Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits

TL;DR

This paper introduces a rigorous numerical method for computing Floquet normal forms of periodic linear systems, enabling precise determination of stable and unstable bundles of periodic orbits with applications to Lorenz and $
73$-models.

Contribution

A novel fixed point based numerical approach for Floquet normal forms that rigorously computes stable and unstable bundles of periodic orbits.

Findings

Successfully applied to Lorenz equations and $
73$-model.

Provides rigorous bounds for stability analysis.

Enhances accuracy in computing Floquet solutions.

Abstract

In this paper, a new rigorous numerical method to compute fundamental matrix solutions of non-autonomous linear differential equations with periodic coefficients is introduced. Decomposing the fundamental matrix solutions $\Phi(t)$ by their Floquet normal forms, that is as product of real periodic and exponential matrices $\Phi(t)=Q(t)e^{Rt}$, one solves simultaneously for $R$ and for the Fourier coefficients of $Q$ via a fixed point argument in a suitable Banach space of rapidly decaying coefficients. As an application, the method is used to compute rigorously stable and unstable bundles of periodic orbits of vector fields. Examples are given in the context of the Lorenz equations and the $\zeta^3$-model.