Dynamics of the universal area-preserving map associated with period doubling: Hyperbolic sets

Nonlinearity

Volumn 22, Issue 10, 2009, Pages 2487-2520

  Gaidashev, Denis ^a   Johnson, Tomas ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350676541     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/22/10/010     Document Type: Article

References (54)

1. Abad J J and Koch H 2000 Renormalization and periodic orbits for Hamiltonian flows Commun. Math. Phys. 212 371-94
2. Abad J J, Koch H and Wittwer P 1998 A renormalization group for Hamiltonians: numerical results Nonlinearity 11 1185-94
3. Benettin G et al 1980 Universal properties in conservative dynamical systems Lett. al Nuovo Cimento 28 1-4
4. Benettin G et al 1980 Properties in conservative dynamical systems Lett. Nuovo Cimento 28 1-4
5. Bountis T 1981 Period doubling bifurcations and universality in conservative Systems Physica D 3 577-89
6. CAPD-Computer Assisted Proofs in Dynamics group, a C++ package for rigorous numerics, http://capd.wsb-nlu.edu.pl
7. de Carvalho A, Lyubich M and Martens M 2005 Renormalization in the Hénon family: I. Universality but non-rigidity J. Stat. Phys. 121 611-69
8. Collet P, Eckmann J-P and Koch H 1980 Period doubling bifurcations for families of maps on J. Stat. Phys. D 3
9. Collet P, Eckmann J-P and Koch H 1981 On universality for area-preserving maps of the plane Physica D 3 457-67
10. Derrida B and Pomeau Y 1980 Feigenbaum's ratios of two dimensional area preserving maps Phys. Lett. A 80 217-9
11. Duarte P 2000 Persistent homoclinic tangencies for conservative maps near identity Ergodic Theory Dynam. Syst. 20 393-438
12. Eckmann J-P, Koch H and Wittwer P 1982 Existence of a fixed point of the doubling transformation for area-preserving maps of the plane Phys. Rev. A 26 720-2
13. Eckmann J-P, Koch H and Wittwer P 1984 A computer-assisted proof of universality for area-preserving maps Mem. Am. Math. Soc. 47 1-121
14. Epstein H 1986 New proofs of the existence of the Feigenbaum functions Commun. Math. Phys. 106 395-426
15. Epstein H 1989 Fixed points of composition operators II Nonlinearity 2 305-10
16. Escande D F and Doveil F 1981 Renormalization method for computing the threshold of the large scale stochastic instability in two degree of freedom Hamiltonian systems J. Stat. Phys. 26 257-84
17. de Faria E 1992 Proof of universality for critical circle mappings Thesis CUNY
18. de Faria E 1999 Asymptotic rigidity of scaling ratios for critical circle mappings Ergodic Theo. Dynam. Syst. 19 995-1035
19. Feigenbaum M J 1978 Quantitative universality for a class of nonlinear transformations J. Stat. Phys. 19 25-52
20. Feigenbaum M J 1979 Universal metric properties of non-linear transformations J. Stat. Phys. 21 669-706
21. Gaidashev D 2005 Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori Discrete Contin. Dyn. Syst. 13 63-102
22. Gaidashev D and Koch H 2004 Renormalization and shearless invariant tori: numerical results Nonlinearity 17 1713-22
23. Gaidashev D and Koch H 2008 Period doubling in area-preserving maps: an associated one-dimensional problem arXiv:mathDS/0811.2588
24. Gaidashev D and Yampolsky M 2007 Cylinder renormalization of Siegel disks Exp. Math. 16 215-26
25. Gaidashev D and Johnson T Dynamics of the universal area-preserving map associated with period doubling: stable sets arXiv:math.DS/0905.4527
26. Galias Z and Zgliczynski P 2001 Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map Nonlinearity 14 909-32
27. Gorodetsky A and Kaloshin V 2008 Conservative homoclinic bifurcations and some applications Steklov Institute Proceedings to be published
28. Helleman R H G 1980 Self-generated chaotic behavior in nonlinear mechanics Fundamental problems in statistical mechanics ed E G D Cohen (Amsterdam: North-Holland) p 165
29. Katok A and Hasselblat B 1995 Introduction to the Modern Theory of Dynamical Systems (Cambridge: Cambridge University Press)
30. Khanin K, Lopes Dias J and Marklof J 2007 Multidimensional continued fractions, dynamic renormalization and KAM theory Commun. Math. Phys. 270 197-231
31. Koch H 2002 On the renormalization of Hamiltonian flows, and critical invariant tori Discrete Contin. Dyn. Syst. 8 633-46
32. Koch H 2004a A renormalization group fixed point associated with the breakup of golden invariant tori Discrete Contin. Dyn. Syst. 11 881-909
33. Koch H 2008 Existence of critical invariant tori Ergod. Theory Dynam. Syst. 28 1879-94
34. Kocić S 2005 Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori Nonlinearity 18 2513-44
35. Kokubu H, Wilczak D and Zgliczyński P 2007 Rigorous verification of cocoon bifurcations in the Michelson system Nonlinearity 20 2147-74
36. Lyubich M 1999 Feigenbaum-Coullet-Tresser universality and Milnor's hairness conjecture Ann. Math. 149 319-420
37. Lyubich M and Martnes M 2008 Renormalization in the Hénon family: II. Homoclinic tangle arXiv:mathDS/0804.0780
38. Lyubich M and Yampolsky M 1997 Dynamics of quadratic polynomials: complex bounds for real maps Ann. Inst. Fourier 47 1219-55
39. MacKay R S 1983 Renormalization approach to invariant circles in area-preserving maps Physica D 7 283-300
40. MacKay R S 1982 Renormalisation in area preserving maps Thesis Princeton (London: World Scientific)
41. Mehr A and Escande D F 1984 Destruction of KAM tori in Hamiltonian systems: link with the distabilization of nearby cycles and calculation of residues Physica D 13 302-38
42. McMullen C 1998 Self-similarity of Siegel disks and Hausdorff dimension of Julia sets Acta Math. 180 247-92
43. Neumaier A 1990 Interval Methods for Systems of Equations (Encyclopedia of Mathematics and its Applications vol 37) (Cambridge: Cambridge University Press)
44. Palis J and Takens F 1993 Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors (Cambridge Studies in Advanced Mathematics vol 35) (Cambridge: Cambridge University Press)
45. Shenker S J and Kadanoff L P 1982 Critical behaviour of KAM surfaces: I. Empirical results J. Stat. Phys. 27 631-56
46. Sparrow C 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (Berlin: Springer)
47. Tresser C and Coullet P 1978 Itérations d'endomorphismes et groupe de renormalisation C. R. Acad. Sci. Paris A 287 577-80
48. Yampolsky M 2002 Hyperbolicity of renormalization of critical circle maps Publ. Math. Inst. Hautes Etudes Sci. 96 1-41
49. Yampolsky M 2003 Renormalization horseshoe for critical circle maps Commun. Math. Phys. 240 75-96
50. Yampolsky M 2008 Siegel disks and renormalization fixed points Holomorphic Dynamics and Renormalization (Fields Inst. Commun. vol 53) (Providence, RI: AMS) pp 377-93 arXiv:math.DS/0602678
51. Zgliczyński P 1997 Computer assisted proof of chaos in the R.ossler equations and in the Hénon map Nonlinearity 10 243-52
52. Zgliczyński P 2009 Covering relations, cone conditions and the stable manifold theorem J. Diff. Eqns 246 1774-819
53. Zgliczyński P and Gidea M 2004 Covering relations for multidimensional dynamical systems J. Diff. Eqns 202 32-58 (Pubitemid 39025510)
54. Programs available at http://math.uu.se/∼johnson and http://math.uu.se/∼gaidash