Transcritical bifurcations in non-integrable Hamiltonian systems

TL;DR

This paper investigates transcritical bifurcations of periodic orbits in non-integrable Hamiltonian systems, providing criteria, normal forms, and implications for semiclassical quantization, supported by numerical and quantum results.

Contribution

It introduces a detailed analysis of transcritical bifurcations in non-integrable Hamiltonians, including normal forms and their role in semiclassical trace formulas.

Findings

Transcritical bifurcations are common in Hamiltonians with straight-line librating orbits.

Normal forms for these bifurcations are derived and used in semiclassical analysis.

Semiclassical and quantum density of states show excellent agreement in examples.

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 H\'enon-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.