Periodic Eigendecomposition and Its Application to Kuramoto--Sivashinsky System

Abstract

Periodic eigendecomposition, to be formulated in this paper, is a numerical method to compute Floquet spectrum and Floquet vectors along periodic orbits in a dynamical system. It is rooted in numerical algorithms advances in computation of 'covariant vectors' of the linearized flow along an ergodic trajectory in a chaotic system. Recent research on covariant vectors strongly strongly suggests that the physical dimension of inertial manifold of a dissipative PDE can be characterized by a finite number of 'entangled modes', dynamically isolated from the residual set of transient degrees of freedom. We anticipate that Floquet vectors display similar properties as covariant vectors. In this paper we incorporate periodic Schur decomposition to the computation of dynamical Floquet vectors, compare it with other methods, and show that the method can yield the full Floquet spectrum of a periodic orbit at every point along the orbit to high accuracy. Its power, and in particular its ability to resolve eigenvalues whose magnitude differs by hundreds of orders magnitude, is demonstrated by applying the algorithm to computation of the full linear stability spectrum of several periodic solutions in one dimensional Kuramoto-Sivashinsky flow.