Bifurcations of magnetic geodesic flows on surfaces of revolution

TL;DR

This paper analyzes the bifurcation structure and topology of magnetic geodesic flows on surfaces of revolution, providing a detailed classification of singularities and the bifurcation diagrams of the integrable system.

Contribution

It offers a comprehensive topological classification of the Liouville fibration and bifurcation diagrams for magnetic geodesic flows on surfaces of revolution, including explicit descriptions of singularities and bifurcation curves.

Findings

Topology of Liouville fibrations near singular orbits is described.

All bifurcation diagrams of the momentum maps are characterized.

The bifurcation diagram consists of two specific curves in the (h,k)-plane.

Abstract

We study magnetic geodesic flows invariant under rotations on the 2-sphere. The dynamical system is given by a generic pair of functions $(f,\Lambda)$ in one variable. Topology of the Liouville fibration of the given integrable system near its singular orbits and singular fibers is described. Types of these singularities are computed. Topology of the Liouville fibration on regular 3-dimensional isoenergy manifolds is described by computing the Fomenko--Zieschang invariant. All possible bifurcation diagrams of the momentum maps of such integrable systems are described. It is shown that the bifurcation diagram consists of two curves in the $(h,k)$-plane. One of these curves is a line segment $h=0$, and the other lies in the half-plane $h\ge0$ and can be obtained from the curve $(a:-1:k) = (f:\Lambda:1)^*$ projectively dual to the curve $(f:\Lambda:1)$ by the transformation $(a:-1:k)\mapsto(a^2/2,k)=(h,k)$.