On the use of stabilising transformations for detecting unstable periodic orbits in the Kuramoto-Sivashinsky equation

TL;DR

This paper enhances a method for detecting unstable periodic orbits in high-dimensional systems by using stabilising transformations based on a small set of known orbits, applied to the Kuramoto-Sivashinsky equation.

Contribution

It extends a stabilising transformation approach to high-dimensional systems, reducing the number of transformations needed by focusing on the unstable manifold dimension.

Findings

The new method outperforms Newton-Armijo and Levernberg-Marquardt in detecting UPOs.

The approach is effective for the high-dimensional Kuramoto-Sivashinsky system.

The number of transformations depends on the unstable manifold, not system dimension.

Abstract

In this paper we develop further a method for detecting unstable periodic orbits (UPOs) by stabilising transformations, where the strategy is to transform the system of interest in such a way that the orbits become stable. The main difficulty of using this method is that the number of transformations, which were used in the past, becomes overwhelming as we move to higher dimensions (Davidchack and Lai 1999; Schmelcher et al. 1997, 1998). We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs (Crofts and Davidchack 2006). The main benefit of using the proposed set is that its cardinality depends on the dimension of the unstable manifold at the UPO rather than the dimension of the system. In a typical situation the dimension of the unstable manifold is much smaller than the dimension of the system so the number of transformations is much smaller. Here we extend this approach to high-dimensional systems of ODEs and apply it to the model example of a chaotic spatially extended system -- the Kuramoto-Sivashinsky equation. A comparison is made between the performance of this new method against the competing methods of Newton-Armijo (NA) and Levernberg-Marquardt (LM). In the latter case, we take advantage of the fact that the LM algorithm is able to solve under-determined systems of equations, thus eliminating the need for any additional constraints.