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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Volume 259, Pages 174–202
(Mi tm576)
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This article is cited in 13 scientific papers (total in 13 papers)
Partial Preservation of Frequencies and Floquet Exponents in KAM Theory
M. B. Sevryuk Institute of Energy Problems of Chemical Physics, Russian Academy of Sciences
Abstract:
Under a small perturbation of a completely integrable Hamiltonian system, invariant tori with Diophantine frequencies of motion are not destroyed but only slightly deformed, provided that the Hessian (with respect to the action variables) of the unperturbed Hamiltonian vanishes nowhere (the Kolmogorov nondegeneracy). The motion on every perturbed torus is quasiperiodic with the same frequencies. In this sense the frequencies of invariant tori of the unperturbed system are preserved. Recently, it has been found that the Kolmogorov nondegeneracy condition can be weakened so as to guarantee the preservation of only some subset of frequencies. Such partial preservation of frequencies can also be defined for lower dimensional invariant tori, whose dimension is less than the number of degrees of freedom. We consider a more general problem of partial preservation not only of the frequencies of invariant tori but also of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus). The results are formulated for Hamiltonian, reversible, and dissipative systems (with a complete proof for the reversible case).
Received in November 2006
Citation:
M. B. Sevryuk, “Partial Preservation of Frequencies and Floquet Exponents in KAM Theory”, Analysis and singularities. Part 2, Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 259, Nauka, MAIK «Nauka/Inteperiodika», M., 2007, 174–202; Proc. Steklov Inst. Math., 259 (2007), 167–195
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https://www.mathnet.ru/eng/tm576 https://www.mathnet.ru/eng/tm/v259/p174
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Abstract page: | 401 | Full-text PDF : | 120 | References: | 61 |
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