Transcritical bifurcations in non-integrable Hamiltonian systems

Abstract

We report on transcritical bifurcations of periodic orbits in non-integrable 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.

LaTeX, 38 pages, 13 figures; substantially revised version v4; to be published in Phys. Rev. E

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Cited by in corpus (9)

• Shell structure and orbit bifurcations in finite fermion systems
• Normal forms and uniform approximations for bridge orbit bifurcations
• Closed orbits and spatial density oscillations in the circular billiard
• Analytic approach to bifurcation cascades in a class of generalized Hénon-Heiles potentials
• Shells, orbit bifurcations and symmetry restorations in Fermi systems
• Semiclassical treatment of symmetry breaking and bifurcations in a non-integrable potential
• Bifurcation cascade, self-similarity and duality in the 3-rotor problem
• Mathematical remarks on transcritical bifurcation in Hamiltonian systems
• Bifurcation of straight-line librations