On the solution manifold of a differential equation with a delay which has a zero

TL;DR

This paper proves that the solution manifold of a differential equation with a state-dependent delay is diffeomorphic to a hyperplane, refining previous results that only provided a covering by two almost graphs.

Contribution

It establishes that the solution manifold is an almost graph over a hyperplane and diffeomorphic to it, improving the understanding of the structure of solution manifolds for such equations.

Findings

Solution manifold is an almost graph over a hyperplane

Solution manifold is diffeomorphic to a hyperplane

Previous results only provided a covering by two almost graphs

Abstract

For a differential equation with a state-dependent delay we show that the associated solution manifold $X_f$ of codimnsion 1 in the space $C^1([-r,0],\mathbb {R})$ is an almost graph over a hyperplane, which implies that $X_f$ is diffeomorphic to the hyperplane. For the case considered previous results only provide a covering by 2 almost graphs.