A numerical method for locating stable periodic orbits in chaotic systems
The knowledge of the stable and weakly unstable periodic orbits of chaotic systems is a subject of increasing importance, both from the theoretical point of view, as well as for experimental studies. Recent work on the periodic orbit expansion in terms of zeta functions has shown that only a few short orbits are needed in order to construct the full classical spectrum [\textit{P. Cvitanović} and \textit{B. Eckhardt}, J. Phys. A, Math. Gen. 24, L 237--L 241 (1991; Zbl 0729.58042)]. The quadratic Zeeman effect (QZE) for atomic hydrogen [\textit{M. Yu. Sumetskij}, Sov. Phys. JETP 56, 959 (1983)] is a well known case study which has provided interesting examples of analysis of high energy atomic spectra in terms of periodic orbits. A numerical method for locating stable orbits in chaotic systems is proposed in the paper under review. The method is based on the selection of trajectory segments which are characterized by a relatively low positive local Lyapunov exponent (LLE). Once a selection is made, convergence to stable (or weakly unstable) periodic orbits is obtained by a Newton method. Previous experience with the LLE distribution for the standard map [\textit{M. A. Sepulveda}, \textit{R. Badii} and the second author, Phys. Rev. Lett. 63, 1226-1229 (1989)] has shown that all trajectory segments with low exponents gather on the boundaries of the stable islands. The new approach is well-suited for the study of systems with more than two degrees of freedom, as compared with the algorithms given in [\textit{B. Mestel} and \textit{I. Percival}, Physica D 24, 172-178 (1987; Zbl 0616.65134), \textit{M. Baranger}, \textit{K. T. R. Davies} and \textit{J. H. Mahoney}, Ann. Phys. 186, 95-110 (1988; Zbl 0651.58031) and \textit{D. Auerbach}, \textit{P. Cvitanović}, \textit{P. Eckmann}, \textit{J.-P. Gunaratne} and \textit{I. Procaccia}, Phys. Rev. Lett. 58, 2387-2389 (1987)], which are also based on the use of Newton's method, or variants of it. The proposed technique is tested on the quartic oscillator model, and on the potential of the hydrogen atom in a strong magnetic field. In the latter case, new stable periodic orbits are found in the region of strong chaotic motion.
scientific article; zbMATH DE number 91659