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

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 propose to 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 dependence of the number of transformations on the dimensionality of the unstable manifold rather than on system size enables us to apply, for the first time, the method of stabilising transformations to high-dimensional systems. Another important aspect of our treatment of high-dimensional flows is that we do not restrict to a Poincare surface of section. This is a particularly nice feature, since the correct placement of such a section in a high-dimensional phase space is a challenging problem in itself. 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 Henon maps and to the Kuramoto-Sivashinsky system in the weakly turbulent regime.