# 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 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.

show all references

##### 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.
 [1] Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080 [2] Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57 [3] Olga Bernardi, Matteo Dalla Riva. Analytic dependence on parameters for Evans' approximated Weak KAM solutions. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4625-4636. doi: 10.3934/dcds.2017199 [4] Guangzhou Chen, Guijian Liu, Jiaquan Wang, Ruzhong Li. Identification of water quality model parameters using artificial bee colony algorithm. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 157-165. doi: 10.3934/naco.2012.2.157 [5] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41 [6] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75 [7] Victor Magron, Marcelo Forets, Didier Henrion. Semidefinite approximations of invariant measures for polynomial systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6745-6770. doi: 10.3934/dcdsb.2019165 [8] Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413 [9] Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069 [10] Elmehdi Amhraoui, Tawfik Masrour. Smoothing approximations for piecewise smooth functions: A probabilistic approach. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021033 [11] Hiroyuki Kobayashi, Shingo Takeuchi. Applications of generalized trigonometric functions with two parameters. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1509-1521. doi: 10.3934/cpaa.2019072 [12] David Yang Gao, Changzhi Wu. On the triality theory for a quartic polynomial optimization problem. Journal of Industrial and Management Optimization, 2012, 8 (1) : 229-242. doi: 10.3934/jimo.2012.8.229 [13] Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693 [14] Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545 [15] Xiaoli Wang, Meihua Yang, Peter E. Kloeden. Sigmoidal approximations of a delay neural lattice model with Heaviside functions. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2385-2402. doi: 10.3934/cpaa.2020104 [16] Dušan M. Stipanović, Claire J. Tomlin, George Leitmann. A note on monotone approximations of minimum and maximum functions and multi-objective problems. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 487-493. doi: 10.3934/naco.2011.1.487 [17] Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems and Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749 [18] Claude Carlet, Serge Feukoua. Three parameters of Boolean functions related to their constancy on affine spaces. Advances in Mathematics of Communications, 2020, 14 (4) : 651-676. doi: 10.3934/amc.2020036 [19] Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135 [20] Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767

2020 Impact Factor: 2.425