Efficient Detection of Periodic Orbits in Chaotic Systems by Stabilising Transformations

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-dimensional kicked double rotor map and the six-dimensional system of three coupled Hénon maps.