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

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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

Helmut Rüssmann ^1,

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

Received July 2009 Revised May 2010 Published August 2010

Abstract
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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.

Keywords: KAM theory, cut functions., artificial parameters, polynomial approximations.

Mathematics Subject Classification: Primary: 37J40; Secondary: 70H08, 70K43, 41A1.

Citation: Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete and 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), 631-640.
[2]	E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,'' McGraw-Hill Book Company, Inc., New York, 1955,
[3]	R. A. DeVore and G. G. Lorentz, "Constructive Approximation,'' Springer-Verlag, Berlin, 1993.
[4]	L. Hörmander and B. Bernhardsson, An extension of Bohr's inequality,, in, 29 ().
[5]	J. Moser, Combination tones for Duffing's equation, Comm. Pure Appl. Math., 18 (1965), 167-181. doi: 10.1002/cpa.3160180116.
[6]	J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. doi: 10.1007/BF01399536.
[7]	H. Rüssmann, On an inequality for trigonometric polynomials in several variables, in "Analysis, et cetera (Research papers published in honor of Jürgen Moser's 60th birthday)'' (eds. P. H. Rabinowitz and E. Zehnder), Academic Press, Boston, MA (1990), 545-562.
[8]	H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204. doi: 10.1070/RD2001v006n02ABEH000169.

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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), 631-640.
[2]	E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,'' McGraw-Hill Book Company, Inc., New York, 1955,
[3]	R. A. DeVore and G. G. Lorentz, "Constructive Approximation,'' Springer-Verlag, Berlin, 1993.
[4]	L. Hörmander and B. Bernhardsson, An extension of Bohr's inequality,, in, 29 ().
[5]	J. Moser, Combination tones for Duffing's equation, Comm. Pure Appl. Math., 18 (1965), 167-181. doi: 10.1002/cpa.3160180116.
[6]	J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. doi: 10.1007/BF01399536.
[7]	H. Rüssmann, On an inequality for trigonometric polynomials in several variables, in "Analysis, et cetera (Research papers published in honor of Jürgen Moser's 60th birthday)'' (eds. P. H. Rabinowitz and E. Zehnder), Academic Press, Boston, MA (1990), 545-562.
[8]	H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204. doi: 10.1070/RD2001v006n02ABEH000169.

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