Fomenko–Zieschang Invariant in the Bogoyavlenskyi Integrable Case

    2000, Volume 5, Number 4, pp.  437-457

    Author(s): Zotev D. B.

    The topology of an integrable Hamiltonian system with two degrees of freedom, occuring in dynamics of the magnetic heavy body with a fixed point [1], is explored. The equations of critical submanifolds of the supplementary integral $f$, restricted to arbitrary isoenergy surface $Q_h^3$, are obtained. In particular, all the phase trajectories of a stable periodic motion are found. It is proved, that $f$ is a Bottean integral. The bifurcation diagram, full Fomenko–Zieschang invariant and the topology of each regular isoenergy surface $Q_h^3$ are calculated, as well as the topology of phase manifold $M^4$, which has a degenerate peculiarity of the symplectic structure. This peculiarity did not appear in dynamics before. A method of the computer visualization of Liouville tori bifurcations is offering.
    Citation: Zotev D. B., Fomenko–Zieschang Invariant in the Bogoyavlenskyi Integrable Case, Regular and Chaotic Dynamics, 2000, Volume 5, Number 4, pp. 437-457

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