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

Previous Article
Finite smooth normal forms and integrability of local families of vector fields
DCDS-S Home
This Issue
Next Article
A parametrised version of Moser's modifying terms theorem

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

Abstract
Related Papers

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 & 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

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. 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

[1]	Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[2]	Guangzhou Chen, Guijian Liu, Jiaquan Wang, Ruzhong Li. Identification of water quality model parameters using artificial bee colony algorithm. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 157-165. doi: 10.3934/naco.2012.2.157
[3]	Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[4]	Olga Bernardi, Matteo Dalla Riva. Analytic dependence on parameters for Evans' approximated Weak KAM solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4625-4636. doi: 10.3934/dcds.2017199
[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 & 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 & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[7]	Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[8]	Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[9]	Hiroyuki Kobayashi, Shingo Takeuchi. Applications of generalized trigonometric functions with two parameters. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1509-1521. doi: 10.3934/cpaa.2019072
[10]	David Yang Gao, Changzhi Wu. On the triality theory for a quartic polynomial optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (1) : 229-242. doi: 10.3934/jimo.2012.8.229
[11]	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 & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749
[12]	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
[13]	Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545
[14]	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 & Optimization, 2011, 1 (3) : 487-493. doi: 10.3934/naco.2011.1.487
[15]	G. A. Swarup. On the cut point conjecture. Electronic Research Announcements, 1996, 2: 98-100.
[16]	Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135
[17]	Xiwang Cao, Hao Chen, Sihem Mesnager. Further results on semi-bent functions in polynomial form. Advances in Mathematics of Communications, 2016, 10 (4) : 725-741. doi: 10.3934/amc.2016037
[18]	Kohei Ueno. Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 985-996. doi: 10.3934/dcds.2011.31.985
[19]	Kohei Ueno. Weighted Green functions of polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2283-2305. doi: 10.3934/dcds.2014.34.2283
[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 & Continuous Dynamical Systems - A, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences