2007/05/31 by Matthias Brack, M. Brack, Kaori Tanaka +1 · 2 citations
Mathematics · Physics and Astronomy · #Nonlinear Photonic Systems #Nuclear physics research studies #Quantum chaos and dynamical systems #math.DS #nlin.CD
paper · pdf · doi:10.1103/physreve.77.046205
published as Phys. Rev. E 77, 046205 (2008) · LaTeX, 38 pages, 13 figures; substantially revised version v4; to be published in Phys. Rev. E
arxiv created 2008/03/07 · openalex publication_date 2008/04/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We report on transcritical bifurcations of periodic orbits in nonintegrable two-dimensional Hamiltonian systems. We discuss their existence criteria and some of their properties using a recent mathematical description of transcritical bifurcations in families of symplectic maps. We then present numerical examples of transcritical bifurcations in a class of generalized Hénon-Heiles Hamiltonians and illustrate their stabilities and unfoldings under various perturbations of the Hamiltonians. We demonstrate that for Hamiltonians containing straight-line librating orbits, the transcritical bifurcation of these orbits is the typical case which occurs also in the absence of any discrete symmetries, while their isochronous pitchfork bifurcation is an exception. We determine the normal forms of both types of bifurcations and derive the uniform approximation required to include transcritically bifurcating orbits in the semiclassical trace formula for the density of states of the quantum Hamiltonian. We compute the coarse-grained density of states in a specific example both semiclassically and quantum mechanically and find excellent agreement of the results.