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2013
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# Mikhail Kharlamov

Gagarina Str., 8, 400131, Volgograd

 Kharlamov M. P., Ryabov P. E., Savushkin A. Y. Topological Atlas of the Kowalevski–Sokolov Top 2016, vol. 21, no. 1, pp.  24-65 Abstract 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 DOI:10.1134/S1560354716010032
 Kharlamov M. P. Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field 2014, vol. 19, no. 2, pp.  226-244 Abstract For the integrable system on $e(3,2)$ found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom. Keywords: generalized two-field gyrostat, critical subsystems, Appelrot classes, invariant relations, types of critical points Citation: Kharlamov M. P.,  Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 226-244 DOI:10.1134/S1560354714020063
 Kharlamov M. P. Separation of Variables in the Generalized 4th Appelrot Class 2007, vol. 12, no. 3, pp.  267-280 Abstract We consider an analogue of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields. The trajectories of this family fill a four-dimensional surface $\mathfrak{O}$ in the six-dimensional phase space. The constants of the three first integrals in involution restricted to this surface fill one of the sheets of the bifurcation diagram in $\mathbb{R}^3$. We point out a pair of partial integrals to obtain explicit parametric equations of this sheet. The induced system on $\mathfrak{O}$ is shown to be Hamiltonian with two degrees of freedom having a thin set of points where the induced symplectic structure degenerates. The region of existence of motions in terms of the integral constants is found. We provide the separation of variables on $\mathfrak{O}$ and algebraic formulae for the initial phase variables. Keywords: Kowalevski top, double field, Appelrot classes, separation of variables Citation: Kharlamov M. P.,  Separation of Variables in the Generalized 4th Appelrot Class, Regular and Chaotic Dynamics, 2007, vol. 12, no. 3, pp. 267-280 DOI:10.1134/S1560354707030021
 Kharlamov M. P. Bifurcation diagrams of the Kowalevski top in two constant fields 2005, vol. 10, no. 4, pp.  381-398 Abstract The Kowalevski top in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in ${\bf R}^3$. A correspondence to the Appelrot classes in the classical Kowalevski problem is established. The admissible regions for the values of the first integrals are found in the form of some inequalities of general character and boundary conditions for the induced diagrams on energy levels. Keywords: Kowalevski top, double field, critical set, bifurcation diagrams Citation: Kharlamov M. P.,  Bifurcation diagrams of the Kowalevski top in two constant fields , Regular and Chaotic Dynamics, 2005, vol. 10, no. 4, pp. 381-398 DOI:10.1070/RD2005v010n04ABEH000321