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Typical integrable Hamiltonian systems on a four-dimensional symplectic manifold. (English. Russian original) Zbl 0931.37027
Izv. Math. 62, No. 2, 261-285 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 2, 49-74 (1998).
Author’s abstract: We study the topology of integrable Hamiltonian systems with two degrees of freedom in the neighbourhood of a degenerate circle. Among all degenerate circles, the class of so-called generic degenerate circles is singled out. These circles cannot be removed from the symplectic manifold by a small perturbation of the Poisson action, and the system remains topologically equivalent to the unperturbed system in their neighbourhood. Moreover, if the unperturbed system has only Bott circles and generic degenerate circles, then, under the condition of simplicity, the perturbed system is globally topologically equivalent to it. It is proved that if an additional condition holds, then there is a small perturbation for which all degenerate circles are generic.

MSC:
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
54H20 Topological dynamics (MSC2010)
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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