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Volumn 20, Issue 2, 2007, Pages 417-460

The quasi-periodic Hamiltonian Hopf bifurcation

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EID: 33947682204     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/20/2/009     Document Type: Article
Times cited : (23)

References (76)
  • 2
    • 0002764805 scopus 로고
    • Proof of a theorem of A N Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian
    • Arnol'd V I 1963 Proof of a theorem of A N Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian Russ. Math. Surv. 18 9-36
    • (1963) Russ. Math. Surv. , vol.18 , Issue.5 , pp. 9-36
    • Arnol'd, V.I.1
  • 3
    • 84927914894 scopus 로고
    • Matrices depending on parameters
    • Arnol'd V I 1971 Matrices depending on parameters Russ. Math. Surv. 26 29-43
    • (1971) Russ. Math. Surv. , vol.26 , Issue.2 , pp. 29-43
    • Arnol'd, V.I.1
  • 5
    • 0039578965 scopus 로고
    • Diophantine approximations on images of mappings
    • Bakhtin V I 1991 Diophantine approximations on images of mappings Dokl. Akad. Nauk BSSR 35 398-400
    • (1991) Dokl. Akad. Nauk BSSR , vol.35 , pp. 398-400
    • Bakhtin, V.I.1
  • 6
    • 0039578965 scopus 로고
    • Diophantine approximations on images of mappings
    • Bakhtin V I 1991 Diophantine approximations on images of mappings Dokl. Akad. Nauk BSSR 35 475
    • (1991) Dokl. Akad. Nauk BSSR , vol.35 , pp. 475
    • Bakhtin, V.I.1
  • 7
    • 0000200040 scopus 로고
    • Degeneration of Hamiltonian monodromy cycles
    • Bates L and Zou M R 1993 Degeneration of Hamiltonian monodromy cycles Nonlinearity 6 313-35
    • (1993) Nonlinearity , vol.6 , Issue.2 , pp. 313-335
    • Bates, L.1    Zou, M.R.2
  • 8
    • 0004223966 scopus 로고
    • Birkhoff G D 1966 Dynamical Systems (American Mathematical Society Colloquium Publications vol IX) (Providence, RI: American Mathematical Society) addendum by Jürgen Moser
    • (1966) Dynamical Systems
    • Birkhoff, G.D.1
  • 9
    • 84963026889 scopus 로고
    • Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE
    • Bourgain J 1994 Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE Int. Math. Res. Not. 1994 475-97
    • (1994) Int. Math. Res. Not. , vol.1994 , Issue.11 , pp. 475-497
    • Bourgain, J.1
  • 10
    • 0031287692 scopus 로고    scopus 로고
    • On Melnikov's persistency problem
    • Bourgain J 1997 On Melnikov's persistency problem Math. Res. Lett. 4 445-58
    • (1997) Math. Res. Lett. , vol.4 , pp. 445-458
    • Bourgain, J.1
  • 13
    • 0002761467 scopus 로고
    • Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case
    • Broer H W 1981 Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980) (Lecture Notes in Mathematics vol 898) (Berlin: Springer) pp 54-74
    • (1981) Dynamical Systems and Turbulence, Warwick 1980 , vol.898 , pp. 54-74
    • Broer, H.W.1
  • 15
    • 0142137606 scopus 로고    scopus 로고
    • Normal-internal resonances in quasi-periodically forced oscillators: A conservative approach
    • Broer H W, Hanmann H, Jorba, Villanueva J and Wagener F O O 2003 Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach Nonlinearity 16 1751-91
    • (2003) Nonlinearity , vol.16 , Issue.5 , pp. 1751-1791
    • Broer, H.W.1    Hanmann, H.2    Jorba3    Villanueva, J.4    Wagener, F.O.O.5
  • 16
    • 21244442290 scopus 로고    scopus 로고
    • Bifurcations of normally parabolic tori in Hamiltonian systems
    • Broer H W, Hanmann H and You J 2005 Bifurcations of normally parabolic tori in Hamiltonian systems Nonlinearity 18 1735-69
    • (2005) Nonlinearity , vol.18 , Issue.4 , pp. 1735-1769
    • Broer, H.W.1    Hanmann, H.2    You, J.3
  • 17
    • 33750612100 scopus 로고    scopus 로고
    • Normal linear stability of quasi-periodic tori
    • Broer H W, Hoo J and Naudot V 2007 Normal linear stability of quasi-periodic tori J. Diff. Eqns 232 355-418
    • (2007) J. Diff. Eqns , vol.232 , Issue.2 , pp. 355-418
    • Broer, H.W.1    Hoo, J.2    Naudot, V.3
  • 18
    • 0002954939 scopus 로고    scopus 로고
    • A reversible bifurcation analysis of the inverted pendulum
    • Broer H W, Hoveijn I and van Noort M 1998 A reversible bifurcation analysis of the inverted pendulum Physica D 112 50-63
    • (1998) Physica , vol.112 , Issue.1-2 , pp. 50-63
    • Broer, H.W.1    Hoveijn, I.2    Van Noort, M.3
  • 19
    • 0033196198 scopus 로고    scopus 로고
    • The inverted pendulum: A singularity theory approach
    • Broer H W, Hoveijn I, van Noort M and Vegter G 1999 The inverted pendulum: a singularity theory approach J. Diff. Eqns 157 120-49
    • (1999) J. Diff. Eqns , vol.157 , Issue.1 , pp. 120-149
    • Broer, H.W.1    Hoveijn, I.2    Van Noort, M.3    Vegter, G.4
  • 22
    • 77957124197 scopus 로고    scopus 로고
    • KAM theory: Quasi-periodicity in dynamical systems
    • Broer H W and Sevryuk M B 2007 KAM theory: quasi-periodicity in dynamical systems Handbook of Dynamical Systems vol 3 ed H W Broer et al (Amsterdam: North-Holland)
    • (2007) Handbook of Dynamical Systems , vol.3
    • Broer, H.W.1    Sevryuk, M.B.2
  • 23
  • 24
    • 52449147455 scopus 로고
    • Smooth prime integrals for quasi-integrable Hamiltonian systems
    • Chierchia L and Gallavotti G 1982 Smooth prime integrals for quasi-integrable Hamiltonian systems Nuovo Cimento B 67 277-95
    • (1982) Nuovo Cimento , vol.67 , pp. 277-295
    • Chierchia, L.1    Gallavotti, G.2
  • 30
    • 0347450519 scopus 로고    scopus 로고
    • Monodromy in the hydrogen atom in crossed fields
    • Cushman R H and Sadovskií D A 2000 Monodromy in the hydrogen atom in crossed fields Physica D 142 166-96
    • (2000) Physica , vol.142 , Issue.1-2 , pp. 166-196
    • Cushman, R.H.1    Sadovskií, D.A.2
  • 31
  • 32
    • 4644273775 scopus 로고    scopus 로고
    • Vanishing twist near focus-focus points
    • Dullin H R and Vu Ngoc San 2004 Vanishing twist near focus-focus points Nonlinearity 17 1777-85
    • (2004) Nonlinearity , vol.17 , Issue.5 , pp. 1777-1785
    • Dullin, H.R.1    Ngoc San, V.2
  • 34
    • 2942585296 scopus 로고    scopus 로고
    • Hamiltonian Hopf bifurcation of the hydrogen atom in crossed fields
    • Efstathiou K, Cushman R H and Sadovskií D A 2004 Hamiltonian Hopf bifurcation of the hydrogen atom in crossed fields Physica D 194 250-74
    • (2004) Physica , vol.194 , Issue.3-4 , pp. 250-274
    • Efstathiou, K.1    Cushman, R.H.2    Sadovskií, D.A.3
  • 35
    • 0000412297 scopus 로고
    • Perturbations of stable invariant tori for Hamiltonian systems
    • Eliasson L H 1988 Perturbations of stable invariant tori for Hamiltonian systems Ann. Sc. Norm. Sup. Pisa, Cl. Sci., Ser.IV 15 115-47
    • (1988) Ann. Sc. Norm. Sup. Pisa, Cl. Sci., Ser.IV , vol.15 , pp. 115-147
    • Eliasson, L.H.1
  • 36
    • 51249174640 scopus 로고
    • Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case
    • Eliasson L H 1990 Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case Comm. Math. Helv. 65 4-35
    • (1990) Comm. Math. Helv. , vol.65 , pp. 4-35
    • Eliasson, L.H.1
  • 39
    • 0003053373 scopus 로고
    • On the Conservation of hyperbolic invariant tori for Hamiltonian systems
    • Graff S M 1974 On the Conservation of hyperbolic invariant tori for Hamiltonian systems J. Diff. Eqns 15 1-69
    • (1974) J. Diff. Eqns , vol.15 , Issue.1 , pp. 1-69
    • Graff, S.M.1
  • 40
    • 0001209620 scopus 로고
    • On constructing formal integrals of a Hamiltonian system near an equilibrium point
    • Gustavson F G 1966 On constructing formal integrals of a Hamiltonian system near an equilibrium point Astron. J. 71 670-86
    • (1966) Astron. J. , vol.71 , pp. 670-686
    • Gustavson, F.G.1
  • 41
    • 0010138127 scopus 로고    scopus 로고
    • The quasi-periodic centre-saddle bifurcation
    • Hanmann H 1998 The quasi-periodic centre-saddle bifurcation J. Diff. Eqns 142 305-70
    • (1998) J. Diff. Eqns , vol.142 , Issue.2 , pp. 305-370
    • Hanmann, H.1
  • 43
    • 77953870876 scopus 로고    scopus 로고
    • On non-degenerate Hamiltonian Hopf bifurcations in 3DOF systems
    • Hanmann H and van der Meer J-C 2005 On non-degenerate Hamiltonian Hopf bifurcations in 3DOF systems Equadiff 2003, Hasselt 2003 ed F Dumortier et al (Singapore: World Scientific) pp 476-81
    • (2005) Equadiff 2003, Hasselt 2003 , pp. 476-481
    • Hanmann, H.1    Van Der Meer, J.-C.2
  • 44
    • 0003610914 scopus 로고
    • Hausdorff F 1962 Set theory 2nd edn (New York: Chelsea) (translated from the German by J R Aumann et al)
    • (1962) Set Theory
    • Hausdorff, F.1
  • 45
    • 33646013219 scopus 로고
    • Huitema G B 1988 Unfoldings of quasi-periodic tori PhD Thesis University of Groningen
    • (1988) PhD Thesis
    • Huitema, G.B.1
  • 46
    • 21244462710 scopus 로고    scopus 로고
    • de Jong H H 1999 Quasiperiodic breathers in systems of weakly coupled pendulums PhD Thesis Rijksuniversiteit Groningen
    • (1999) PhD Thesis
    • De Jong, H.H.1
  • 47
    • 0001246972 scopus 로고
    • On conservation of conditionally periodic motions for a small change in Hamilton's function
    • Kolmogorov A N 1954 On conservation of conditionally periodic motions for a small change in Hamilton's function Dokl. Akad. Nauk SSSR (N.S.) 98 527-30
    • (1954) Dokl. Akad. Nauk SSSR (N.S.) , vol.98 , pp. 527-530
    • Kolmogorov, A.N.1
  • 48
    • 0347473179 scopus 로고
    • Perturbation theory of conditionally periodic solutions of infinite-dimensional Hamiltonian systems and its applications to the Korteweg-de Vries equation
    • Kuksin S B 1989 Perturbation theory of conditionally periodic solutions of infinite-dimensional Hamiltonian systems and its applications to the Korteweg-de Vries equation Math. USSR-Sb. 64 397-413
    • (1989) Math. USSR-Sb. , vol.64 , Issue.2 , pp. 397-413
    • Kuksin, S.B.1
  • 49
    • 33947655612 scopus 로고
    • Landau L D and Lifshitz E M 1960 Mechanics (Course of Theoretical Physics) vol 1 (Oxford: Pergamon) (translated from the Russian by J B Bell)
    • (1960) Mechanics , vol.1
    • Landau, L.D.1    Lifshitz, E.M.2
  • 50
    • 16244377519 scopus 로고    scopus 로고
    • Persistence of lower dimensional tori of general types in Hamiltonian systems
    • Li Y and Yi Y 2005 Persistence of lower dimensional tori of general types in Hamiltonian systems Trans. AMS 357 1565-600
    • (2005) Trans. AMS , vol.357 , Issue.4 , pp. 1565-1600
    • Li, Y.1    Yi, Y.2
  • 51
    • 0002893048 scopus 로고
    • Lagrangian reduction and the double spherical pendulum
    • Marsden J E and Scheurle J 1993 Lagrangian reduction and the double spherical pendulum Z. Angew. Math. Phys. 44 17-43
    • (1993) Z. Angew. Math. Phys. , vol.44 , Issue.1 , pp. 17-43
    • Marsden, J.E.1    Scheurle, J.2
  • 53
    • 0001203818 scopus 로고
    • Hamiltonian Hopf bifurcation with symmetry
    • van der Meer J-C 1990 Hamiltonian Hopf bifurcation with symmetry Nonlinearity 3 1041-56
    • (1990) Nonlinearity , vol.3 , Issue.4 , pp. 1041-1056
    • Van Der Meer, J.-C.1
  • 54
    • 0039034734 scopus 로고
    • On certain cases of conservation of almost periodic motions with a small change of the Hamiltonian function
    • Mel'nikov V K 1965 On certain cases of conservation of almost periodic motions with a small change of the Hamiltonian function Dokl. Akad. Nauk SSSR 165 1245-8
    • (1965) Dokl. Akad. Nauk SSSR , vol.165 , pp. 1245-1248
    • Mel'nikov, V.K.1
  • 55
    • 0039034681 scopus 로고
    • A certain family of conditionally periodic solutions of a Hamiltonian system
    • Mel'nikov V K 1968 A certain family of conditionally periodic solutions of a Hamiltonian system Dokl. Akad. Nauk SSSR 181 546-9
    • (1968) Dokl. Akad. Nauk SSSR , vol.181 , pp. 546-549
    • Mel'nikov, V.K.1
  • 57
    • 3042570326 scopus 로고
    • Convergent series expansions for quasi-periodic motions
    • Moser J K 1967 Convergent series expansions for quasi-periodic motions Math. Ann. 169 136-76
    • (1967) Math. Ann. , vol.169 , Issue.1 , pp. 136-176
    • Moser, J.K.1
  • 58
    • 0041581444 scopus 로고    scopus 로고
    • Kolmogorov condition for integrable systems with focus-focus singularities
    • Nguyen Tien Zung 1996 Kolmogorov condition for integrable systems with focus-focus singularities Phys. Lett. A 215 40-4
    • (1996) Phys. Lett. , vol.215 , Issue.1-2 , pp. 40-44
    • Tien Zung, N.1
  • 59
    • 9344225119 scopus 로고    scopus 로고
    • A note on focus-focus singularities
    • Nguyen Tien Zung 1997 A note on focus-focus singularities Diff. Geom. Appl. 7 123-30
    • (1997) Diff. Geom. Appl. , vol.7 , Issue.2 , pp. 123-130
    • Tien Zung, N.1
  • 60
    • 33947708628 scopus 로고    scopus 로고
    • van Noort M 2001 Global coherent dynamics of the parametrically forced pendulum: a case study in one-a-half degree of freedom PhD Thesis University of Groningen
    • (2001) PhD Thesis
    • Van Noort, M.1
  • 61
    • 23844497722 scopus 로고    scopus 로고
    • Dynamics close to a non semi-simple 1: -1 resonance periodic orbit
    • Ollé M, Pacha J R and Villanueva J 2005 Dynamics close to a non semi-simple 1 : -1 resonance periodic orbit Discrete. Contin. Dyn. Syst. B 5 799-816
    • (2005) Discrete. Contin. Dyn. Syst. , vol.5 , pp. 799-816
    • Ollé, M.1    Pacha, J.R.2    Villanueva, J.3
  • 62
    • 78650359235 scopus 로고    scopus 로고
    • Pacha J R 2002 On the quasi-periodic Hamiltonian Andronov-Hopf bifurcation PhD Thesis Universitat Politècnica de Catalunya
    • (2002) PhD Thesis
    • Pacha, J.R.1
  • 63
    • 84990623105 scopus 로고
    • Integrability of Hamiltonian systems on Cantor sets
    • Pöschel J 1982 Integrability of Hamiltonian systems on Cantor sets Commun. Pure Appl. Math. 35 653-96
    • (1982) Commun. Pure Appl. Math. , vol.35 , pp. 653-696
    • Pöschel, J.1
  • 64
    • 0000882204 scopus 로고
    • On elliptic lower-dimensional tori in Hamiltonian systems
    • Pöschel J 1989 On elliptic lower-dimensional tori in Hamiltonian systems Math. Z. 202 559-608
    • (1989) Math. Z. , vol.202 , Issue.4 , pp. 559-608
    • Pöschel, J.1
  • 67
    • 0347301749 scopus 로고    scopus 로고
    • A Cantor set of tori with monodromy near a focus-focus singularity
    • Rink B W 2004 A Cantor set of tori with monodromy near a focus-focus singularity Nonlinearity 17 347-56
    • (2004) Nonlinearity , vol.17 , Issue.1 , pp. 347-356
    • Rink, B.W.1
  • 68
    • 0000285454 scopus 로고    scopus 로고
    • Invariant tori in non-degenerate nearly integrable Hamiltonian systems
    • Rüssmann H 2001 Invariant tori in non-degenerate nearly integrable Hamiltonian systems Regul. Chaotic Dyn. 6 119-204
    • (2001) Regul. Chaotic Dyn. , vol.6 , Issue.2 , pp. 119-204
    • Rüssmann, H.1
  • 69
    • 33947663233 scopus 로고    scopus 로고
    • Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method
    • Sevryuk M B 2007 Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method Discrete Contin. Dyn. Syst. A at press
    • (2007) Discrete Contin. Dyn. Syst.
    • Sevryuk, M.B.1
  • 70
    • 4043050765 scopus 로고    scopus 로고
    • Quantizing the classical cat
    • Stewart I 2004 Quantizing the classical cat Nature 430 731-2
    • (2004) Nature , vol.430 , Issue.7001 , pp. 731-732
    • Stewart, I.1
  • 72
    • 0037332741 scopus 로고    scopus 로고
    • On semi-global invariants for focus-focus singularities
    • Vu Ngoc San 2003 On semi-global invariants for focus-focus singularities Topology 42 365-80
    • (2003) Topology , vol.42 , Issue.2 , pp. 365-380
    • Ngoc San, V.1
  • 74
    • 0001316020 scopus 로고
    • On the algebraic problem concerning the normal form of linear dynamical systems
    • Williamson J 1936 On the algebraic problem concerning the normal form of linear dynamical systems Am. J. Math. 58 141-63
    • (1936) Am. J. Math. , vol.58 , Issue.1 , pp. 141-163
    • Williamson, J.1
  • 75
    • 0035604021 scopus 로고    scopus 로고
    • Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition
    • Xu J and You J 2001 Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition J. Math. Pures Appl. 80 1045-67
    • (2001) J. Math. Pures Appl. , vol.80 , Issue.10 , pp. 1045-1067
    • Xu, J.1    You, J.2
  • 76
    • 0033540589 scopus 로고    scopus 로고
    • Perturbations of lower-dimensional tori for Hamiltonian systems
    • You J 1999 Perturbations of lower-dimensional tori for Hamiltonian systems J. Diff. Eqns 152 1-29
    • (1999) J. Diff. Eqns , vol.152 , Issue.1 , pp. 1-29
    • You, J.1


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