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Structurally stable non-degenerate singularities of integrable systems

E. A. Kudryavtseva, A. A. Oshemkov

TL;DR

This work investigates the structural stability of singularities in integrable Hamiltonian systems via the Liouville fibration. It proves that non-degenerate semilocal singularities satisfying a connectedness condition remain topologically invariant under small real-analytic perturbations, by leveraging strengthened local normal forms (Vey-type) and semilocal classifications. The results unify local and semilocal stability phenomena, providing criteria to verify the connectedness condition and showing analytic robustness in many classical cases, while highlighting instabilities under purely smooth perturbations. The Kovalevskaya top serves as a prominent illustration, where saddle-saddle singularities are analytically stable but not under $C^\infty$ smooth perturbations, underscoring the delicate nature of analytic versus smooth stability in integrable systems.

Abstract

In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a non-degenerate singular fibre satisfying the so-called connectedness condition is structurally stable under (small enough) real-analytic integrable perturbations of the system. In other words, the topology of the fibration in a neighbourhood of such a fibre is preserved after any such perturbation. As an illustration, we show that a saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations, but structurally unstable under $C^\infty$ smooth integrable perturbations.

Structurally stable non-degenerate singularities of integrable systems

TL;DR

This work investigates the structural stability of singularities in integrable Hamiltonian systems via the Liouville fibration. It proves that non-degenerate semilocal singularities satisfying a connectedness condition remain topologically invariant under small real-analytic perturbations, by leveraging strengthened local normal forms (Vey-type) and semilocal classifications. The results unify local and semilocal stability phenomena, providing criteria to verify the connectedness condition and showing analytic robustness in many classical cases, while highlighting instabilities under purely smooth perturbations. The Kovalevskaya top serves as a prominent illustration, where saddle-saddle singularities are analytically stable but not under smooth perturbations, underscoring the delicate nature of analytic versus smooth stability in integrable systems.

Abstract

In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a non-degenerate singular fibre satisfying the so-called connectedness condition is structurally stable under (small enough) real-analytic integrable perturbations of the system. In other words, the topology of the fibration in a neighbourhood of such a fibre is preserved after any such perturbation. As an illustration, we show that a saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations, but structurally unstable under smooth integrable perturbations.
Paper Structure (12 sections, 9 theorems, 37 equations, 3 figures)

This paper contains 12 sections, 9 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Structurally stable singularities
  3. Non-degenerate singularities: local symplectic normal form
  4. The connectedness condition
  5. Main result
  6. Symplectic structural stability of local singularities
  7. Structural stability of semilocal singularities
  8. Proof of Theorem \ref{['thm:stab']}
  9. Structurally stable non-degenerate singularities of the Kovalevskaya top
  10. Proof of Principle Lemma \ref{['lem:3']}, Theorem \ref{['thm:stab:help']} and Corollary \ref{['cor:stab:help']}
  11. Local normal form and its rigidity
  12. Proof of Theorem \ref{['thm:2']}

Key Result

Theorem 1.3

For each non-degenerate singular point $m_0\in M$, the fibration is locally symplectically equivalent to the direct product of a regular fibration and several copies of elliptic, hyperbolic and focus-focus singularities, i.e., to a canonical system

Figures (3)

  • Figure 1: Elementary semilocal singularities: elliptic, some hyperbolic and focus-focus singularities.
  • Figure 2: Bifurcation diagram and singularity types for the Kovalevskaya top with (a) $g=0$, (b) $0<g^2<1$, (c) $1<g^2<8/(3\sqrt 3)$, (d) $8/(3\sqrt 3)<g^2<2$, (e) $g^2>2$.
  • Figure 3: Semilocal singularity ${\mathcal{C}}_2$ and its perturbed fibration (with notations due to BF). White circles transform to gray ones through a singular fiber, and after that gray circles transform to black ones.

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2: kud:toric
  • Theorem 1.3: Smooth local normal form
  • Theorem 1.4: Real-analytic local normal form
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Definition 1.10: zung96a, BF, bol:osh06
  • ...and 16 more