Bulletin of the American Mathematical Society

Author(s): Henk W. Broer
Journal: Bull. Amer. Math. Soc. 41 (2004), 507-521.
MSC (2000): Primary 37C55, 37C70, 37A60, 34C15
Posted: February 9, 2004
Retrieve article in: PDF DVI PostScript

Abstract: Kolmogorov-Arnold-Moser (or KAM) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov's pioneering work.

1.: V.I. Arnold, Proof of a theorem by A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Usp. Math. Nauk 18(5) (1963) 13-40; Russ. Math. Surv. 18(5) (1963) 9-36. MR 29:328
2.: V.I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys 18(6) (1963) 85-191. [Corrigenda (in Russian): Uspekhi Mat. Nauk 23 (1968) 216.] MR 30:943
3.: V.I. Arnold, Instability of dynamical systems with many degrees of freedom, Soviet Mathematics 5(3) (1964) 581-585. MR 29:329
4.: V.I. Arnold, Small divisors I: On mappings of the circle onto itself, Amer. Math. Soc. Transl., Ser.2 46 (1965) 213-284. [Russian original: Izvest. Akad. Nauk SSSR, Ser Mat. 25(1) (1961) 21-86; Corrigenda: ibid. 28(2) (1964) 479-480.] MR 25:4113
5.: V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. Springer Verlag 1988. [Russian original: Nauka 1978.] MR 89h:58049
6.: V.I. Arnold, Mathematical Methods in Classical Mechanics. Springer Verlag 1978, 2nd edition 1989. [Russian original: Nauka 1974.] MR 57:14033b, MR 90c:58046
7.: V.I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics. Addison-Wesley 1989. [French original: Gauthier-Villars 1968.] MR 38:1233
8.: J. Bricmont, Science of Chaos or Chaos in Science? In P.R. Gross, N. Levitt and M.W. Lewis (eds.), The Flight from Science and Reason, Ann. of the New York Academy of Sciences 775 131-175, New York Academy of Sciences, New York 1996. [Also appeared in Physicalia Magazine 17 (1995) 159-208.] MR 99h:00012
9.: J. Bricmont, K. Gawedzki and A. Kupiainen, KAM Theorem and Quantum Field Theory, Comm. Math. Phys. 201 (1999) 699-727. MR 2000d:37070
10.: P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11(1) (1984) 85-141. MR 85h:58001
11.: H.W. Broer, A.N. Kolmogorov: la `K' de KAM, Bull. de la Societat Catalana de Matemàtiques (2004), to appear.
12.: H.W. Broer, H. Hanßmann, À. Jorba, J. Villanueva and F.O.O. Wagener, Normal-internal resonances in quasiperiodically forced oscillators: a conservative approach, Nonlinearity 16 (2003) 1751-1791.
13.: H.W. Broer, H. Hanßmann and J. You, Bifurcations of normally parabolic tori in Hamiltonian systems, Preprint University of Groningen. Submitted for publication.
14.: H.W. Broer, H. Hanßmann and J. You, Umbilical torus bifurcations in Hamiltonian systems, Preprint University of Groningen. Submitted for publication.
15.: H.W. Broer and G.B. Huitema, A proof of the iso-energetic KAM-theorem from the `ordinary' one, Journ. Diff. Eqns. 90(1) (1991) 52-60. MR 92a:58117
16.: H.W. Broer and G.B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, Journ. Dynamics and Differential Equations 7(1) (1995) 191-212. MR 96b:58099
17.: H.W. Broer, G.B. Huitema and M.B. Sevryuk, Quasi-periodic motions in families of dynamical systems: order amidst chaos, LNM 1645. Springer Verlag 1996. MR 99d:58142
18.: H.W. Broer, G.B. Huitema, F. Takens and B.L.J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. AMS 83(421) (1990) 1-170. MR 91e:58156
19.: H.W. Broer, J. Puig and C. Simó, Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation, Commun. Math. Phys. 641 (2003) 467-503.
20.: A.D. Bruno, Convergence of transformations of differential equations to the normal forms, Dokl. Akad. Nauk. SSSR. 165 (1965) 987-989. MR 33:325
21.: L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. Henri Poincaré, Physique Théorique 60(1) (1994) 1-144. MR 95b:58056
22.: M.C. Ciocci, Bifurcation of periodic orbits and persistence of quasi periodic solutions in families of reversible systems. Ph.D. Thesis, Gent 2003.
23.: M.C. Ciocci, A. Litvak-Hinenzon and H.W. Broer, Survey on dissipative KAM theory including quasi-periodic bifurcations, based on lectures by Henk Broer. To appear in J. Montaldi and T. Ratiu (eds.): Peyresq Lectures on Geometric Mechanics and Symmetry, LMS Lecture Notes Series 306. Cambridge University Press.
24.: H. Cremer, Zum Zentrumproblem, Math. Ann. 98 (1927) 151-163.
25.: R.L. Devaney, An Introduction to Chaotic Dynamical Systems. Addison Wesley 1989. MR 91a:58114
26.: F. Diacu and P. Holmes, Celestial Encounters, the origins of chaos and stability. Princeton University Press 1996. MR 98e:70006
27.: E.I. Dinaburg and Y.G. Sinai, The one-dimensional Schrödinger equation with a quasiperiodic potential. Func. Anal. and Appl. 9 (1975) 179-189. MR 57:10076
28.: L.H. Eliasson, S.B. Kuksin, S. Marmi and J.-C. Yoccoz, Dynamical Systems and Small Divisors, Cetraro 1998, LNM 1784. Springer Verlag 2002. MR 2003c:37001
29.: J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson model for large disorder or low energy, Comm. Math. Phys. 88 (1983) 151-189. MR 85c:82004
30.: G. Gallavotti, Statistical Mechanics, A Short Treatise. Springer Verlag 1999. MR 2001j:82001
31.: G. Gallavotti, F. Bonetto and G. Gentile, Aspects of the ergodic, qualitative and statistical theory of motion. To appear Springer Verlag. Preliminary version at http://ipparco.roma1.infn.it/libri.html.
32.: G. Gallavotti, G. Gentile and V. Mastropietro, Field theory and KAM tori, MPEJ 1 (1995) paper 5. MR 97b:81072
33.: H. Hanßmann, The quasi-periodic centre-saddle bifurcation, Journ. Diff. Eqns. 142(2) (1998) 305-370. MR 98m:58099
34.: M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES 49 (1979) 5-234. MR 81h:58039
35.: E. Hopf, A mathematical example displaying features of turbulence. Commun. Appl. Math. 1 (1948) 303-322. MR 10:716a
36.: H.H. de Jong, Quasiperiodic breathers in systems of weakly coupled pendulums: Applications of KAM theory to classical and statistical mechanics. Ph.D. Thesis, Groningen 1999.
37.: A. Jorba and J. Villanueva, On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems, Nonlinearity 10 (1997) 783-822. MR 98h:58169
38.: T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge 45. Springer Verlag 2003.
39.: A.N. Kolmogorov, On the conservation of conditionally periodic motions for a small change in Hamilton's function [in Russian], Dokl. Akad. Nauk SSSR 98 (1954) 525-530; English translation in LNP 93 (1979) 51-56. MR 16:924c
40.: A.N. Kolmogorov, The general theory of dynamical systems and classical mechanics. Proceedings of the International Congress of Mathematicians (Amsterdam, 1954), Vol. 1, pages 315-333, North Holland, Amsterdam, 1957 [in Russian]. [Reprinted in: International Mathematical Congress in Amsterdam, 1954 (Plenary Lectures), pages 187-208. Fizmatgiz, Moscow 1961; English translation as Appendix D in R.H. Abraham, Foundations of Mechanics, pages 263-279. Benjamin, 1967. Reprinted as Appendix in R.H. Abraham and J.E. Marsden, Foundations of Mechanics, Second Edition, pages 741-757. Benjamin / Cummings 1978.] MR 20:4066
41.: S.B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, LNM 1556 Springer Verlag 1993. MR 95k:58145
42.: S.B. Kuksin, V.F. Lazutkin and J. Pöschel (eds.), Seminar on Dynamical Systems (St. Petersburg, 1991) Birkhäuser 1994. MR 94m:58002
43.: L.D. Landau, On the problem of turbulence, Akad. Nauk. 44 (1944) 311-314. MR 6:246c
44.: L.D. Landau and E.M. Lifschitz, Fluid Mechanics. Pergamon, Oxford 1959. MR 21:6839
45.: J. Laskar, Large scale chaos and marginal stability in the solar system, Proceedings XIIth Int. Congress of Math. Phys. (Paris 1994), Int. Press Cambridge 1995. MR 96j:70015
46.: R. de la Llave, A tutorial on KAM Theory. In A. Katok et al., eds., Proceedings of Symposia in Pure Mathematics 69, Amer. Math. Soc. (2001), 175-292. MR 2002h:37123
47.: R.S. MacKay and S. Aubry, Proof of existence for breathers for time reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7(6) (1994) 1623-1643. MR 96a:34085
48.: S. Marmi, An introduction to small divisor problems. Dipartimento di Matematica dell'Università di Pisa 2000.
49.: J.K. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II. 1 (1962) 1-20. MR 26:5255
50.: J.K. Moser, On the theory of quasiperiodic motions, SIAM Review 8(2) (1966) 145-172. MR 34:3013
51.: J.K. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967) 136-176. MR 34:7888
52.: J.K. Moser, Lectures on Hamiltonian systems, Mem. AMS 81 (1968) 1-60. MR 37:6060
53.: J.K. Moser, Stable and random motions in dynamical systems, with special emphasis to celestial mechanics, Ann. Math. Studies 77. Princeton University Press 1973. MR 56:1355
54.: N.N. Nekhoroshev, Exponential estimate on the stability time of near integrable Hamiltonian systems, Russ. Math. Surveys 32(6) (1977) 1-65. MR 58:18570
55.: S.E. Newhouse, D. Ruelle and F. Takens, Occurrence of strange Axiom A attractors near quasi-periodic flows on $\mathbb{T}^m,$ $m\le 3,$ Commun. Math. Phys. 64 (1978) 35-40. MR 80f:58029
56.: J. Oxtoby, Measure and Category. Springer Verlag 1971. MR 81j:28003
57.: J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35(5) (1982) 653-696. MR 84d:58039
58.: J. Pöschel, On elliptic lower dimensional tori in Hamiltonian systems, Math. Z. 202 (1989) 559-608. MR 91a:58065
59.: J. Pöschel. A lecture on the classical KAM theorem. In A. Katok et al., eds., Proceedings of Symposia in Pure Mathematics 69. Amer. Math. Soc. (2001), 707-732. MR 2002g:37081
60.: J. Puig, Cantor Spectrum for the Almost Mathieu Operator. Corollaries of localization, reducibility and duality. Preprint archive mp_arc 03-396 (2003).
61.: D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press 1989. MR 90f:58048
62.: D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys. 20 (1971) 167-192; 23 (1971) 343-4. MR 44:1297
63.: H. Rüssmann, Konvergente Reihenentwicklungen in der Störungstheorie der Himmelsmechanik. In K. Jacobs (ed.), Selecta Mathematica V, Heidelberger Taschenbücher 201 (1979) 93-260, Springer Verlag. MR 82j:70011
64.: M.B. Sevryuk, New results in the reversible KAM theory. In: S.B. Kuksin, V.F. Lazutkin and J. Pöschel (eds.), Seminar on Dynamical Systems (St. Petersburg, 1991) 184-199, Birkhäuser 1994. MR 96d:58120
65.: C.L. Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942) 607-612. MR 4:76c
66.: H. Whitney, Differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36(2) (1934) 369-387.
67.: J.-C. Yoccoz, -conjugaisons des difféomorphismes du cercle. In: J. Palis (ed), Geometric Dynamics, Proceedings, Rio de Janeiro 1981, LNM 1007 (1983) 814-827. MR 85h:58102
68.: J.-C. Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque 231 (1995) 3-88. MR 96m:58214
69.: J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms. In: S. Marmi, L.H. Eliasson, S.B. Kuksin and J.-C. Yoccoz (eds.), Dynamical Systems and Small Divisors, LNM 1784 (2002) 125-174, Springer Verlag.
70.: E. Zehnder, An implicit function theorem for small divisor problems, Bull. Amer. Math. Soc. 80(1) (1974) 174-179. MR 49:4019
71.: E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems, I and II, Comm. Pure Appl. Math. 28(1) (1975) 91-140; 29(1) (1976) 49-111. MR 54:14001

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 37C55, 37C70, 37A60, 34C15

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS