American Institute of Mathematical Sciences

December  2010, 3(4): 683-718. doi: 10.3934/dcdss.2010.3.683

KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character

 1 Institut für Mathematik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany

Received  July 2009 Revised  May 2010 Published  August 2010

In this paper we present a new variant of the KAM theory, containing an artificial parameter $q$, $0 < q < 1$, which makes the steps of the KAM iteration infinitely small in the limit $q$↗$1$. This KAM procedure can be compared for $q<1$ with a Riemann sum which tends, for $q$↗$1$, to the corresponding Riemann integral. As a consequence this limit has all advantages of an integration process compared with its preliminary stages: Simplification of the conditions for the involved parameters and global linearization which therefore improves numerical results. But there is a difference from integrals: The KAM iteration itself works only for $q<1$, however, $q$ can be chosen as near to $1$ as we want and the limit $q$↗$1$ exists for all involved parameters. Hence, the mentioned advantages remain mainly preserved. The new technique of estimation differs completely from all what has appeared about KAM theory in the literature up to date. Only Kolmogorov's idea of local linearization and Moser's modifying terms are left. The basic idea is to use the polynomial structure in order to transfer, at least partially, the whole KAM procedure outside of the original domain of definition of the given dynamical system.
Citation: Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683
References:
 [1] K. I. Babenko, Best approximations to a class of analytic functions,, Izv. Akad. Nauk SSSR Ser. Mat., 22 (1958), 631.   Google Scholar [2] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,'', McGraw-Hill Book Company, (1955).   Google Scholar [3] R. A. DeVore and G. G. Lorentz, "Constructive Approximation,'', Springer-Verlag, (1993).   Google Scholar [4] L. Hörmander and B. Bernhardsson, An extension of Bohr's inequality,, in, 29 ().   Google Scholar [5] J. Moser, Combination tones for Duffing's equation,, Comm. Pure Appl. Math., 18 (1965), 167.  doi: 10.1002/cpa.3160180116.  Google Scholar [6] J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1967), 136.  doi: 10.1007/BF01399536.  Google Scholar [7] H. Rüssmann, On an inequality for trigonometric polynomials in several variables,, in, (1990), 545.   Google Scholar [8] H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems,, Regul. Chaotic Dyn., 6 (2001), 119.  doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar

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References:
 [1] K. I. Babenko, Best approximations to a class of analytic functions,, Izv. Akad. Nauk SSSR Ser. Mat., 22 (1958), 631.   Google Scholar [2] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,'', McGraw-Hill Book Company, (1955).   Google Scholar [3] R. A. DeVore and G. G. Lorentz, "Constructive Approximation,'', Springer-Verlag, (1993).   Google Scholar [4] L. Hörmander and B. Bernhardsson, An extension of Bohr's inequality,, in, 29 ().   Google Scholar [5] J. Moser, Combination tones for Duffing's equation,, Comm. Pure Appl. Math., 18 (1965), 167.  doi: 10.1002/cpa.3160180116.  Google Scholar [6] J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1967), 136.  doi: 10.1007/BF01399536.  Google Scholar [7] H. Rüssmann, On an inequality for trigonometric polynomials in several variables,, in, (1990), 545.   Google Scholar [8] H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems,, Regul. Chaotic Dyn., 6 (2001), 119.  doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar
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