On solution manifolds of differential systems with discrete state-dependent delays

Hans-Otto Walther

doi:10.3934/dcdss.2022108

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2023, Volume 16, Issue 3&4: 627-638. Doi: 10.3934/dcdss.2022108

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On solution manifolds of differential systems with discrete state-dependent delays

Hans-Otto Walther

Universität Gießen, Mathematisches Institut, Arndtstr. 2, 35392 Gießen, Germany

For Jibin Li, on the occasion of his 80th birthday

Received: December 2021

Early access: May 2022

Published: March 2023

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Abstract

For differential equations with state-dependent delays the associated initial value problem is well-posed, with differentiable solution operators, on submanifolds of the space $ C^1_n=C^1([-r,0],\mathbb{R}^n) $, under mild smoothness assumptions. We study these solution manifolds and find that for a large class of equations their solution manifolds are nearly as simple as a graph over the subspace $ X_0\subset C^1_n $ defined by $ \phi'(0)=0 $. The result supplements recent work on finite atlases of solution manifolds and is related to the open problem whether in some cases the constructed atlases are minimal.

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Mathematics Subject Classification: Primary: 34K05, 34K43; Secondary: 58B99.

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