Structurally stable non-degenerate singularities of 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.

arXiv comment25 pages, 3 figures

References in corpus (6)

• Torus Actions and Integrable Systems
• Bifurcation of common levels of first integrals of the Kovalevskaya problem
• Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities
• Hidden toric symmetry and structural stability of singularities in integrable systems
• The Liouville-type theorem for integrable Hamiltonian systems with incomplete flows
• Existence of a smooth Hamiltonian circle action near parabolic orbits

Cited by in corpus (1)

• Symplectic classification for universal unfoldings of $A_n$ singularities in integrable systems