Progress in the analysis of experimental chaos through periodic orbits

TLDR: In this article, the analysis of chaotic systems can be considerably improved with the knowledge of their periodic-orbit structure, which has led to an extension of the conventional Bloch-Kirchhoff equations of motion, to the construction of approximate generating partitions, and to an efficient control of the chaotic system around various unstable periodic orbits.

Abstract: The understanding of chaotic systems can be considerably improved with the knowledge of their periodic-orbit structure. The identification of the low-order unstable periodic orbits embedded in a strange attractor induces a hierarchical organization of the dynamics which is invariant under smooth coordinate changes. The applicability of this technique is by no means limited to analytical or numerical calculations, but has been recently extended to experimental time series. As a specific example, the authors review some of the major results obtained on a nuclear-magnetic-resonance laser which have led to an extension of the conventional (Bloch-Kirchhoff) equations of motion, to the construction of approximate generating partitions, and to an efficient control of the chaotic system around various unstable periodic orbits. The determination of the symbolic dynamics, with the precision achieved by recording all unstable cycles up to order 9, improves the topological and metric characterization of a heteroclinic crisis. The periodic-orbit approach permits detailed study of chaotic motion, thereby leading to an improved classification scheme which subsumes the older ones, based on estimates of scalar quantities such as fractal dimensions and metric entropies.