Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2002-07-24
J. Phys. A 36,1095 (2003)
Nonlinear Sciences
Chaotic Dynamics
15 pages, LaTeX, 7 figures; revised and extended version, to appear in J. Phys. A final version 3; error in eq. (33) corrected
Scientific paper
10.1088/0305-4470/36/4/317
We present an analytical calculation of periodic orbits in the homogeneous quartic oscillator potential. Exploiting the properties of the periodic Lam{\'e} functions that describe the orbits bifurcated from the fundamental linear orbit in the vicinity of the bifurcation points, we use perturbation theory to obtain their evolution away from the bifurcation points. As an application, we derive an analytical semiclassical trace formula for the density of states in the separable case, using a uniform approximation for the pitchfork bifurcations occurring there, which allows for full semiclassical quantization. For the non-integrable situations, we show that the uniform contribution of the bifurcating period-one orbits to the coarse-grained density of states competes with that of the shortest isolated orbits, but decreases with increasing chaoticity parameter $\alpha$.
Brack Matthias
Fedotkin S. N.
Magner Alexander G.
Mehta Mridul
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