Efficient method for detection of periodic orbits in chaotic maps and flows

TL;DR

This paper presents an improved algorithm for detecting unstable periodic orbits in chaotic systems, especially high-dimensional ones, by reducing the number of stabilising transformations needed, thus enhancing efficiency.

Contribution

The authors develop a new method that uses known stability matrices of existing orbits to significantly decrease the transformations required for orbit detection in high-dimensional systems.

Findings

Efficient detection of periodic orbits in high-dimensional chaotic systems.

Successful application to four-dimensional and six-dimensional maps.

Reduced computational complexity in orbit detection.

Abstract

An algorithm for detecting unstable periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp. 4733-4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when applied to low-dimensional system. The difficulty in applying the algorithm to higher dimensional systems is mainly due to the fact that the number of stabilising transformations grows extremely fast with increasing system dimension. In this thesis, we construct stabilising transformations based on the knowledge of the stability matrices of already detected periodic orbits (used as seeds). The advantage of our approach is in a substantial reduction of the number of transformations, which increases the efficiency of the detection algorithm, especially in the case of high-dimensional systems. The performance of the new approach is illustrated by its application to the four-dimensional kicked double rotor map, a six-dimensional system of three coupled H\'enon maps and to the Kuramoto-Sivashinsky system in the weakly turbulent regime.