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

Abstract

<p style='text-indent:20px;'>For differential equations with state-dependent delays the associated initial value problem is well-posed, with differentiable solution operators, on submanifolds of the space <inline-formula><tex-math id="M1">\begin{document}$ C^1_n=C^1([-r,0],\mathbb{R}^n) $\end{document}</tex-math></inline-formula>, under mild smoothness assumptions. We study these <i>solution manifolds</i> and find that for a large class of equations their solution manifolds are nearly as simple as a graph over the subspace <inline-formula><tex-math id="M2">\begin{document}$ X_0\subset C^1_n $\end{document}</tex-math></inline-formula> defined by <inline-formula><tex-math id="M3">\begin{document}$ \phi'(0)=0 $\end{document}</tex-math></inline-formula>. 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.</p>