Fast numerical computation of quasi-periodic equilibrium states in 1D statistical mechanics, including twist maps

Nonlinearity

Volumn 22, Issue 6, 2009, Pages 1311-1336

  Calleja, Renato ^a   De La Llave, Rafael ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67650714419     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/22/6/004     Document Type: Article

References (54)

1. Anderson C and Dahleh M D 1996 Rapid computation of the discrete Fourier transform SIAM J. Sci. Comput. 17 913-9 (Pubitemid 126621184)
2. Allgower E L and Georg K 2003 Introduction to Numerical Continuation Methods (Classics in Applied Mathematics vol 45) vol 45 (Philadelphia: SIAM) (Berlin: Springer) (Reprint of the 1990 edition)
3. Abad J J, Koch H and Wittwer P 1998 A renormalization group for Hamiltonians: numerical results Nonlinearity 11 1185-94 (Pubitemid 128126175)
4. Aubry S and Le Daeron P Y 1983 The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states Physics D 8 381-422
5. Amit D J 1978 Field Theory, the Renormalization Group, and Critical Phenomena (International Series in Pure and Applied Physics) (New York: McGraw-Hill)
6. Angenent S B 1990 Monotone recurrence relations, their Birkhoff orbits and topological entropy Ergodic Theory Dyn. Syst. 10 15-41
7. Abramowitz M and Stegun I A 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards Applied Mathematics Series vol 55) (Washington, DC: US Government Printing Office) For sale by the Superintendent of Documents
8. Blügel S and Bihlmayer G 2007 Magnetism of low dimensional systems: theory Handbook of Magnetism and Advanced Magnetic Materials: Vol I Fundamentals and Theory ed H Kronmüller and S S P Parkin (Chichester: Wiley)
9. Berretti A, Celletti A, Chierchia L and Falcolini C 1992 Natural boundaries for area-preserving twist maps J. Stat. Phys. 66 1613-30
10. Braun O M and Kivshar Y S 2004 The Frenkel-Kontorova model: Concepts, Methods, and Applications (Berlin: Springer)
11. Bates Peter W 2006 On some nonlocal evolution equations arising in materials science Nonlinear dynamics and evolution equations (Field Inst. Commun. vol 48) (Providence, RI: AMS) pp 13-52
12. Celletti A and Chierchia L 1998 Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics Regul. Chaotic Dyn. 3 107-21 (J. Moser at 70) (in Russian)
13. Celletti A and Chierchia L 2007 KAM stability and celestial mechanics Mem. Am. Math. Soc. 187 viii+134
14. Cardaliaguet P, Da Lio F, Forcadel N and Monneau R 2007 Dislocation dynamics: a non-local moving boundary Free Boundary Problems (International Series of Numerical Mathematics vol 154) (Basel: Birkhäuser) pp 125-35
15. Candel A and de la Llave R 1998 On the Aubry-Mather theory in statistical mechanics Commun. Math. Phys. 192 649-69 (Pubitemid 128359710)
16. Calleja R and de la Llave R 2008 A criterion for the breakdown of quasi-periodic solutions Preprint mp-arc 09-56
17. de la Llave R 1992 A renormalization group explanation of numerical observations of analyticity domains J. Stat. Phys. 66 1631-4
18. de la Llave R 2001 A tutorial on KAM theory Smooth Ergodic Theory and its Applications Proc. Symp. Pure Mathematics (Seattle, WA, 1999) vol 69 (Providence, RI: American Mathematical Society) pp 175-292
19. de la Llave R 2008 KAM theory for equilibrium states in 1D statistical mechanics models Ann. Henri Poincare 9 835-80
20. de la Llave R and Petrov N P 2002 Regularity of conjugacies between critical circle maps: an experimental study Exp. Math. 11 219-41
21. de la Llave R and Rana D 1991 Accurate strategies for K.A.M. bounds and their implementation Computer Aided Proofs in Analysis (Cincinnati, OH, 1989) (IMA Volumes in Mathematics and its Application vol 28) (New York: Springer) pp 127-46
22. de la Llave R and Tompaidis S 1994 Computation of domains of analyticity for some perturbative expansions of mechanics Physics D 71 55-81
23. de la Llave R and Tompaidis S 1995 On the singularity structure of invariant curves of symplectic mappings Chaos 5 227-37
24. de la Llave R and Valdinoci E 2007 Critical points inside the gaps of ground state laminations for some models in statistical mechanics J. Stat. Phys. 129 81-119
25. de la Llave R and Valdinoci E 2007 Ground states and critical points for generalized Frenkel-Kontorova models in Nonlinearity 20 2409-24
26. Dutt A and Rokhlin V 1993 Fast Fourier transforms for nonequispaced data SIAM J. Sci. Comput. 14 1368-93
27. Dyson F J 1971 An Ising ferromagnet with discontinuous long-range order Commun. Math. Phys. 21 269-83
28. Frenkel J and Kontorova T 1939 On the theory of plastic deformation and twinning Acad. Sci. USSR J. Phys. 1 137-49
29. Greene J M 1979 A method for determining a stochastic transition J. Math. Phys. 20 1183-201
30. Golub G H and Van Loan C F 1996 Matrix Computations (Johns Hopkins Studies in the Mathematical Sciences) 3rd edn (Baltimore, MD: Johns Hopkins University Press)
31. Hald O H 1975 On a Newton-Moser type method Numer. Math. 23 411-26
32. Huguet G, de la Llave R and Sire Y 2008 Fast numerical algorithms for the computation of invariant tori in hamiltonian systems Preprint mp-arc 09-2
33. Herman M-R 1986 Sur les courbes invariantes par les diffé omorphismes de l'anneau Astérisque 2 248 (With a correction to: On the curves invariant under diffeomorphisms of the annulus, Vol 1 (French) [Astérisque No 103-104, Soc. Math. France, Paris, 1983; MR0728564 (85m:58062)])
34. Kadanoff L P 2000 Statics, dynamics and renormalization Statistical Physics (River Edge, NJ: World Scientific)
35. Kook H and Meiss J D 1989 Periodic orbits for reversible, symplectic mappings Physics D 35 65-86
36. Knuth D E 1981 The art of computer programming Seminumerical Algorithms (Series in Computer Science and Information Processing) vol 2 2nd edn (Reading, MA: Addison-Wesley)
37. Koch H 1999 A renormalization group for Hamiltonians, with applications to KAM tori Ergodic Theory Dyn. Syst. 19 475-521
38. Koch H 2004 A renormalization group fixed point associated with the breakup of golden invariant tori Discrete Contin. Dyn. Syst. 11 881-909 (Pubitemid 40195131)
39. MacKay R S 1982 Renormalisation in area preserving maps PhD Thesis Princeton University
40. MacKay R S 1983 A renormalisation approach to invariant circles in area-preserving maps Physics D 7 283-300 (Order inchaos Los Alamos, NM 1982)
41. Mather J N 1982 Existence of quasiperiodic orbits for twist homeomorphisms of the annulus Topology 21 457-67
42. Mather J 1986 A criterion for the nonexistence of invariant circles Inst. Hautes Itudes Sci. Publ. Math. 63 153-204
43. Mattis D C 2006 The Theory of Magnetism Made Simple (Hackensack, NJ: World Scientific)
44. Meiss J D 1992 Symplectic maps, variational principles, and transport Rev. Mod. Phys. 64 795-848
45. Mather J N and Forni G 1994 Action minimizing orbits in Hamiltonian systems Transition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991) (Lecture Notes in Mathematics vol 1589) (Berlin: Springer) pp 92-186
46. Percival I C 1974 Variational principles for the invariant toroids of classical dynamics J. Phys. A: Math. Gen. 7 794-802
47. Percival I C 1979 A variational principle for invariant tori of fixed frequency J. Phys. A: Math. Gen. 12 L57-60
48. Rana D 1987 Proof of accurate upper and lower bounds to stability domains in small denominator problems PhD Thesis Princeton University
49. Rado G T and Shul H (ed) 1963 Magnetism vol IIB (New York: Academic)
50. Ruelle D 1999 Statistical Mechanics (River Edge, NJ: World Scientific) (Rigorous results, Reprint of the 1989 edition)
51. Rüssmann H 1975 On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) (Lecture Notes in Physics vol 38) (Berlin: Springer) pp 598-624
52. Saad Y 2003 Iterative methods for sparse linear systems 2nd edn (Philadelphia, PA: SIAM)
53. Shraiman B I 1984 Transition from quasiperiodicity to chaos: a perturbative renormalization-group approach Phys. Rev. A 29 3464-6
54. Veselov A P 1991 Integrable mappings Usp. Mat. Nauk 46 3-45