Impact Factor

Pavel Ryabov

Institute of Procise Mechanics RAS , Volgograd State University


Ryabov P. E., Oshemkov A. A., Sokolov S. V.
The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram
2016, vol. 21, no. 5, pp.  581-592
The Adler – van Moerbeke integrable case of the Euler equations on the Lie algebra $so(4)$ is investigated. For the $L-A$ pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler – van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler – van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on $so(4)$.
Keywords: integrable Hamiltonian systems, spectral curve, bifurcation diagram
Citation: Ryabov P. E., Oshemkov A. A., Sokolov S. V.,  The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 581-592
Kharlamov M. P., Ryabov P. E., Savushkin A. Y.
Topological Atlas of the Kowalevski–Sokolov Top
2016, vol. 21, no. 1, pp.  24-65
We investigate the phase topology of the integrable Hamiltonian system on $e(3)$ found by V. V. Sokolov (2001) and generalizing the Kowalevski case. This generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. The relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the isoenergy manifolds of the reduced systems with two degrees of freedom are classified. The set of critical points of the momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the diagram of the momentum map and give a classification of isoenergy and isomomentum diagrams equipped with the description of regular integral manifolds and their bifurcations. We construct the Smale–Fomenko diagrams which, when considered in the enhanced space of the energy-momentum constants and the essential physical parameters, separate 25 different types of topological invariants called the Fomenko graphs. We find all marked loop molecules of rank 0 nondegenerate critical points and of rank 1 degenerate periodic trajectories. Analyzing the cross-sections of the isointegral equipped diagrams, we get a complete list of the Fomenko graphs. The marks on them producing the exact topological invariants of Fomenko–Zieschang can be found from previous investigations of two partial cases with some additions obtained from the loop molecules or by a straightforward calculation using the separation of variables.
Keywords: integrable Hamiltonian systems, relative equilibria, isoenergy surfaces, critical subsystems, bifurcation diagrams, rough topological invariants
Citation: Kharlamov M. P., Ryabov P. E., Savushkin A. Y.,  Topological Atlas of the Kowalevski–Sokolov Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 24-65
Ryabov P. E.
Bifurcation Sets in an Integrable Problem on Motion of a Rigid Body in Fluid
1999, vol. 4, no. 4, pp.  59-76
In the paper we obtain the bifurcation sets for a family of Liouville integrable Hamiltonian systems with the additional integral of fourth degree.
Citation: Ryabov P. E.,  Bifurcation Sets in an Integrable Problem on Motion of a Rigid Body in Fluid, Regular and Chaotic Dynamics, 1999, vol. 4, no. 4, pp. 59-76
Orel O. E., Ryabov P. E.
Bifurcation sets in a problem on motion of a rigid body in fluid and in the generalization of this problem
1998, vol. 3, no. 2, pp.  82-91
In the paper, topology of energy surfaces is described and bifurcation sets is constructed for the classical Chaplygin problem and its generalization. We also describe bifurcations of Liouville tori and calculate the Fomenko invariant (for the classical case this result is obtained analytically and for the generalized case it is obtained with the help of computer modeling). Topological analysis shows that some topological characteristics (such as the form of the bifurcation set) change continuously and some of them (such as topology of energy surfaces) change drastically as $g\to0$.
Citation: Orel O. E., Ryabov P. E.,  Bifurcation sets in a problem on motion of a rigid body in fluid and in the generalization of this problem, Regular and Chaotic Dynamics, 1998, vol. 3, no. 2, pp. 82-91
Kharlamov M. P., Ryabov P. E.
The Bifurcations of the First Integrals in the Case of Kowalewski-Yehia
1997, vol. 2, no. 2, pp.  25-40
The bifurcation set in case of Kowalewski-Yehia integrability has been determined in this paper.
Citation: Kharlamov M. P., Ryabov P. E.,  The Bifurcations of the First Integrals in the Case of Kowalewski-Yehia, Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, pp. 25-40

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