Select language: En
0
2013
Impact Factor

    Pavel Ryabov

    Leningradsky prospect, 49, Moscow, 125993
    Financial University

    Publications:

    Ryabov P., Biryucheva E.
    Abstract
    The paper presents explicitly the spectral curve and the discriminant set of the integrable case of M. Adler and P. van Moerbeke. For critical points of rank 0 and 1 of the momentum map we explicitly calculate the characteristic values defining their type. An algorithm is proposed for finding the bifurcation diagram from the real part of the discriminant set with the help of critical points of rank 0 and 1. The algorithm works under the condition that the real part of the discriminant set contains the bifurcation diagram.
    Keywords: integrable Hamiltonian systems, spectral curve, discriminant set, bifurcation diagram
    Citation: Ryabov P., Biryucheva E.,  The discriminant set and bifurcation diagram of the integrable case of M. Adler and P. van Moerbeke, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp.  633–650
    DOI:10.20537/nd1604007
    Ryabov P., Savushkin A. Y.
    Phase topology of the Kowalevski–Sokolov top
    2015, Vol. 11, No. 2, pp.  287-317
    Abstract
    The phase topology of the integrable Hamiltonian system on $e(3)$ found by V. V. Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. 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 classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a oneparameter 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 surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations).
    Keywords: integrable Hamiltonian systems, relative equilibria, iso-energy surfaces, critical subsystems, bifurcation diagrams, rough topology
    Citation: Ryabov P., Savushkin A. Y.,  Phase topology of the Kowalevski–Sokolov top, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 2, pp.  287-317
    DOI:10.20537/nd1502006

    Back to the list