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Volumn 237, Issue 13, 2008, Pages 1773-1799

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance 'bubble'

Author keywords

3D diffeomorphisms; Quasi periodic bifurcations; Two torus attractor

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL MODELS; RESONANCE;

EID: 45049085469     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2008.01.026     Document Type: Article
Times cited : (58)

References (69)
  • 1
    • 45049084763 scopus 로고    scopus 로고
    • V.S. Afrajmovich, L.P. Shil'nikov, On invariant two-dimensional tori, their disintegration and stochasticity, in: Methods of the Qualitative Theory of Differential Equations, Gor'kov. Gos. University, Gor'kij, 1983, pp. 3-26
    • V.S. Afrajmovich, L.P. Shil'nikov, On invariant two-dimensional tori, their disintegration and stochasticity, in: Methods of the Qualitative Theory of Differential Equations, Gor'kov. Gos. University, Gor'kij, 1983, pp. 3-26
  • 2
    • 45049086468 scopus 로고
    • Arnol'd V.I. Mathematical Methods of Classical Mechanics. Springer GTM vol. 60 (1980)
    • (1980) Springer GTM , vol.60
    • Arnol'd, V.I.1
  • 5
    • 0000009552 scopus 로고
    • Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study
    • Aronson D.G., Chory M.A., Hall G.R., and McGehee R.P. Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study. Comm. Math. Phys. 83 3 (1982) 303-354
    • (1982) Comm. Math. Phys. , vol.83 , Issue.3 , pp. 303-354
    • Aronson, D.G.1    Chory, M.A.2    Hall, G.R.3    McGehee, R.P.4
  • 6
    • 44949274027 scopus 로고
    • Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos
    • Baesens C., Guckenheimer J., Kim S., and McKay R.S. Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos. Physica D 49 3 (1991) 387-475
    • (1991) Physica D , vol.49 , Issue.3 , pp. 387-475
    • Baesens, C.1    Guckenheimer, J.2    Kim, S.3    McKay, R.S.4
  • 8
    • 0002761467 scopus 로고
    • Formal normal forms for vector fields and some consequences for bifurcations in the volume preserving case
    • Dynamical Systems and Turbulence, Warwick 1980. Dold A., and Eckmann B. (Eds)
    • Broer H.W. Formal normal forms for vector fields and some consequences for bifurcations in the volume preserving case. In: Dold A., and Eckmann B. (Eds). Dynamical Systems and Turbulence, Warwick 1980. Springer LNM vol. 898 (1980) 54-74
    • (1980) Springer LNM , vol.898 , pp. 54-74
    • Broer, H.W.1
  • 10
    • 0001874618 scopus 로고
    • Unfoldings and bifurcations of quasi-periodic tori
    • Broer H.W., Huitema G.B., Takens F., and Braaksma B.L.J. Unfoldings and bifurcations of quasi-periodic tori. Mem. AMS 83 421 (1990) 1-175
    • (1990) Mem. AMS , vol.83 , Issue.421 , pp. 1-175
    • Broer, H.W.1    Huitema, G.B.2    Takens, F.3    Braaksma, B.L.J.4
  • 13
    • 0030355090 scopus 로고    scopus 로고
    • Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms
    • Broer H.W., Roussarie R., and Simó C. Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms. ETDS 16 (1996) 1147-1172
    • (1996) ETDS , vol.16 , pp. 1147-1172
    • Broer, H.W.1    Roussarie, R.2    Simó, C.3
  • 14
    • 0006963160 scopus 로고    scopus 로고
    • Towards global models near homoclinic tangencies of dissipative diffeomorphisms
    • Broer H.W., Simó C., and Tatjer J.C. Towards global models near homoclinic tangencies of dissipative diffeomorphisms. Nonlinearity 11 (1998) 667-770
    • (1998) Nonlinearity , vol.11 , pp. 667-770
    • Broer, H.W.1    Simó, C.2    Tatjer, J.C.3
  • 15
    • 0042384678 scopus 로고    scopus 로고
    • Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing
    • Broer H.W., Simó C., and Vitolo R. Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing. Nonlinearity 15 4 (2002) 1205-1267
    • (2002) Nonlinearity , vol.15 , Issue.4 , pp. 1205-1267
    • Broer, H.W.1    Simó, C.2    Vitolo, R.3
  • 16
    • 56349155062 scopus 로고    scopus 로고
    • H.W. Broer, C. Simó, R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol'd resonance web, Bull. Belgian Math. Soc. Simon Stevin, 2008 (in press)
    • H.W. Broer, C. Simó, R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol'd resonance web, Bull. Belgian Math. Soc. Simon Stevin, 2008 (in press)
  • 17
    • 45049085621 scopus 로고    scopus 로고
    • H.W. Broer, C. Simó, R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: A computer assisted dynamical inventory, 2007 (in preparation)
    • H.W. Broer, C. Simó, R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: A computer assisted dynamical inventory, 2007 (in preparation)
  • 18
    • 27944461314 scopus 로고
    • Infinitely many moduli of strong stability in divergence free unfoldings of singularities of vector fields
    • Geometric Dynamics, Proceedings, Rio de Janeiro 1981. Palis J. (Ed)
    • Broer H.W., and van Strien S.J. Infinitely many moduli of strong stability in divergence free unfoldings of singularities of vector fields. In: Palis J. (Ed). Geometric Dynamics, Proceedings, Rio de Janeiro 1981. Springer LNM vol. 1007 (1983) 39-59
    • (1983) Springer LNM , vol.1007 , pp. 39-59
    • Broer, H.W.1    van Strien, S.J.2
  • 19
    • 0002700073 scopus 로고
    • Formally symmetric normal forms and genericity
    • Broer H.W., and Takens F. Formally symmetric normal forms and genericity. Dynamics Reported 2 (1989) 36-60
    • (1989) Dynamics Reported , vol.2 , pp. 36-60
    • Broer, H.W.1    Takens, F.2
  • 20
    • 53149138221 scopus 로고    scopus 로고
    • Integrable and non-integrable deformations of the skew Hopf bifurcation
    • Broer H.W., Takens F., and Wagener F.O.O. Integrable and non-integrable deformations of the skew Hopf bifurcation. Regular and Chaotic Dynamics 4 2 (1999) 17-43
    • (1999) Regular and Chaotic Dynamics , vol.4 , Issue.2 , pp. 17-43
    • Broer, H.W.1    Takens, F.2    Wagener, F.O.O.3
  • 21
    • 0001231858 scopus 로고
    • Subordinate Shil'nikov bifurcations near some singularities of vector fields having low codimension
    • Broer H.W., and Vegter G. Subordinate Shil'nikov bifurcations near some singularities of vector fields having low codimension. ETDS 4 (1984) 509-525
    • (1984) ETDS , vol.4 , pp. 509-525
    • Broer, H.W.1    Vegter, G.2
  • 22
    • 51649147118 scopus 로고
    • Bifurcations de points fixes elliptiques. I. Courbes invariantes
    • Chenciner A. Bifurcations de points fixes elliptiques. I. Courbes invariantes. Publ. Math. IHES 61 (1985) 67-127
    • (1985) Publ. Math. IHES , vol.61 , pp. 67-127
    • Chenciner, A.1
  • 23
    • 0013369791 scopus 로고
    • Bifurcations de points fixes elliptiques. II. Orbites périodiques et ensembles de Cantor invariants
    • Chenciner A. Bifurcations de points fixes elliptiques. II. Orbites périodiques et ensembles de Cantor invariants. Invent. Math. 80 (1985) 81-106
    • (1985) Invent. Math. , vol.80 , pp. 81-106
    • Chenciner, A.1
  • 24
    • 51649147531 scopus 로고
    • Bifurcations de points fixes elliptiques. III. Orbites périodiques de petites périodes et élimination résonnante des couples de courbes invariantes
    • Chenciner A. Bifurcations de points fixes elliptiques. III. Orbites périodiques de petites périodes et élimination résonnante des couples de courbes invariantes. Publ. Math. IHES 66 (1987) 5-91
    • (1987) Publ. Math. IHES , vol.66 , pp. 5-91
    • Chenciner, A.1
  • 26
    • 45049084202 scopus 로고    scopus 로고
    • M.C. Ciocci, A. Litvak-Hinenzon, H.W. Broer, Survey on dissipative kam theory including quasi-periodic bifurcation theory based on lectures by Henk Broer, in: J. Montaldi, T. Ratiu (Eds.), Geometric Mechanics and Symmetry: The Peyresq Lectures, in: LMS Lecture Notes Series, vol. 306, Cambridge University Press, ISBN-13: 9780521539579, 2005, pp. 303-355
    • M.C. Ciocci, A. Litvak-Hinenzon, H.W. Broer, Survey on dissipative kam theory including quasi-periodic bifurcation theory based on lectures by Henk Broer, in: J. Montaldi, T. Ratiu (Eds.), Geometric Mechanics and Symmetry: The Peyresq Lectures, in: LMS Lecture Notes Series, vol. 306, Cambridge University Press, ISBN-13: 9780521539579, 2005, pp. 303-355
  • 27
    • 0347108885 scopus 로고    scopus 로고
    • Strange attractors in saddle cycles: Prevalence and globality
    • Díaz L., Rocha J., and Viana M. Strange attractors in saddle cycles: Prevalence and globality. Inv. Math. 125 (1996) 37-74
    • (1996) Inv. Math. , vol.125 , pp. 37-74
    • Díaz, L.1    Rocha, J.2    Viana, M.3
  • 28
    • 45049086070 scopus 로고    scopus 로고
    • F. Dumortier, S. Ibáñez, H. Kokubu, C. Simó, About the unfolding of a Hopf-zero singularity, 2007 (in preparation)
    • F. Dumortier, S. Ibáñez, H. Kokubu, C. Simó, About the unfolding of a Hopf-zero singularity, 2007 (in preparation)
  • 29
    • 84974232147 scopus 로고
    • Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3
    • Dumortier F., Roussarie R., and Sotomayor J. Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3. ETDS 7 3 (1987) 375-413
    • (1987) ETDS , vol.7 , Issue.3 , pp. 375-413
    • Dumortier, F.1    Roussarie, R.2    Sotomayor, J.3
  • 30
    • 0002036676 scopus 로고
    • Bifurcations of Planar Vector Fields
    • Springer, Berlin
    • Dumortier F., Roussarie R., Sotomayor J., and Zoladek H. Bifurcations of Planar Vector Fields. LNM vol. 1480 (1991), Springer, Berlin
    • (1991) LNM , vol.1480
    • Dumortier, F.1    Roussarie, R.2    Sotomayor, J.3    Zoladek, H.4
  • 31
    • 45049086130 scopus 로고    scopus 로고
    • E. Castellà, Sobre la dinàmica prop dels punts de Lagrange del sistema Terra-Luna, Ph.D. Thesis (in Catalan), Univ. of Barcelona, 2003
    • E. Castellà, Sobre la dinàmica prop dels punts de Lagrange del sistema Terra-Luna, Ph.D. Thesis (in Catalan), Univ. of Barcelona, 2003
  • 32
    • 0040845842 scopus 로고    scopus 로고
    • On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem
    • Castellà E., and Jorba À. On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem. Celestial Mech. 76 (2000) 35-54
    • (2000) Celestial Mech. , vol.76 , pp. 35-54
    • Castellà, E.1    Jorba, À.2
  • 33
    • 0001356311 scopus 로고
    • Persistence and smoothness of invariant manifolds for flows
    • Fenichel N. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21 (1971/72) 193-226
    • (1971) Indiana Univ. Math. J. , vol.21 , pp. 193-226
    • Fenichel, N.1
  • 34
    • 0002895479 scopus 로고
    • Local birth of homoclinic chaos
    • Gaspard P. Local birth of homoclinic chaos. Physica D 62 1-4 (1993) 94-122
    • (1993) Physica D , vol.62 , Issue.1-4 , pp. 94-122
    • Gaspard, P.1
  • 35
    • 45049087050 scopus 로고    scopus 로고
    • G. Gomez, J.M. Mondelo, C. Simó, Refined Fourier analysis: Procedures, error estimates and applications, preprint Universitat de Barcelona 2001
    • G. Gomez, J.M. Mondelo, C. Simó, Refined Fourier analysis: Procedures, error estimates and applications, preprint Universitat de Barcelona 2001
  • 38
    • 33750532700 scopus 로고    scopus 로고
    • A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms
    • Haro A., and de la Llave R. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete and continuous dynamical systems-B 6 6 (2006) 1261-1300
    • (2006) Discrete and continuous dynamical systems-B , vol.6 , Issue.6 , pp. 1261-1300
    • Haro, A.1    de la Llave, R.2
  • 39
    • 45049087511 scopus 로고    scopus 로고
    • A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Explorations and mechanisms for the breakdown of hyperbolicity
    • Haro A., and de la Llave R. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Explorations and mechanisms for the breakdown of hyperbolicity. SIAM J. Appl. Dynam. Syst. 6 1 (2007) 1-292
    • (2007) SIAM J. Appl. Dynam. Syst. , vol.6 , Issue.1 , pp. 1-292
    • Haro, A.1    de la Llave, R.2
  • 41
    • 0347550175 scopus 로고
    • Quasi-genericity of bifurcations to high-dimensional invariant tori for maps
    • Iooss G., and Los J.E. Quasi-genericity of bifurcations to high-dimensional invariant tori for maps. Comm. Math. Phys. 119 (1988) 453-500
    • (1988) Comm. Math. Phys. , vol.119 , pp. 453-500
    • Iooss, G.1    Los, J.E.2
  • 42
    • 0033243813 scopus 로고    scopus 로고
    • A methodology for the numerical computation of normal forms, centre manifolds and first integrals of hamiltonian systems
    • Jorba À. A methodology for the numerical computation of normal forms, centre manifolds and first integrals of hamiltonian systems. Experim. Math. 8 (1999) 155-195
    • (1999) Experim. Math. , vol.8 , pp. 155-195
    • Jorba, À.1
  • 43
    • 0040483472 scopus 로고    scopus 로고
    • Numerical computation of the normal behaviour of invariant curves of n-dimensional maps
    • Jorba À. Numerical computation of the normal behaviour of invariant curves of n-dimensional maps. Nonlinearity 14 (2001) 943-976
    • (2001) Nonlinearity , vol.14 , pp. 943-976
    • Jorba, À.1
  • 44
    • 0001343753 scopus 로고
    • Breaking of symmetry in the saddle-node Hopf bifurcation
    • Kirk V. Breaking of symmetry in the saddle-node Hopf bifurcation. Phys. Lett. A 154 (1991) 243-248
    • (1991) Phys. Lett. A , vol.154 , pp. 243-248
    • Kirk, V.1
  • 45
    • 3142771892 scopus 로고    scopus 로고
    • A planar model system for the saddle-node Hopf bifurcation with global reinjection
    • Krauskopf B., and Oldeman B. A planar model system for the saddle-node Hopf bifurcation with global reinjection. Nonlinearity 17 4 (2004) 1119-1152
    • (2004) Nonlinearity , vol.17 , Issue.4 , pp. 1119-1152
    • Krauskopf, B.1    Oldeman, B.2
  • 46
    • 33747418994 scopus 로고    scopus 로고
    • Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation
    • Krauskopf B., and Oldeman B. Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation. Nonlinearity 19 9 (2006) 2149-2167
    • (2006) Nonlinearity , vol.19 , Issue.9 , pp. 2149-2167
    • Krauskopf, B.1    Oldeman, B.2
  • 47
    • 0000868937 scopus 로고    scopus 로고
    • Codimension-three unfoldings of reflectionally symmetric planar vector fields
    • Krauskopf B., and Rousseau C. Codimension-three unfoldings of reflectionally symmetric planar vector fields. Nonlinearity 10 (1997) 1115-1150
    • (1997) Nonlinearity , vol.10 , pp. 1115-1150
    • Krauskopf, B.1    Rousseau, C.2
  • 49
    • 33750534582 scopus 로고    scopus 로고
    • Remarks on interacting Neimark-Sacker bifurcations
    • Kuznetsov Yu., and Meijer H.G.E. Remarks on interacting Neimark-Sacker bifurcations. J. Difference Equ. Appl. 12 10 (2006) 1009-1035
    • (2006) J. Difference Equ. Appl. , vol.12 , Issue.10 , pp. 1009-1035
    • Kuznetsov, Yu.1    Meijer, H.G.E.2
  • 50
    • 45049087395 scopus 로고    scopus 로고
    • H.G.E. Meijer, Codimension 2 bifurcations of iterated maps, Ph.D. Thesis, University of Utrecht, 2006
    • H.G.E. Meijer, Codimension 2 bifurcations of iterated maps, Ph.D. Thesis, University of Utrecht, 2006
  • 51
    • 45049087361 scopus 로고    scopus 로고
    • J.M. Mondelo, Contribution to the study of fourier methods for quasi-periodic functions and the vicinity of the collinear libration points, Ph.D. Thesis, Universitat de Barcelona, 2001
    • J.M. Mondelo, Contribution to the study of fourier methods for quasi-periodic functions and the vicinity of the collinear libration points, Ph.D. Thesis, Universitat de Barcelona, 2001
  • 52
    • 0021660205 scopus 로고
    • The separation of motions in systems with rapidly rotating phase
    • Neishtadt A.I. The separation of motions in systems with rapidly rotating phase. Prikladnaja Mat. Mekh. 48 (1984) 133-139
    • (1984) Prikladnaja Mat. Mekh. , vol.48 , pp. 133-139
    • Neishtadt, A.I.1
  • 53
    • 51649164983 scopus 로고
    • Bifurcations and stability of families of diffeomorphisms
    • Newhouse S., Palis J., and Takens F. Bifurcations and stability of families of diffeomorphisms. Publ. Math. IHES 57 (1983) 5-71
    • (1983) Publ. Math. IHES , vol.57 , pp. 5-71
    • Newhouse, S.1    Palis, J.2    Takens, F.3
  • 54
    • 0020810173 scopus 로고
    • Universal properties of the transition from quasiperiodicity to chaos in dissipative systems
    • Östlund S., Rand D., Sethna J., and Siggia E. Universal properties of the transition from quasiperiodicity to chaos in dissipative systems. Physica D 8 3 (1983) 303-342
    • (1983) Physica D , vol.8 , Issue.3 , pp. 303-342
    • Östlund, S.1    Rand, D.2    Sethna, J.3    Siggia, E.4
  • 57
    • 34250451946 scopus 로고
    • On the nature of turbulence
    • Ruelle D., and Takens F. On the nature of turbulence. Comm. Math. Phys. 20 (1971) 167-192
    • (1971) Comm. Math. Phys. , vol.20 , pp. 167-192
    • Ruelle, D.1    Takens, F.2
  • 59
    • 29144490560 scopus 로고    scopus 로고
    • On the use of Lyapunov exponents to detect global properties of the dynamics
    • Dumortier F., Broer H.W., Mahwin J., and Verduyn-Lunel S.M. (Eds), World Scientific, Singapore. 981 256 169 2
    • Simó C. On the use of Lyapunov exponents to detect global properties of the dynamics. In: Dumortier F., Broer H.W., Mahwin J., and Verduyn-Lunel S.M. (Eds). Equadiff 2003, Proceedings International Conference on Differential Equations, Hasselt 2003 (2005), World Scientific, Singapore. 981 256 169 2 714-719
    • (2005) Equadiff 2003, Proceedings International Conference on Differential Equations, Hasselt 2003 , pp. 714-719
    • Simó, C.1
  • 60
    • 0039802126 scopus 로고
    • A numerical survey on the Takens-Bogdanov bifurcation for diffeomorphisms
    • Mira C., et al. (Ed), World Scientific, Singapore
    • Simó C., Broer H.W., and Roussarie R. A numerical survey on the Takens-Bogdanov bifurcation for diffeomorphisms. In: Mira C., et al. (Ed). European Conference on Iteration Theory 89 (1992), World Scientific, Singapore 320-334
    • (1992) European Conference on Iteration Theory 89 , pp. 320-334
    • Simó, C.1    Broer, H.W.2    Roussarie, R.3
  • 61
    • 0002533366 scopus 로고
    • Forced Oscillations and Bifurcations
    • Applications of Global Analysis 1 reprinted in: H.W. Broer B. Krauskopf G. Vegter (Eds.), Global Analysis of Dynamical Systems: Festschrift Dedicated to Floris Takens for His 60th Birthday, IOP Publish. Bristol, 2001
    • Takens F. Forced Oscillations and Bifurcations. Applications of Global Analysis 1. Communications of the Mathematical Institute Rijksuniversiteit Utrecht vol. 3 (1974) reprinted in: H.W. Broer B. Krauskopf G. Vegter (Eds.), Global Analysis of Dynamical Systems: Festschrift Dedicated to Floris Takens for His 60th Birthday, IOP Publish. Bristol, 2001
    • (1974) Communications of the Mathematical Institute Rijksuniversiteit Utrecht , vol.3
    • Takens, F.1
  • 63
    • 51649178043 scopus 로고
    • Singularities of vector fields
    • Takens F. Singularities of vector fields. Publ. Math. IHES 43 (1974) 47-100
    • (1974) Publ. Math. IHES , vol.43 , pp. 47-100
    • Takens, F.1
  • 64
    • 0039621597 scopus 로고    scopus 로고
    • Resonances in skew and reducible quasi-periodic Hopf bifurcations
    • Takens F., and Wagener F.O.O. Resonances in skew and reducible quasi-periodic Hopf bifurcations. Nonlinearity 13 2 (2000) 377-396
    • (2000) Nonlinearity , vol.13 , Issue.2 , pp. 377-396
    • Takens, F.1    Wagener, F.O.O.2
  • 65
    • 45049085877 scopus 로고
    • Varadarajan V.S. Lie Groups, Lie Algebras, and their Representations. Springer GTM 102 (1984)
    • (1984) Springer GTM , vol.102
    • Varadarajan, V.S.1
  • 66
    • 45049086042 scopus 로고    scopus 로고
    • R. Vitolo, Bifurcations of attractors in 3D diffeomorphisms, Ph.D. Thesis, University of Groningen, 2003
    • R. Vitolo, Bifurcations of attractors in 3D diffeomorphisms, Ph.D. Thesis, University of Groningen, 2003
  • 67
    • 45049085224 scopus 로고    scopus 로고
    • The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms
    • Gaeta G., Vitolo R., and Walcher S. (Eds). Otranto, Italy 2-9, June 2007, World Scientific. 978-981-277-616-7
    • Vitolo R., Broer H.W., and Simo' C. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms. In: Gaeta G., Vitolo R., and Walcher S. (Eds). SPT 2007, Proceedings of the International Conference Symmetry and Perturbation Theory. Otranto, Italy 2-9, June 2007 (2007), World Scientific. 978-981-277-616-7
    • (2007) SPT 2007, Proceedings of the International Conference Symmetry and Perturbation Theory
    • Vitolo, R.1    Broer, H.W.2    Simo', C.3
  • 68
    • 0142239172 scopus 로고    scopus 로고
    • Semi-local analysis of the k:1 and k:2 resonances in quasi-periodically forced systems
    • Broer H.W., Krauskopf B., and Vegter G. (Eds), IOP, Bristol, Philadelphia
    • Wagener F.O.O. Semi-local analysis of the k:1 and k:2 resonances in quasi-periodically forced systems. In: Broer H.W., Krauskopf B., and Vegter G. (Eds). Global Analysis of Dynamical Systems. Festschrift Dedicated to Floris Takens for his 60th Birthday (2001), IOP, Bristol, Philadelphia 113-129
    • (2001) Global Analysis of Dynamical Systems. Festschrift Dedicated to Floris Takens for his 60th Birthday , pp. 113-129
    • Wagener, F.O.O.1
  • 69
    • 45049085850 scopus 로고    scopus 로고
    • F.O.O. Wagener, On the skew Hopf bifurcation, PhD Thesis, University of Groningen, 1998
    • F.O.O. Wagener, On the skew Hopf bifurcation, PhD Thesis, University of Groningen, 1998


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