solution manifold of certain differential systems with discrete state-dependent delays is structurally simple, resembling a graph over a closed subspace, under specific smoothness and linearity conditions.

Contribution

It establishes that the solution manifold for these delay differential systems is almost a graph, simplifying the analysis of solution operators in such systems.

Findings

Solution manifold is nearly a graph over a closed subspace.

Differentiability of solution operators is established.

Structural simplicity aids in understanding delay differential systems.

Abstract

We show that for a system $$ x'(t)=g(x(t-d_1(Lx_t)),\dots,x(t-d_k(Lx_t))) $$ 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.