Floquet-like theory and gauge transformations for general smooth dynamical systems

TL;DR

This paper extends Floquet theory to non-autonomous, non-periodic systems using gauge transformations, providing a geometric framework for analyzing general smooth dynamical systems.

Contribution

It introduces a novel geometric approach employing gauge transformations to generalize Floquet theory beyond periodic systems.

Findings

Provides a new method for analyzing non-periodic dynamical systems

Connects gauge transformations with the analysis of smooth dynamical systems

Extends classical Floquet theory to a broader class of equations

Abstract

The classical Floquet theory allows to map a time-periodic system of linear differential equations into an autonomous one. By looking at it in a geometrical way, we extend the theory to a class of non-autonomous non-periodic equations. This is obtained by considering a change of variables which depends on time in a non-trivial way, i.e. introducing gauge transformations, well known in fundamental Physics and Field Theory -- but which seems to have received little attention in this context.