Structurally stable non-degenerate singularities of integrable systems

TL;DR

This paper proves that certain non-degenerate singular fibers in integrable systems are topologically stable under small real-analytic perturbations, with implications for the stability of specific singularities like the Kovalevskaya top.

Contribution

It establishes the structural stability of non-degenerate singular fibers satisfying the connectedness condition under real-analytic perturbations in integrable systems.

Findings

Non-degenerate singular fibers with connectedness are stable under real-analytic perturbations.

Kovalevskaya top's saddle-saddle singularity is stable under real-analytic but not smooth perturbations.

Topological stability depends on the regularity of the perturbation.

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.