Parabolicity of degenerate singularities of axisymmetric Zhukovsky case

TL;DR

This paper analyzes the nature of degenerate singularities in axisymmetric Zhukovsky systems, proving they are structurally stable and of parabolic or cuspidal type under generic conditions.

Contribution

It establishes the parabolic and cuspidal nature of degenerate singularities in axisymmetric Zhukovsky systems, extending understanding of their stability in integrable rigid body dynamics.

Findings

Degenerate singularities are of parabolic and cuspidal type.

These singularities are structurally stable under small perturbations.

All such points satisfy specific bifurcation criteria.

Abstract

The degenerate singularities of systems from one well-known multiparameter family of integrable systems of rigid body dynamics are studied. Axisymmetric Zhukovsky systems are considered, i.e. axisymmetric Euler tops after adding a constant gyrostatic moment. For all values of the set of parameters, excluding some hypersurfaces, it is proved that the degenerate local and semi-local singularities of the system are of the parabolic and cuspidal type, respectively. Thus these singularities are structurally stable under small perturbations of the system in the class of integrable systems. Note that all these degenerate points lie in the preimage of the cusp point of the parametric bifurcation curve and satisfy the criterion of A.~Bolsinov, L.~Guglielmi and E.~Kudryavtseva.