메뉴 건너뛰기




Volumn 17, Issue 8, 2007, Pages 2547-2570

The cusp-Hopf bifurcation

Author keywords

Bistability; Bursting oscillations; Codimension three; Cusp; Hopf bifurcation

Indexed keywords

COMPUTER SIMULATION; NEURONS; SYSTEMS ANALYSIS;

EID: 34748909407     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127407018622     Document Type: Article
Times cited : (26)

References (37)
  • 3
    • 0009271127 scopus 로고
    • Classification of ℤ(2)-equivariant imperfect bifurcations with corank 2
    • Dangelmayr, G. & Armbruster, D. [1983] "Classification of ℤ(2)-equivariant imperfect bifurcations with corank 2," Proc. London Math. Soc. 46, 517-546.
    • (1983) Proc. London Math. Soc , vol.46 , pp. 517-546
    • Dangelmayr, G.1    Armbruster, D.2
  • 4
    • 0037004616 scopus 로고    scopus 로고
    • A note on the triple-zero linear degeneracy: Normal forms, dynamical and bifurcation behaviors of an unfolding
    • Freire, E., Gamero, E., Rodriguez-Luis, A. J. & Algaba, A. [2002] "A note on the triple-zero linear degeneracy: Normal forms, dynamical and bifurcation behaviors of an unfolding," Int. J. Bifurcation and Chaos, 12 2799-2820.
    • (2002) Int. J. Bifurcation and Chaos , vol.12 , pp. 2799-2820
    • Freire, E.1    Gamero, E.2    Rodriguez-Luis, A.J.3    Algaba, A.4
  • 5
    • 0012144124 scopus 로고
    • On some bifurcations of an equilibrium with one zero and a pair of pure imaginary roots
    • Gorkii State University, Gorkii, USSR, pp, in Russian
    • Gavrilov, N. K. [1978] "On some bifurcations of an equilibrium with one zero and a pair of pure imaginary roots," Methods of the Qualitative Theory of Differential Equations (Gorkii State University, Gorkii, USSR), pp. 33-40 (in Russian).
    • (1978) Methods of the Qualitative Theory of Differential Equations , pp. 33-40
    • Gavrilov, N.K.1
  • 6
    • 34748892411 scopus 로고
    • On stability of an equilibrium state with one zero and a pair of purely imaginary eigenvalues
    • Gorkii State University, Gorkii, USSR, pp, in Russian
    • Gavrilov, N. K. & Roschin, N. V. [1983] "On stability of an equilibrium state with one zero and a pair of purely imaginary eigenvalues," Methods of the Qualitative Theory of Differential Equations (Gorkii State University, Gorkii, USSR), pp. 41-49 (in Russian).
    • (1983) Methods of the Qualitative Theory of Differential Equations , pp. 41-49
    • Gavrilov, N.K.1    Roschin, N.V.2
  • 7
    • 0043112545 scopus 로고
    • Bifurcations of an equilibrium state with one zero root and a pair of purely imaginary roots and additional degeneration
    • ed. Leontovich-Andronova, E. A, Gorkii State University, Gorkii, USSR, pp, in Russian
    • Gavrilov, N. K. [1987] "Bifurcations of an equilibrium state with one zero root and a pair of purely imaginary roots and additional degeneration," Methods of the Qualitative Theory of Differential Equations, ed. Leontovich-Andronova, E. A. (Gorkii State University, Gorkii, USSR), pp. 43-51 (in Russian).
    • (1987) Methods of the Qualitative Theory of Differential Equations , pp. 43-51
    • Gavrilov, N.K.1
  • 8
    • 33751096546 scopus 로고
    • Classification and unfoldings of degenerate Hopf bifurcations
    • Golubitsky, M. & Langford, W. F. [1981] "Classification and unfoldings of degenerate Hopf bifurcations," J. Diff. Eqs. 41, 375-415.
    • (1981) J. Diff. Eqs , vol.41 , pp. 375-415
    • Golubitsky, M.1    Langford, W.F.2
  • 9
    • 34748847487 scopus 로고    scopus 로고
    • Golubitsky, M. & Schaeffer, D. G. [1985] Singularities and Groups in Bifurcation Theory, I, Applied Mathematical Sciences, 51 (Springer-Verlag, NY).
    • Golubitsky, M. & Schaeffer, D. G. [1985] Singularities and Groups in Bifurcation Theory, Volume I, Applied Mathematical Sciences, Vol. 51 (Springer-Verlag, NY).
  • 10
    • 0001101472 scopus 로고    scopus 로고
    • Defining functions for multiple Hopf bifurcations
    • Govaerts, W., Guckenheimer, J. & Khibnik, A. [1997] "Defining functions for multiple Hopf bifurcations," SIAM J. Num. Anal. 34, 1269-1288.
    • (1997) SIAM J. Num. Anal , vol.34 , pp. 1269-1288
    • Govaerts, W.1    Guckenheimer, J.2    Khibnik, A.3
  • 11
    • 0008340999 scopus 로고
    • On a codimension two bifurcation
    • eds. Rand, D. & Young, L, Dynamical Systems and Turbulence, Springer-Verlag, NY, pp
    • Guckenheimer, J. [1980] "On a codimension two bifurcation," in eds. Rand, D. & Young, L., Dynamical Systems and Turbulence, Lecture Notes in Mathematics, Vol. 898 (Springer-Verlag, NY), pp. 99-142.
    • (1980) Lecture Notes in Mathematics , vol.898 , pp. 99-142
    • Guckenheimer, J.1
  • 12
    • 34748859787 scopus 로고    scopus 로고
    • Guckenheimer, J. & Holmes, P. [1986] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 70 (Springer-Verlag, NY), Corrected second printing.
    • Guckenheimer, J. & Holmes, P. [1986] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, Vol. 70 (Springer-Verlag, NY), Corrected second printing.
  • 13
    • 34748838451 scopus 로고    scopus 로고
    • Codimension three Hopf and cusp bifurcations,
    • Master's thesis, University of Guelph
    • Harlim, J. [2001] "Codimension three Hopf and cusp bifurcations," Master's thesis, University of Guelph.
    • (2001)
    • Harlim, J.1
  • 14
    • 84985452394 scopus 로고    scopus 로고
    • Holmes, P. [1980] Unfolding a degenerate nonlinear oscillator: A codimension two bifurcation, in Nonlinear Dynamics, ed. Helleman, R. G. H., Ann. N. Y. Acad. Sci., 357, pp. 473-488.
    • Holmes, P. [1980] "Unfolding a degenerate nonlinear oscillator: A codimension two bifurcation," in Nonlinear Dynamics, ed. Helleman, R. G. H., Ann. N. Y. Acad. Sci., Vol. 357, pp. 473-488.
  • 15
    • 84985461759 scopus 로고    scopus 로고
    • Iooss, G. & Langford, W. F. [1980] Conjectures on the routes to turbulence via bifurcations, in Nonlinear Dynamics, ed. Helleman, R. G. H., Ann. N. Y. Acad. Sci., 357, pp. 489-505.
    • Iooss, G. & Langford, W. F. [1980] "Conjectures on the routes to turbulence via bifurcations," in Nonlinear Dynamics, ed. Helleman, R. G. H., Ann. N. Y. Acad. Sci., Vol. 357, pp. 489-505.
  • 16
    • 0003327863 scopus 로고
    • Topics in Bifurcation Theory and Applications
    • World Scientific, Singapore
    • Iooss, G. & Adelmeyer, M. [1992] Topics in Bifurcation Theory and Applications, Advanced Series in Nonlinear Dynamics, Vol. 3 (World Scientific, Singapore).
    • (1992) Advanced Series in Nonlinear Dynamics , vol.3
    • Iooss, G.1    Adelmeyer, M.2
  • 17
    • 0000303940 scopus 로고    scopus 로고
    • Neural excitability, spiking, and bursting
    • Izhikevich, E. [2000] "Neural excitability, spiking, and bursting," Int. J. Bifurcation and Chaos 10, 1171-1266.
    • (2000) Int. J. Bifurcation and Chaos , vol.10 , pp. 1171-1266
    • Izhikevich, E.1
  • 18
    • 0000868937 scopus 로고    scopus 로고
    • Codimension-three unfoldings of reflectionally symmetric planar vector fields
    • Krauskopf, B. & Rousseau, C. [1997] "Codimension-three unfoldings of reflectionally symmetric planar vector fields," Nonlinearity 10, 1115-1150.
    • (1997) Nonlinearity , vol.10 , pp. 1115-1150
    • Krauskopf, B.1    Rousseau, C.2
  • 20
    • 32044433268 scopus 로고    scopus 로고
    • Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations
    • Kuznetsov, Yu. [2005] "Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations," Int. J. Bifurcation and Chaos 15, 3535-3546.
    • (2005) Int. J. Bifurcation and Chaos , vol.15 , pp. 3535-3546
    • Kuznetsov, Y.1
  • 21
    • 0018505697 scopus 로고
    • Periodic and steady-state mode interactions lead to tori
    • Langford, W. F. [1979] "Periodic and steady-state mode interactions lead to tori," SIAM J. Appl. Math. 37, 649-686.
    • (1979) SIAM J. Appl. Math , vol.37 , pp. 649-686
    • Langford, W.F.1
  • 22
    • 0020300080 scopus 로고
    • Chaotic dynamics in the unfoldings of degenerate bifurcations
    • Kyoto University, Japan, pp
    • Langford, W. F. [1982] "Chaotic dynamics in the unfoldings of degenerate bifurcations," in Proc. Int. Symp. Applied Mathematics and Information Science (Kyoto University, Japan), pp. 2.41-2.47.
    • (1982) Proc. Int. Symp. Applied Mathematics and Information Science
    • Langford, W.F.1
  • 23
    • 0002333747 scopus 로고
    • A review of interactions of Hopf and steady-state bifurcations
    • eds. Barenblatt, G. I, Iooss, G. & Joseph, D. D, Pitman Advanced Publishing Program, pp
    • Langford, W. F. [1983] "A review of interactions of Hopf and steady-state bifurcations," in Nonlinear Dynamics and Turbulence, eds. Barenblatt, G. I., Iooss, G. & Joseph, D. D. (Pitman Advanced Publishing Program), pp. 215-237.
    • (1983) Nonlinear Dynamics and Turbulence , pp. 215-237
    • Langford, W.F.1
  • 24
    • 34748923543 scopus 로고
    • Hopf bifurcation at a hysteresis point
    • Colloq. Math. Soc. János Bolyai, North Holland, pp
    • Langford, W. F. [1984a] "Hopf bifurcation at a hysteresis point," in Differential Equations: Qualitative Theory, Colloq. Math. Soc. János Bolyai, Vol. 47 (North Holland), pp. 649-686.
    • (1984) Differential Equations: Qualitative Theory , vol.47 , pp. 649-686
    • Langford, W.F.1
  • 25
    • 0005998024 scopus 로고
    • Numerical studies of torus bifurcations
    • Numerical Methods for Bifurcation Problems, eds. Küpper, T, Mittelmann, H. D. & Weber, H, Birkhäuser Verlag, Basel, pp
    • Langford, W. F. [1984b] "Numerical studies of torus bifurcations," in Numerical Methods for Bifurcation Problems, eds. Küpper, T., Mittelmann, H. D. & Weber, H., ISNM, Vol. 70 (Birkhäuser Verlag, Basel), pp. 285-295.
    • (1984) ISNM , vol.70 , pp. 285-295
    • Langford, W.F.1
  • 26
    • 0242280424 scopus 로고    scopus 로고
    • Interactions of Andronov-Hopf and Bogdanov-Takens bifurcations
    • Amer. Math. Soc, pp
    • Langford, W. F. & Zhan, K. [1999] "Interactions of Andronov-Hopf and Bogdanov-Takens bifurcations," Fields Institute Communications, Vol. 24 (Amer. Math. Soc.), pp. 365-383.
    • (1999) Fields Institute Communications , vol.24 , pp. 365-383
    • Langford, W.F.1    Zhan, K.2
  • 27
    • 0030298636 scopus 로고    scopus 로고
    • Classification and unfoldings of 1 : 2 resonant Hopf bifurcation
    • LeBlanc, V. G. & Langford, W. F. [1996] "Classification and unfoldings of 1 : 2 resonant Hopf bifurcation," Arch. Rat. Mech. Anal. 136, 649-686.
    • (1996) Arch. Rat. Mech. Anal , vol.136 , pp. 649-686
    • LeBlanc, V.G.1    Langford, W.F.2
  • 28
    • 0003932428 scopus 로고
    • Clarendon Press, Oxford, Chaps. 3 and 4
    • Mullin, T. [1993] The Nature of Chaos (Clarendon Press, Oxford), Chaps. 3 and 4.
    • (1993) The Nature of Chaos
    • Mullin, T.1
  • 29
    • 0000357907 scopus 로고
    • A formal classification of bursting mechanisms in excitable systems
    • Mathematical Topics in Population Biology, Morphogenesis and Neurosciences, Springer-Verlag, NY, pp
    • Rinzel, J. [1987] "A formal classification of bursting mechanisms in excitable systems," in Mathematical Topics in Population Biology, Morphogenesis and Neurosciences, Lecture Notes in Biomathematics, Vol. 71 (Springer-Verlag, NY), pp. 267-281.
    • (1987) Lecture Notes in Biomathematics , vol.71 , pp. 267-281
    • Rinzel, J.1
  • 30
    • 34748893015 scopus 로고
    • Chaos in experimental systems: Two examples
    • ed. Pnevmatikos, S. N, Elsevier Science Publ, pp
    • Roux, J. C. [1985] "Chaos in experimental systems: Two examples," in Singularities and Dynamical Systems, ed. Pnevmatikos, S. N. (Elsevier Science Publ.), pp. 345-352.
    • (1985) Singularities and Dynamical Systems , pp. 345-352
    • Roux, J.C.1
  • 31
    • 0346040258 scopus 로고    scopus 로고
    • Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity
    • Sieber, J. & Krauskopf, B. [2004] "Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity," Nonlinearity 17, 85-103.
    • (2004) Nonlinearity , vol.17 , pp. 85-103
    • Sieber, J.1    Krauskopf, B.2
  • 33
    • 0012781340 scopus 로고
    • A note on abelian integrals and bifurcation theory
    • van Gils, S. A. [1985] "A note on abelian integrals and bifurcation theory," J. Diff. Eqs. 59, 437-441.
    • (1985) J. Diff. Eqs , vol.59 , pp. 437-441
    • van Gils, S.A.1
  • 34
    • 0000173986 scopus 로고
    • Hopf bifurcation with non-semisimple 1:1 resonance
    • van Gils, S. A., Krupa, M. & Langford, W. F. [1990] "Hopf bifurcation with non-semisimple 1:1 resonance," Nonlinearity 3, 825-850.
    • (1990) Nonlinearity , vol.3 , pp. 825-850
    • van Gils, S.A.1    Krupa, M.2    Langford, W.F.3
  • 35
    • 0041531260 scopus 로고
    • Hopf bifurcations with non-simple eigenvalues
    • eds. Golubitsky, M. & Guckenheimer, J, Contemp. Math
    • Vanderbauwhede, A. [1986] "Hopf bifurcations with non-simple eigenvalues," in Multiparameter Bifurcation Theory, eds. Golubitsky, M. & Guckenheimer, J., Contemp. Math., Vol. 56, pp. 343-353.
    • (1986) Multiparameter Bifurcation Theory , vol.56 , pp. 343-353
    • Vanderbauwhede, A.1
  • 37
    • 0002562820 scopus 로고
    • Bifurcations of certain family of planar vector fields tangent to axes
    • Zoladek, H. [1987] "Bifurcations of certain family of planar vector fields tangent to axes," J. Diff. Eqs. 67, 1-55.
    • (1987) J. Diff. Eqs , vol.67 , pp. 1-55
    • Zoladek, H.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.