Classification of saddle-focus singularities

TL;DR

This paper introduces an algorithm for classifying saddle-focus singularities in integrable Hamiltonian systems with three degrees of freedom, providing a comprehensive list of singularities up to complexity three.

Contribution

It develops a new algorithm for topological classification and enumerates all saddle-focus singularities of low complexity, extending previous work.

Findings

Algorithm for classification of saddle-focus singularities

Complete list of singularities of complexity 1, 2, and 3

Representation of singularities as almost direct products with cyclic groups

Abstract

The paper presents an algorithm for topological classification of nondegenerate saddle-focus singularities of integrable Hamiltonian systems with three degrees of freedom up to semi-local equivalence. In particular, we prove that any singularity of saddle-focus type can be represented as an almost direct product in which the acting group is cyclic. Based on the constructed algorithm, a complete list of singularities of saddle-focus type of complexity $1$, $2$, and $3$, i.e., singularities whose leaf contains one, two, or three singular points of rank $0$, is obtained. Earlier both singularities of saddle-focus type of complexity $1$ were also described by L.M. Lerman.