arXiv subjects

A. A. Oshemkov

Publications and source records attributed to A. A. Oshemkov.

3 recordsLinked to original sources

arXiv2023

Classification of saddle-focus singularities

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.

math.DG↗

arXiv2023

Integrable systems with linear periodic integral for the Lie algebra $\mathrm{e}(3)$

Integrable systems with a linear periodic integral for the Lie algebra $\mathrm{e}(3)$ are considered. One investigates singulariries of the Liouville foliation, bifurcation diagram of the momentum mapping, transformations of Liouville tori, topology of isoenergy surfaces and other topological properties of such systems.

math.DG↗

arXiv2022

Structurally stable non-degenerate singularities of integrable systems

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.

math.DS↗