Efficient detection of periodic orbits in chaotic systems by stabilising transformations

TL;DR

This paper presents an improved method for detecting periodic orbits in chaotic systems by reducing the number of stabilising transformations needed, demonstrated on higher-dimensional systems.

Contribution

It introduces an alternative approach to construct fewer stabilising transformations, enhancing efficiency in higher-dimensional chaotic systems.

Findings

Effective in low-dimensional systems

Reduced number of transformations for higher dimensions

Demonstrated on 4D and 6D chaotic maps

Abstract

An algorithm for detecting 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 systems. The difficulty in applying the algorithm to higher-dimensional systems is mainly due to the fact that the number of the stabilising transformations grows extremely fast with increasing system dimension. Here we analyse the properties of stabilising transformations and propose an alternative approach for constructing a smaller set of transformations. The performance of the new approach is illustrated on the four-dimentional kicked double rotor map and the six-dimensional system of three coupled Henon maps.