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

TL;DR

This paper constructs a finite atlas of manifold charts for solution manifolds of differential systems with discrete state-dependent delays, revealing their geometric structure and dependence on delay functions.

Contribution

It introduces a finite atlas for solution manifolds of delay differential equations with state-dependent delays, detailing how the atlas size depends on delay functions' zerosets.

Findings

Finite atlas of at most 2^k charts for solution manifolds.

Charts are almost graphs over a base manifold.

Atlas size depends only on zerosets of delay functions.

Abstract

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 a finite-dimensional vectorspace $F$, each $d_k:F\supset W\to[0,r]$, $k=1,\ldots,{\bf k}$, continuously differentiable, and $x_t(s)=x(t+s)$. The solutions define a semiflow of continuously differentiable solution operators on the submanifold $X_f\subset C^1([-r,0],\mathbb{R}^n)$ which is given by the 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 a finite atlas of at most $2^{{\bf k}}$ manifold charts, whose domains are almost graphs over $X_0$. The size of the atlas depends solely on the zerosets of the delay functions $d_k$.