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

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