A direct approach to simplicity of solution manifolds

TL;DR

This paper introduces a novel, straightforward method for analyzing the structure of solution manifolds in differential equations with state-dependent delays, avoiding complex prior hypotheses and algebraic delay system theory.

Contribution

It constructs a diffeomorphism near the solution manifold that simplifies its structure without relying on previous restrictive assumptions.

Findings

Constructs a diffeomorphism transforming the solution manifold

Avoids hypotheses beyond smoothness

Simplifies analysis of solution manifolds in delay systems

Abstract

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} $X\subset C^1([-h,0],\mathbb{R}^n)$. For systems with discrete state-dependent delays we construct a diffeomorphism on a neighbourhood of $X$ which takes $X$ to an open subset of the subspace given by $\phi'(0)=0$. This is in line with earlier work on the nature of solution manifolds. The present approach, however, is new and dismisses all hypotheses beyond smoothness which have been instrumental so far. Compared to a recent case study it is more direct in the sense that theory of {\it algebraic delay systems} is avoided.