# Floquet Problem and Center Manifold Reduction for Ordinary Differential Operators with Periodic Coefficients in Hilbert Spaces

## TL;DR

This paper develops a spectral splitting and center manifold reduction for first-order differential equations with periodic operator coefficients in Hilbert spaces, extending existing results to more general periodic settings.

## Contribution

It introduces a pointwise projector and spectral splitting technique for periodic operator systems, enabling center manifold reduction for nonlinear equations with periodic coefficients.

## Key findings

- Spectral splitting into finite and infinite dimensional parts.
- Construction of a pointwise projector for periodic operator systems.
- Extension of center manifold theory to periodic coefficient settings.

## Abstract

A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite dimensional system of ordinary differential equations with constant coefficients and an infinite dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders (Kuchment) and for elliptic problems in quasicylinders (Nazarov). As an application we give a center manifold reduction for a class of non-linear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients (Mielke).