Numerical Methods in Floquet Theory

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

This paper discusses numerical methods of computing a particular normal form, called a Floquet normal form, of solutions for linear, time-periodic differential equations. This paper is interesting for the following reasons. First, it gives a nice introduction to the key theorems of Floquet theory, with proofs and/or references. Second, we may model legged animals, robots, and other systems exhibiting “rhythmic” behavior by oscillators – ordinary differential equations exhibiting limit cycle behavior. It turns out that linear, time-periodic differential equations arise naturally in the analysis of such systems (when linearizing the dynamics about the periodic orbit). If the Floquet normal form for such a system can be computed, this may allow insights into the mathematical models describing legged animals, robots, and other rhythmic systems, which leads to insight into these systems themselves.

Link to Paper

Abstract

In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution \(\Phi(t)\) is a canonical decomposition of the form \(\Phi(t)=Q(t)e^{Rt}\), where \(Q(t)\) is a real periodic matrix and \(R\) is a constant matrix. To rigorously compute the Floquet normal form, the idea is to use the regularity of \(Q(t)\) and to simultaneously solve for \(R\) and \(Q(t)\) with the contraction mapping theorem in a Banach space of rapidly decaying coefficients. The explicit knowledge of \(R\) and \(Q\) can then be used to construct, in a rigorous computer-assisted way, stable and unstable bundles of periodic orbits of vector fields. The new proposed method does not require rigorous numerical integration of the ODE.

Authors : R. Castelli and J. Lessard