# Floquet Theory for Quaternion-valued Differential Equations

## TL;DR

This paper extends Floquet theory to quaternion-valued differential equations, providing a normal form for solutions, analyzing stability of periodic systems, and applying results to quaternionic Hill's equation with illustrative examples.

## Contribution

It introduces Floquet theory for quaternion-valued differential equations, including a normal form and stability analysis, which was not previously established.

## Key findings

- Floquet normal form for quaternion-valued differential equations derived.
- Stability criteria for quaternionic periodic systems established.
- Application to quaternionic Hill's equation demonstrated.

## Abstract

This paper describes the Floquet theory for quaternion-valued differential equations (QDEs). The Floquet normal form of fundamental matrix for linear QDEs with periodic coefficients is presented and the stability of quaternionic periodic systems is accordingly studied. As an important application of Floquet theory, we give a discussion on the stability of quaternion-valued Hill's equation. Examples are presented to illustrate the proposed results.