Classification of hyperbolic singularities of rank zero of integrable Hamiltonian systems. (English. Russian original) Zbl 1205.37072
Sb. Math. 201, No. 8, 1153-1191 (2010); translation from Mat. Sb. 201, No. 8, 63-102 (2010).
Semilocal classification of saddle singularities of integrable Hamiltonian systems is considered. For this aim a certain combinatorial object (an f-graph) is associated with every nondegenerate saddle singularity of rank zero is entered. As a result, the problem of semilocal classification is reduced to the problem of enumeration of the f-graphs. The correspondence between nondegenerate saddle singularities of rank zero and f-graph is based on the following supervision: the operation of direct product of simplest singularities conform to the operation of product of f-graph, and the factorization of direct product of singularities by a free component-wise action of a finite group conform to the analogous factorization of f-graphs. This enables to describe a simple algorithm for obtaining the lists of saddle singularities of rank zero for a given number of degrees of freedom and a given complexity.
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |
| 37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
| 57R45 | Singularities of differentiable mappings in differential topology |
| 58K45 | Singularities of vector fields, topological aspects |
| 70G40 | Topological and differential topological methods for problems in mechanics |
