319 research outputs found
Bifurcation cascades and self-similarity of periodic orbits with analytical scaling constants in Henon-Heiles type potentials
We investigate the isochronous bifurcations of the straight-line librating
orbit in the Henon-Heiles and related potentials. With increasing scaled energy
e, they form a cascade of pitchfork bifurcations that cumulate at the critical
saddle-point energy e=1. The stable and unstable orbits created at these
bifurcations appear in two sequences whose self-similar properties possess an
analytical scaling behavior. Different from the standard Feigenbaum scenario in
area preserving two-dimensional maps, here the scaling constants \alpha and
\beta corresponding to the two spatial directions are identical and equal to
the root of the scaling constant \delta that describes the geometric
progression of bifurcation energies e_n in the limit n --> infinity. The value
of \delta is given analytically in terms of the potential parameters.Comment: 20 pages, 10 figures, LaTeX. Contribution to Festschrift "To Martin
C. Gutzwiller on His Seventy-Fifth Birthday", eds. A. Inomata et al., final
revised version (updated references, note added in proof
Semiclassical description of shell effects in finite fermion systems
Since its first appearance in 1971, Gutzwiller's trace formula has been
extended to systems with continuous symmetries, in which not all periodic
orbits are isolated. In order to avoid the divergences occurring in connection
with symmetry breaking and orbit bifurcations (characteristic of systems with
mixed classical dynamics), special uniform approximations have been developed.
We first summarize some of the recent developments in this direction. Then we
present applications of the extended trace formulae to describe prominent
gross-shell effects of various finite fermion systems (atomic nuclei, metal
clusters, and a mesoscopic device) in terms of the leading periodic orbits of
their suitably modeled classical mean-field Hamiltonians.Comment: LaTeX, 12 pages, 9 figures; invited contribution to Symposium "30
Jahre Gutzwiller Spurformel" at DPG spring meeting, Hamburg, March 28, 2001.
To appear in Advances in Solid State Physic
Analyzing shell structure from Babylonian and modern times
We investigate ``shell structure'' from Babylonian times: periodicities and
beats in computer-simulated lunar data corresponding to those observed by
Babylonian scribes some 2500 years ago. We discuss the mathematical similarity
between the Babylonians' recently reconstructed method of determining one of
the periods of the moon with modern Fourier analysis and the interpretation of
shell structure in finite fermion systems (nuclei, metal clusters, quantum
dots) in terms of classical closed or periodic orbits.Comment: LaTeX2e, 13pp, 8 figs; contribution to 10th Nuclear Physics Workshop
"Marie and Pierre Curie", 24 - 28 Sept. 2003, Kazimierz Dolny (Poland); final
version accepted for J. Mod. Phys.
Normal forms and uniform approximations for bridge orbit bifurcations
We discuss various bifurcation problems in which two isolated periodic orbits
exchange periodic ``bridge'' orbit(s) between two successive bifurcations. We
propose normal forms which locally describe the corresponding fixed point
scenarios on the Poincar\'e surface of section. Uniform approximations for the
density of states for an integrable Hamiltonian system with two degrees of
freedom are derived and successfully reproduce the numerical quantum-mechanical
results.Comment: 25 pages, 18 figures, version published in Journal of Physics A:
Mathematical and Theoretica
- …