A renormalization group for Hamiltonians, with applications to KAM tori

Ergodic Theory and Dynamical Systems

Volumn 19, Issue 2, 1999, Pages 475-521

  Koch, Hans ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033414175     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0143385799130128     Document Type: Article

References (32)

1. V. I. Arnold. Proof of A. N. Kolmogorov's theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian. Usp. Mat. Nauk. 18(5) (1963), 13-40; Russ. Math. Surv. 18 (1963), 9-36.
2. V. I. Arnold. Proof of A. N. Kolmogorov's theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian. Usp. Mat. Nauk. 18(5) (1963), 13-40; Russ. Math. Surv. 18 (1963), 9-36.
3. M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J. P. Schreiber. Pisot and Salem Numbers. Birkhäuser, Basel, 1992.
4. C. Chandre, M. Govin and H. R. Jauslin. KAM-renormalization group analysis of stability in Hamiltonian flows. Phys. Rev. Lett. 79 (1997), 3881-3884.
5. C. Chandre, M. Govin and H. R. Jauslin. KAM-renormalization group approach to the break-up of invariant tori in Hamiltonian systems. Phys. Rev. E 57 (1998), 1536-1543.
6. L. H. Eliasson. Absolutely convergent series expansions for quasi periodic motions. Report 2-88, Department of Mathematics, University of Stockholm (1988). Math. Phys. Electron. J. 2 (1996), 33pp.
7. L. H. Eliasson. Absolutely convergent series expansions for quasi periodic motions. Report 2-88, Department of Mathematics, University of Stockholm (1988). Math. Phys. Electron. J. 2 (1996), 33pp.
8. D. F. Escande and F. Doveil. Renormalisation method for computing the threshold of the large scale stochastic instability in two degree of freedom Hamiltonian systems. J. Stat. Phys. 26 (1981), 257-284.
9. M. J. Feigenbaum, L. P. Kadanoff and S. J. Shenker. Quasi-periodicity in dissipative systems: a renormalization group analysis. Physica 5D (1982), 370-386.
10. J. Feldman and E. Trubowitz. Renormalization in classical mechanics and many body quantum field theory. J. d'Analyse Math. 58 (1992), 213-247.
11. G. Gallavotti. Perturbation theory. In Mathematical Physics Towards the XXI Century. Eds. R. Sen and A Gersten. Ben Gurion University Press, Ber Sheva, 1994, pp. 275-294.
12. G. Gallavotti, G. Gentile and V. Mastropietro. Field theory and KAM tori. Math. Phys. Electron. J. 1(5) (1995), 13pp.
13. G. Gentile and V. Mastropietro. Tree expansion and multiscale decomposition for KAM tori. Nonlinearity 8 (1995), 1-20.
14. E. Hille and R. S. Phillips. Functional analysis and semi-groups. Amer. Math. Soc. Colloquium Publications 31 (1957).
15. M. W. Hirsch, C. C. Pugh and M. Shub. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.
16. K. Khanin and Ya. G. Sinai. The renormalization group method and KAM theory. In Nonlinear Phenomena in Plasma Physics and Hydrodynamics. Ed. R. Z. Sagdeev. Mir, 1986, pp. 93-118.
17. K. Khanin and Ya. G. Sinai. Renormalization group methods in the theory of dynamical systems. Int. J. Mod. Phys. B2 (1988), 147-165.
18. H. Koch. Unpublished Numerical Results, 1997.
19. A. N. Kolmogorov. On conservation of conditionally periodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR 98 (1954), 527-530.
20. D. Kosygin. Multidimensional KAM theory from the renormalization group viewpoint. In Dynamical Systems and Statistical Mechanics (Adv. Sov. Math., 3). Ed. Ya. G. Sinai. American Mathematical Society, 1991, pp. 99-129.
21. R. de la Llave. Introduction to KAM theory. Preprint, University of Texas, mp_arc 93-98, 1993.
22. R. S. MacKay. Renormalisation in area preserving maps. Thesis, Princeton, 1982. Also Renormalisation in Area Preserving Maps. World Scientific, London, 1993.
23. R. S. MacKay. Renormalisation in area preserving maps. Thesis, Princeton, 1982. Also Renormalisation in Area Preserving Maps. World Scientific, London, 1993.
24. R. S. MacKay. Three topics in Hamiltonian dynamics. Preprint, University of Warwick, 1994. Also in Dynamical Systems and Chaos, Vol. 2. Eds. Y. Aizawa, S. Saito and K. Shiraiwa. World Scientific, London, 1995.
25. R. S. MacKay. Three topics in Hamiltonian dynamics. Preprint, University of Warwick, 1994. Also in Dynamical Systems and Chaos, Vol. 2. Eds. Y. Aizawa, S. Saito and K. Shiraiwa. World Scientific, London, 1995.
26. A. Mehr and D. F. Escande. Destruction of KAM tori in Hamiltonian systems: link with the destabilization of nearby cycles and calculation of residues. Physica 13D (1984), 302-338.
27. J. Moser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Gött., II. Math. Phys. Kl (1962), 1-20.
28. S. Ostlund, D. A. Rand, J. Sethna and E. Siggia. Universal properties of the transition from quasiperiodicity to chaos in dissipative systems. Physica 8D (1983), 303-342.
29. D. A. Rand. Existence, non-existence and universal breakup of dissipative golden invariant tori. Nonlinearity 5 (1992), 639-706.
30. A. Stirnemann. Renormalization for golden circles. Commun. Math. Phys. 152 (1993), 369-431.
31. S. Tompaidis. Numerical study of invariant sets of a volume-preserving map. Preprint, University de Rennes, mp_arc 95-441, 1995.
32. J. J. Abad, H. Koch and P. Wittwer. A renormalization group for Hamiltonians: numerical results. Nonlinearity 11 (1998), 1185-1194.