Normal forms and uniform approximations for bridge orbit bifurcations

2008/05/31 by Ken-ichiro Arita, Matthias Brack, M. Brack · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Bifurcation #Bridge (graph theory) #Classical mechanics #Engineering #Hamiltonian (control theory) #Hamiltonian system #Integrable system #Mathematical analysis #Mathematics #Nonlinear system #Nuclear physics research studies #Numerical methods for differential equations #Orbit (dynamics) #Periodic orbits #Physics #Poincaré map #Quantum #Quantum chaos and dynamical systems #Quantum mechanics #nlin.CD #nucl-th

paper · pdf · doi:10.1088/1751-8113/41/38/385207

published as J. Phys. A 41, 385207 (2008) · 25 pages, 18 figures, version published in Journal of Physics A: Mathematical and Theoretical

openalex publication_date 2008/08/28 · arxiv created 2008/09/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 Poincaré 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.

Citations

• Periodic Orbits and Classical Quantization Conditions
• Bifurcations of periodic orbits and uniform approximations
• Transcritical bifurcations in nonintegrable Hamiltonian systems
• Semiclassical interference of bifurcations
• Uniform approximation for period-quadrupling bifurcations
• Periodic orbits near bifurcations of codimension two: Classical mechanics, semiclassics and Stokes transitions

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• The symmetric 1:2 resonance