A renormalization group fixed point associated with the breakup of golden invariant tori

Discrete and Continuous Dynamical Systems

Volumn 11, Issue 4, 2004, Pages 881-909

  Koch, Hans ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Breakup of invariant tori; Computer assisted proof; Hamiltonian flows

Indexed keywords

EID: 13244256997     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2004.11.881     Document Type: Article

References (37)

1. 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.
2. S.J. Shenker and L.P. Kadanoff, Critical Behavior of a KAM Surface. I. Empirical Results, J. Stat. Phys. 27 (1982), 631-656.
3. R.S. MacKay, Renormalisation in Area Preserving Maps, Thesis, Princeton (1982). World Scientific, London (1993).
4. R.S. MacKay, Renormalisation Approach to Invariant Circles in Area-Preserving Maps, Physica D 7 (1983), 283-300.
5. 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 D 13 (1984), 302-338.
6. K. Khanin, Ya.G. Sinai, The Renormalization Group Method and KAM Theory. In "Nonlinear Phenomena in Plasma Physics and Hydrodynamics", R.Z. Sagdeev (ed), Mir (1986), 93-118.
7. J.M. Greene, H. Johannesson, B. Schaub and H. Suhl, Scaling Anomaly at the Critical Transition of an Incommensurate Structure, Phys. Rev. A 36 (1987), 5858-5861.
8. J. Wilbrink, New Fixed Point of the Renormalization Operator for Invariant Circles, Phys. Lett. A 131 (1988), 251-255.
9. K. Khanin, Ya.G. Sinai, Renormalization Group Methods in the Theory of Dynamical Systems. Int. J. Mod. Phys. B 2 (1988), 147-165.
10. J.A. Ketoja and R.S. MacKay, Fractal Boundary for the Existence of Invariant Circles for Area-Preserving Maps: Observations and Renormalisation Explanation, Physica D 35 (1989), 318-334.
11. J. M. Mao and R. H. G. Helleman, Existence of KAM Tori for Four-Dimensional Volume Preserving Maps. Nuovo Cimento B 104 (1989), 177-182.
12. J.M. Greene and J. Mao, Higher-Order Fixed Points of the Renormalization Operator for Invariant Circles, Nonlinearity 3 (1990), 69-78.
13. J. Wilbrink, New Fixed Point of the Renormalisation Operator Associated with the Recurrence of Invariant Circles in Generic Hamiltonian Maps, Nonlinearity 3 (1990), 567-584.
14. R. Artuso, G. Casati, D.L. Shepelyansky, Breakdown of Universality in Renormalization Dynamics for Critical Invariant Torus. Europhys. Lett. 15 (1991), 381-386.
15. R.S. MacKay, J.D. Meiss, and J. Stark, An Approximate Renormalization for the Break-up of Invariant Tori with Three Frequencies, Phys. Lett. A 100 (1994), 417-424.
16. R.S. MacKay, Three Topics in Hamiltonian Dynamics. Preprint U. Warwick (1994). Also in: "Dynamical Systems and Chaos", Vol.2, Y. Aizawa, S. Saito, K. Shiraiwa (eds), World Scientific, London (1995).
17. R.S. MacKay, Three Topics in Hamiltonian Dynamics. Preprint U. Warwick (1994). Also in: "Dynamical Systems and Chaos", Vol.2, Y. Aizawa, S. Saito, K. Shiraiwa (eds), World Scientific, London (1995).
18. J. del Castillo-Negrete, J. M. Greene, P. Morrison, Area Preserving Non-Twist Maps: Periodic Orbits and Transition to Chaos. Physica D 91 (1996), 1-23.
19. A. Stirnemann, Towards an Existence Proof of MacKay's Fixed Point, Comm. Math. Phys. 188 (1997), 723-735.
20. M. Govin, C. Chandre, H.R. Jauslin, KAM-Renormalization Group Approach to the Breakup of Invariant Tori in Hamiltonian Systems. Phys. Rev. E 57, 1536-1543 (1998).
21. C. Chandre, M. Govin, H.R. Jauslin, and H. Koch, Universality for the Breakup of Invariant Tori in Hamiltonian Flows, Phys. Rev. E 57 (1998), 6612-6617.
22. J.J. Abad, H. Koch, and P. Wittwer, A Renormalization Group for Hamiltonians: Numerical Results, Nonlinearity 11 (1998), 1185-1194.
23. H. Koch, A Renormalization Group for Hamiltonians, with Applications to KAM Tori, Erg. Theor. Dyn. Syst. 19 (1999), 1-47.
24. C. Chandre, H. R. Jauslin, G. Benfatto, A. Celletti, An Approximate Renormalization-Group Transformation for Hamiltonian Systems with Three Degrees of Freedom, Phys. Rev. E 60 (1999), 5412-5421
25. C. Chandre and R.S. MacKay, Approximate Renormalization with Codimension-One Fixed Point for The Break-up of some Three-Frequency Tori, Phys. Lett. A 275 (2000), 394-400.
26. J.J. Abad and H. Koch, Renormalization and Periodic Orbits for Hamiltonian Flows, Commun. Math. Phys. 212 (2000), 371-394.
27. J. Lopes Dias, Renormalisation of Flows on the Multidimensional Torus Close to a KT Frequency Vector, Nonlinearity 15 (2002), 647-664.
28. J. Lopes Dias, Renormalisation Scheme for Vector Fields on T with a Diophantine Frequency, Nonlinearity 15 (2002), 665-679.
29. C. Chandre and H.R. Jauslin, Renormalization-Group Analysis for the Transition to Chaos in Hamiltonian Systems, Physics Reports 365 (2002), 1-64.
30. C. Chandre, Renormalization for Cubic Frequency Invariant Tori in Hamiltonian Systems with Two Degrees of Freedom, Discrete Contin. Dynam. Systems B 2 (2002), 457-465.
31. H. Koch, On the Renormalization of Hamiltonian Flows, and Critical Invariant Tori, Discrete Contin. Dynam. Systems A, 8, 633-646 (2002).
32. The preprint mp_arc 02-175 of this paper includes the source code for the programs used in our proof of Theorem 5.2. See also ftp://ftp.ma.utexas.edu/ pub/papers/koch/kam-fixpt/.
33. H. Koch, A. Schenkel and P. Wittwer, Computer-Assisted Proofs in Analysis and Programming in Logic: A Case Study, SIAM Rev. 38 (1996), 565-604.
34. S.T. Taft and R.A. Duff (eds), Ada 95 Reference Manual: Language and Standard Libraries, International Standard ISO/IEC 8652:1995(E), Lecture Notes in Computer Science 1246 (1997). See also http://www.adahome.com/rm95/.
35. Ada Core Technologies, 73 Fifth Avenue, New York, NY 10003, USA. See also ftp://cs.nyu.edu/pub/gnat.
36. Intel Corp., Intel Architecture Software Developer's Manual, Volume 1: Basic Architecture, http://developer.intel.com/design/pentiumii/manuals/243190. htm.
37. The Institute of Electrical and Electronics Engineers, Inc., IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std 754-1985.