Normal forms and uniform approximations for bridge orbit bifurcations

TL;DR

This paper develops normal forms and uniform approximations to analyze bridge orbit bifurcations in Hamiltonian systems, successfully matching quantum-mechanical results and advancing understanding of complex bifurcation scenarios.

Contribution

It introduces new normal forms for bridge orbit bifurcations and derives uniform approximations that accurately reproduce quantum results.

Findings

Normal forms effectively describe bifurcation scenarios.

Uniform approximations match quantum-mechanical density of states.

Method enhances analysis of bifurcations in integrable systems.

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 Poincar\'e 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.