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Bifurcations of magnetic geodesic flows on surfaces of revolution

Ivan F. Kobtsev, Elena A. Kudryavtseva

TL;DR

The paper advances the topological classification of magnetic geodesic flows on S^2 with rotational symmetry by computing the semi-local and semi-global 4D singularities, and by fully determining the Fomenko–Zieschang isoenergy invariants for the corresponding Liouville foliations. Central to the approach is the reduction to a one-parameter family of planar curves (f, Λ) that define the metric and magnetic field, with the topology encoded in two key constructs: the bifurcation curves γ1 and γ2 and their projective duals γ and Γ. The authors prove a complete realizability correspondence between planar dual curves and the bifurcation diagrams, show that all such diagrams lie on two curves in the (h, k) plane (including a line h = 0 and a dual-transformed curve), and reveal rich phenomena such as asymmetric elliptic forks and cuspidal tori, while providing explicit invariant data (atoms, gluing matrices, and edge marks) that classify Liouville foliations up to rough Liouville equivalence. The results connect geometric properties of Γ and γ to dynamical behaviors of magnetic geodesics, offering a constructive pathway to assemble bifurcation complexes from the curve data and to reconstruct the system from the bifurcation diagram. This yields a rigorous, geometry-driven atlas-like framework for a broad class of integrable magnetic geodesic flows on surfaces of revolution with S^1-symmetry, with potential extensions to more general manifolds and perturbations.

Abstract

We study magnetic geodesic flows invariant under rotations on the 2-sphere. The dynamical system is given by a generic pair of functions $(f,Λ)$ 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 $(a:-1:k) = (f:Λ:1)^*$ projectively dual to the curve $(f:Λ:1)$ by the transformation $(a:-1:k)\mapsto(a^2/2,k)=(h,k)$.

Bifurcations of magnetic geodesic flows on surfaces of revolution

TL;DR

The paper advances the topological classification of magnetic geodesic flows on S^2 with rotational symmetry by computing the semi-local and semi-global 4D singularities, and by fully determining the Fomenko–Zieschang isoenergy invariants for the corresponding Liouville foliations. Central to the approach is the reduction to a one-parameter family of planar curves (f, Λ) that define the metric and magnetic field, with the topology encoded in two key constructs: the bifurcation curves γ1 and γ2 and their projective duals γ and Γ. The authors prove a complete realizability correspondence between planar dual curves and the bifurcation diagrams, show that all such diagrams lie on two curves in the (h, k) plane (including a line h = 0 and a dual-transformed curve), and reveal rich phenomena such as asymmetric elliptic forks and cuspidal tori, while providing explicit invariant data (atoms, gluing matrices, and edge marks) that classify Liouville foliations up to rough Liouville equivalence. The results connect geometric properties of Γ and γ to dynamical behaviors of magnetic geodesics, offering a constructive pathway to assemble bifurcation complexes from the curve data and to reconstruct the system from the bifurcation diagram. This yields a rigorous, geometry-driven atlas-like framework for a broad class of integrable magnetic geodesic flows on surfaces of revolution with S^1-symmetry, with potential extensions to more general manifolds and perturbations.

Abstract

We study magnetic geodesic flows invariant under rotations on the 2-sphere. The dynamical system is given by a generic pair of functions 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 -plane. One of these curves is a line segment , and the other lies in the half-plane and can be obtained from the curve projectively dual to the curve by the transformation .

Paper Structure

This paper contains 14 sections, 15 theorems, 52 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. The history of the problem under investigation
  3. Magnetic geodesic flow $S(f,\Lambda)$ on a spherical surface of revolution
  4. Semi-local and semi-global 4D singularities of the magnetic geodesic flow $S(f,\Lambda)$
  5. Semi-local 4D singularities
  6. Semi-global 4D singularities
  7. Topology of the Liouville foliation on 3D isoenergy manifolds of the system $S(f,\lambda)$
  8. Invariants of rough Liouville equivalence
  9. Calculation of the Fomenko--Zieschang isoenergy marks (global 3D topological invariants)
  10. Description of all bifurcation diagrams for magnetic geodesic flows $S(f,\Lambda)$. Global 4D topological invariants
  11. Projectively dual projective curves. Singular points and inflection points of good projective curves
  12. Geometric characterization of the planar curve $\Gamma=(f,\Lambda,1)$ determining the system $S(f,\Lambda)$
  13. Geometric and dynamical properties of the bifurcation curve $\gamma_1=(h,k)$
  14. The criterion for realizability of a planar curve as the bifurcation curve of a system $S(f,\Lambda)$

Key Result

Theorem 3.3

If the functions $f,\Lambda$ satisfy the genericity conditions 1--5 from Assumption assump2.4 then the singular points of the magnetic geodesic flow $S(f,\Lambda)$ on the sphere are exhausted by the following list:

Figures (10)

  • Figure 1: a) Bifurcation diagram $\Sigma^\circ$ in the space $(\lambda_0,\lambda_1,\lambda_2)=(-h,\varepsilon,p_2(K))$ for a singularity of type $A_3$ (swallowtail); b) appearance of a cuspidal torus in the magnetic geodesic flow with the potential $\varepsilon U$ near an "asymmetric elliptic fork" of the system $S(f,\Lambda)$ with a small $\varepsilon=\lambda_1>0$
  • Figure 2: Liouville foliation near degenerate singular fibers of the magnetic geodesic flow $S(f,\Lambda)$: a) near a cuspidal torus, b) near an elliptic fork
  • Figure 3: The curves $\Gamma$, $\gamma$ and $\gamma_1$ near the saddle 3-atom $V_{\sigma_1\sigma_2}\times S^1$ (a, c, e) and the corresponding bifurcation of the Fomenko molecule (b, d, f), where $(\sigma_1,\sigma_2)=(+,+)$, $(-,-)$ and $(-,+)$ respectively, with notation $\gamma^{(1)}_1=\gamma_1|_{(\rho_1-\varepsilon,\rho_1+\varepsilon)}$, $\gamma^{(2)}_1=\gamma_1|_{(\rho_2-\varepsilon,\rho_2+\varepsilon)}$
  • Figure 4: a) Definition of a marked cross, b) the rules for gluing crosses
  • Figure 5: Examples of atoms glued from crosses: a)$V_{++}=D_1$, b)$V_{+-}=D_2$
  • ...and 5 more figures

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Remark
  • Definition 3.1
  • Theorem 3.3: KO:20
  • Remark 3.4
  • proof
  • Proposition 3.5
  • Remark 3.6
  • ...and 48 more