We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
64/2015
On the regularity of lumped nonlinear dynamics in Banach spaces
Fatihcan M. Atay and Lavinia Roncoroni
Abstract
We study the regularity properties of the lumping problem for differential equations in Banach spaces, namely the projection of dynamics by a reduction operator onto a reduced state space in which a self-contained dynamical description exists. We study dynamics generated by a nonlinear operator $F$ and a linear and bounded reduction operator $M$. We first show, using quotient space methods, that the reduced operator is $C^1$, provided that $F$ itself is $C^1$ in the original state space. We further prove that a particular {lumping relation} holds between the Fréchet differentials of $F$ and the reduced operator. In this way, by smoothness, the linearization principle applies and it is possible to use results from linear theory to study the local behavior of the system.
