Research Group Prof. F. Haake

Theoretical Physics


Classical and semiclassical description of systems with a mixed phase space

figure 1
Figure 1: Phase space portraits of the kicked top, illustrating the transition from integrable dynamics to well-developed chaos as an external parameter is varied.

The phase space of generic Hamiltonian systems displays a mixture of regions with predominantly regular dynamics (islands) and chaotic parts (seas, cf. Fig. 1). The character of a region is determined by the nature of its periodic orbits. Chaotic parts are inhabited by instable (hyperbolic) orbits while in the regular regions there are also stable (elliptic) orbits.

When a control parameter is varied then orbits move through phase space and the shape of the regions changes. Dramatic changes are encountered when orbits change in their nature, disappear or are created. This happens in bifurcations, when orbits become degenerate (they meet in phase space) at certain values of the control parameter.

figure 2
Figure 2: Typical sequence of bifurcations: In the neighbourhood of a period--one orbit located at the centre of an island of stability, two satellite orbits of period three come into existence in a bifurcation.

Subsequently, the three fixed points of the stable satellite meet the central orbit in another bifurcation and emerge on the opposite side.

In the classical part of this study we classified bifurcations of different degree of complexity. We studied bifurcations of codimension one and two. (The codimension giving the number of parameters to be controlled in order to encounter the bifurcation generically). Bifurcations of codimension two tell about possible sequences of bifurcation of codimension one, like the sequence of bifurcations shown in Fig. 2.

Periodic orbits arise naturally in the semiclassical treatment of spectral properties of the quantum-mechanical counterpart of a Hamiltonian system. This connection was established for fully chaotic (hyperbolic) systems by Gutzwiller and in the integrable case Berry and Tabor, as well as by Balian and Bloch.

For mixed dynamics, a collective treatment of the periodic orbits turns out to be mandatory, in particular when orbits are about to bifurcate (and such orbits typically constitute a considerable portion). Conventional periodic-orbit theory diverges there. For each bifurcation of codimension one and two we obtained appropriate regular semiclassical expressions (uniform approximations). With these expression at hand, complete energy spectra of our model system, the kicked top, could be obtained from the periodic orbits, and good agreement was found to exact quantum mechanics.

Selected Publications

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