메뉴 건너뛰기




Volumn 2, Issue 4, 2009, Pages 851-872

Homoclinic orbits of the fitzhugh-nagumo equation: The singular-limit

Author keywords

Geometric singular perturbation theory; Homoclinic bifurcation; Invariant manifolds

Indexed keywords


EID: 77954584186     PISSN: 19371632     EISSN: 19371179     Source Type: Journal    
DOI: 10.3934/dcdss.2009.2.851     Document Type: Article
Times cited : (57)

References (46)
  • 3
    • 0002568232 scopus 로고    scopus 로고
    • Tracking invariant manifolds without differential forms
    • P. Brunovsky, Tracking invariant manifolds without differential forms, Acta Math. Univ. Comenianae, LXV (1996), 23-32.
    • (1996) Acta Math. Univ. Comenianae, LXV , pp. 23-32
    • Brunovsky, P.1
  • 4
    • 0010863128 scopus 로고
    • A geometric approach to singular perturbation problems with applications to nerve impulse equations
    • Gail A. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations, Journal of Differential Equations, 23 (1977), 335-367.
    • (1977) Journal of Differential Equations , vol.23 , pp. 335-367
    • Carpenter, G.A.1
  • 6
    • 0000997882 scopus 로고
    • The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations
    • Bo Deng, The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations, SIAM J. Appl. Math., 22 (1991), 1631-1650.
    • (1991) SIAM J. Appl. Math. , vol.22 , pp. 1631-1650
    • Deng, B.1
  • 7
    • 0002294440 scopus 로고
    • The canard unchained or how fast/slow dynamical systems bifurcate
    • M. Diener, The canard unchained or how fast/slow dynamical systems bifurcate, The Mathematical Intelligencer, 6 (1984), 38-48.
    • (1984) The Mathematical Intelligencer , vol.6 , pp. 38-48
    • Diener, M.1
  • 9
    • 0002336002 scopus 로고
    • Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations
    • Freddy Dumortier, Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations, in: Bifurcations and Periodic Orbits of Vector Fields D. Schlomiuk (ed.), 19-73, 1993.
    • (1993) Bifurcations and Periodic Orbits of Vector Fields D. Schlomiuk (Ed.) , vol.19-73
    • Dumortier, F.1
  • 11
    • 0003297685 scopus 로고
    • Relaxation oscillations including a standard chase on french ducks
    • Wiktor Eckhaus, Relaxation oscillations including a standard chase on french ducks, Lecture Notes in Mathematics, 985 (1983), 449-494.
    • (1983) Lecture Notes in Mathematics , vol.985 , pp. 449-494
    • Eckhaus, W.1
  • 15
    • 0001477332 scopus 로고
    • Double impulse solutions in nerve axon equations
    • John W. Evans, Neil Fenichel and John A. Feroe, Double impulse solutions in nerve axon equations, SIAM J. Appl. Math., 42 (1982), 219-234.
    • (1982) SIAM J. Appl. Math. , vol.42 , pp. 219-234
    • Evans, J.W.1    Fenichel, N.2    Feroe, J.A.3
  • 16
    • 0001356311 scopus 로고
    • Persistence and smoothness of invariant manifolds for flows
    • Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana University Mathematical Journal, 21 (1971), 193-225.
    • (1971) Indiana University Mathematical Journal , vol.21 , pp. 193-225
    • Fenichel, N.1
  • 19
    • 34250627892 scopus 로고
    • Geometric singular perturbation theory for ordinary differential equations
    • Neil Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98.
    • (1979) Journal of Differential Equations , vol.31 , pp. 53-98
    • Fenichel, N.1
  • 20
    • 51249192747 scopus 로고
    • Mathematical models of threshold phenomena in the nerve membrane
    • R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-269.
    • (1955) Bull. Math. Biophysics , vol.17 , pp. 257-269
    • Fitzhugh, R.1
  • 21
    • 0010928907 scopus 로고
    • Stability analysis for the slow traveling pulse of the FitzHugh-Nagumo system
    • Gilberto Flores, Stability analysis for the slow traveling pulse of the FitzHugh-Nagumo system, SIAM J. Math. Anal., 22 (1991), 392-399.
    • (1991) SIAM J. Math. Anal. , vol.22 , pp. 392-399
    • Flores, G.1
  • 24
    • 84896853375 scopus 로고    scopus 로고
    • Bifurcation and degenerate decomposition in multiple time scale dynamical systems
    • Eds.: John Hogan, Alan Champneys and Bernd Krauskopf
    • John Guckenheimer, Bifurcation and degenerate decomposition in multiple time scale dynamical systems, in "Nonlinear Dynamics and Chaos: Where do We Go from Here?" Eds.: John Hogan, Alan Champneys and Bernd Krauskopf, 1-20, 2002.
    • (2002) Nonlinear Dynamics and Chaos: Where Do We Go from Here? , pp. 1-20
    • Guckenheimer, J.1
  • 25
    • 0041452017 scopus 로고    scopus 로고
    • Global bifurcations of periodic orbits in the forced van der pol equation
    • Eds.: Henk W. Broer, Bernd Krauskopf and Gert Vegter
    • John Guckenheimer, Global bifurcations of periodic orbits in the forced van der Pol equation, in "Global Analysis of Dynamical Systems-Festschrift dedicated to Floris Takens," Eds.: Henk W. Broer, Bernd Krauskopf and Gert Vegter, 1-16, 2003.
    • (2003) Global Analysis of Dynamical Systems-Festschrift Dedicated to Floris Takens , pp. 1-16
    • Guckenheimer, J.1
  • 26
    • 67349099196 scopus 로고    scopus 로고
    • Bifurcations of relaxation oscillations, Normal forms, bifurcations and finiteness problems in differential equations
    • John Guckenheimer, Bifurcations of relaxation oscillations, Normal forms, bifurcations and finiteness problems in differential equations, NATO Sci. Ser. II Math. Phys. Chem., 137 (2004), 295-316.
    • (2004) NATO Sci. Ser. II Math. Phys. Chem. , vol.137 , pp. 295-316
    • Guckenheimer, J.1
  • 28
    • 0000880114 scopus 로고
    • Some mathematical problems from neurobiology
    • S. P. Hastings, Some mathematical problems from neurobiology, The American Mathematical Monthly, 82 (1975), 881-895.
    • (1975) The American Mathematical Monthly , vol.82 , pp. 881-895
    • Hastings, S.P.1
  • 29
    • 0001371673 scopus 로고
    • On the existence of homoclinic and periodic orbits in the FitzHugh-Nagumo equations
    • S. P. Hastings, On the existence of homoclinic and periodic orbits in the FitzHugh-Nagumo equations, Quart. J. Math. Oxford, 2 (1976), 123-134.
    • (1976) Quart. J. Math. Oxford , vol.2 , pp. 123-134
    • Hastings, S.P.1
  • 30
    • 35649001607 scopus 로고
    • A quantitative description of membrane current and its application to conduction and excitation in nerve
    • A. L. Hodgin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-505.
    • (1952) J. Physiol. , vol.117 , pp. 500-505
    • Hodgin, A.L.1    Huxley, A.F.2
  • 31
    • 0002809936 scopus 로고
    • Tracking invariant manifolds with differential forms in singularly perturbed systems
    • C. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, Journal of Differential Equations, 108 (1994), 64-88.
    • (1994) Journal of Differential Equations , vol.108 , pp. 64-88
    • Jones, C.1    Kopell, N.2
  • 32
    • 0000263519 scopus 로고
    • Construction of the FitzHugh-Nagumo pulse using differential forms
    • IMA Vol. Math. Appl. 37, Springer, New York
    • C. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, Patterns and dynamics in reactive media (Minneapolis, MN, 1989), 101-115, IMA Vol. Math. Appl., 37, Springer, New York, 1991.
    • (1991) Patterns and Dynamics in Reactive Media (Minneapolis, MN 1989) , pp. 101-115
    • Jones, C.1    Kopell, N.2    Langer, R.3
  • 36
    • 47849115552 scopus 로고    scopus 로고
    • A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits
    • B. Krauskopf and T. Riess, A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690.
    • (2008) Nonlinearity , vol.21 , pp. 1655-1690
    • Krauskopf, B.1    Riess, T.2
  • 37
    • 0036052772 scopus 로고    scopus 로고
    • Extending geometric singular perturbation theory to nonhyperbolic points - fold and canard points in two dimensions
    • PII S0036141099360919
    • M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyper-bolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314. (Pubitemid 33787285)
    • (2001) SIAM Journal on Mathematical Analysis , vol.33 , Issue.2 , pp. 286-314
    • Krupa, M.1    Szmolyan, P.2
  • 38
    • 0030637321 scopus 로고    scopus 로고
    • Fast and slow waves in the FitzHugh-Nagumo equation
    • Martin Krupa, Bjoern Sandstede and Peter Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation, Journal of Differential Equations, 133 (1997), 49-97. (Pubitemid 127164522)
    • (1997) Journal of Differential Equations , vol.133 , Issue.1 , pp. 49-97
    • Krupa, M.1    Sandstede, B.2    Szmolyan, P.3
  • 39
    • 0035839416 scopus 로고    scopus 로고
    • Relaxation oscillation and canard explosion
    • DOI 10.1006/jdeq.2000.3929, PII S0022039600939299
    • Martin Krupa and Peter Szmolyan, Relaxation oscillation and canard explosion, Journal of Differential Equations, 174 (2001), 312-368. (Pubitemid 33378729)
    • (2001) Journal of Differential Equations , vol.174 , Issue.2 , pp. 312-368
    • Krupa, M.1    Szmolyan, P.2
  • 40
    • 84972438215 scopus 로고
    • Using Melnikov's method to solve Shilnikov's problems
    • X-B Lin, Using Melnikov's method to solve Shilnikov's problems, Proc. Roy. Soc. Edinburgh, 116 (1990), 295-325.
    • (1990) Proc. Roy. Soc. Edinburgh , vol.116 , pp. 295-325
    • Lin, X.-B.1
  • 41
    • 0001188228 scopus 로고    scopus 로고
    • Multiple time scales and canards in a chemical oscillator
    • Eds.: Christopher K. R. T. Jones (Editor) and Alexander I. Khibnik (Editor)
    • Alexandra Milik and Peter Szmolyan, Multiple time scales and canards in a chemical oscillator, in "Multiple-Time-Scale Dynamical Systems," Eds.: Christopher K. R. T. Jones (Editor) and Alexander I. Khibnik (Editor), 2001.
    • (2001) Multiple-Time-Scale Dynamical Systems
    • Milik, A.1    Szmolyan, P.2
  • 43
    • 6344238035 scopus 로고
    • An active pulse transmission line simulating nerve axon
    • J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.
    • (1962) Proc. IRE , vol.50 , pp. 2061-2070
    • Nagumo, J.1    Arimoto, S.2    Yoshizawa, S.3
  • 45
    • 44949272344 scopus 로고
    • Transversal heteroclinic and homoclinic orbits in singular perturbation problems
    • Peter Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, Journal of Differential Equations, 92 (1991), 252-281.
    • (1991) Journal of Differential Equations , vol.92 , pp. 252-281
    • Szmolyan, P.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.