메뉴 건너뛰기




Volumn 225, Issue 2, 2006, Pages 381-410

Turning points and traveling waves in FitzHugh-Nagumo type equations

Author keywords

Exchange lemma; Traveling wave; Turning point

Indexed keywords


EID: 33646342480     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2005.10.006     Document Type: Article
Times cited : (33)

References (42)
  • 1
    • 0005235678 scopus 로고
    • A topological invariant arising in the stability analysis of traveling waves
    • Alexander J., Gardner R., and Jones C. A topological invariant arising in the stability analysis of traveling waves. J. Reine Angew. Math. 410 (1990) 167-212
    • (1990) J. Reine Angew. Math. , vol.410 , pp. 167-212
    • Alexander, J.1    Gardner, R.2    Jones, C.3
  • 3
    • 4243665470 scopus 로고
    • Euclidean symmetry and the dynamics of rotating spiral waves
    • Barkley D. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. E 72 (1994) 164-168
    • (1994) Phys. Rev. E , vol.72 , pp. 164-168
    • Barkley, D.1
  • 4
    • 0010863128 scopus 로고
    • A geometric approach to singular perturbation problems with application to nerve impulse equations
    • Carpenter G. A geometric approach to singular perturbation problems with application to nerve impulse equations. J. Differential Equations 23 (1977) 152-173
    • (1977) J. Differential Equations , vol.23 , pp. 152-173
    • Carpenter, G.1
  • 5
    • 3543088932 scopus 로고    scopus 로고
    • Ordinary Differential Equations with Applications
    • Springer-Verlag, New York
    • Chicone C. Ordinary Differential Equations with Applications. Texts Appl. Math. vol. 34 (1999), Springer-Verlag, New York
    • (1999) Texts Appl. Math. , vol.34
    • Chicone, C.1
  • 7
    • 0034634862 scopus 로고    scopus 로고
    • Center manifolds for general invariant sets
    • Chow S.-N., Liu W., and Yi Y. Center manifolds for general invariant sets. J. Differential Equations 168 (2000) 355-385
    • (2000) J. Differential Equations , vol.168 , pp. 355-385
    • Chow, S.-N.1    Liu, W.2    Yi, Y.3
  • 8
    • 0000997882 scopus 로고
    • The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations
    • Deng B. The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations. SIAM J. Math. Anal. 22 (1991) 1631-1650
    • (1991) SIAM J. Math. Anal. , vol.22 , pp. 1631-1650
    • Deng, B.1
  • 9
    • 0016578299 scopus 로고
    • Nerve axon equations: IV. The stable and the unstable impulse
    • Evans J.W. Nerve axon equations: IV. The stable and the unstable impulse. Indiana Univ. Math. J. 24 (1975) 1169-1190
    • (1975) Indiana Univ. Math. J. , vol.24 , pp. 1169-1190
    • Evans, J.W.1
  • 10
    • 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) 193-226
    • (1971) Indiana Univ. Math. J. , vol.21 , pp. 193-226
    • Fenichel, N.1
  • 11
    • 34250627892 scopus 로고
    • Geometric singular perturbation theory for ordinary differential equations
    • Fenichel N. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations 31 (1979) 53-98
    • (1979) J. Differential Equations , vol.31 , pp. 53-98
    • Fenichel, N.1
  • 12
    • 84975973968 scopus 로고
    • Diffusive waves in inhomogeneous media
    • Fife P.C. Diffusive waves in inhomogeneous media. Proc. Edinburgh Math. Soc. 32 (1989) 291-315
    • (1989) Proc. Edinburgh Math. Soc. , vol.32 , pp. 291-315
    • Fife, P.C.1
  • 13
    • 0002626726 scopus 로고
    • The generation and propagation of internal layers
    • Fife P.C., and Hsiao L. The generation and propagation of internal layers. J. Nonlinear Anal. TMA 12 (1988) 19-41
    • (1988) J. Nonlinear Anal. TMA , vol.12 , pp. 19-41
    • Fife, P.C.1    Hsiao, L.2
  • 14
    • 53349102813 scopus 로고
    • Impulses and physiological states in theoretical models of nerve membrane
    • FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1 (1961) 445-466
    • (1961) Biophys. J. , vol.1 , pp. 445-466
    • FitzHugh, R.1
  • 15
    • 0010928907 scopus 로고
    • Stability analysis for the slow traveling pulse of the FitzHugh-Nagumo system
    • Flores G. 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
  • 17
    • 0003285244 scopus 로고
    • Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
    • Springer-Verlag, New York
    • Gukenheimer J., and Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Appl. Math. Sci. vol. 42 (1983), Springer-Verlag, New York
    • (1983) Appl. Math. Sci. , vol.42
    • Gukenheimer, J.1    Holmes, P.2
  • 18
    • 0001371673 scopus 로고
    • On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations
    • Hastings S. On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations. Quart. J. Math. 27 (1976) 123-134
    • (1976) Quart. J. Math. , vol.27 , pp. 123-134
    • Hastings, S.1
  • 19
  • 20
    • 0000263519 scopus 로고
    • Construction of the FitzHugh-Nagumo pulse using differential forms
    • Patterns and Dynamics in Reactive Media. Minneapolis, MN, 1989, Springer, New York
    • Jones C., Kopell N., and Langer R. Construction of the FitzHugh-Nagumo pulse using differential forms. Patterns and Dynamics in Reactive Media. Minneapolis, MN, 1989. IMA Vol. Math. Appl. vol. 37 (1991), Springer, New York 101-115
    • (1991) IMA Vol. Math. Appl. , vol.37 , pp. 101-115
    • Jones, C.1    Kopell, N.2    Langer, R.3
  • 21
    • 84967743294 scopus 로고
    • Stability of the traveling wave solution of the FitzHugh-Nagumo system
    • Jones C. Stability of the traveling wave solution of the FitzHugh-Nagumo system. Trans. Amer. Math. Soc. 286 (1984) 431-469
    • (1984) Trans. Amer. Math. Soc. , vol.286 , pp. 431-469
    • Jones, C.1
  • 22
    • 0002316532 scopus 로고
    • Geometric singular perturbation theory
    • Springer-Verlag, New York
    • Jones C. Geometric singular perturbation theory. Lecture Notes in Math. vol. 1069 (1995), Springer-Verlag, New York 44-118
    • (1995) Lecture Notes in Math. , vol.1069 , pp. 44-118
    • Jones, C.1
  • 23
    • 0030528987 scopus 로고    scopus 로고
    • Tracking invariant manifolds up to exponentially small errors
    • Jones C., Kaper T., and Kopell N. Tracking invariant manifolds up to exponentially small errors. SIAM J. Math. Anal. 27 (1996) 558-577
    • (1996) SIAM J. Math. Anal. , vol.27 , pp. 558-577
    • Jones, C.1    Kaper, T.2    Kopell, N.3
  • 24
    • 0002809936 scopus 로고
    • Tracking invariant manifolds with differential forms in singularly perturbed systems
    • Jones C., and Kopell N. Tracking invariant manifolds with differential forms in singularly perturbed systems. J. Differential Equations 108 (1994) 64-88
    • (1994) J. Differential Equations , vol.108 , pp. 64-88
    • Jones, C.1    Kopell, N.2
  • 26
    • 38249018789 scopus 로고
    • Heteroclinic bifurcation and singularly perturbed boundary value problems
    • Lin X.-B. Heteroclinic bifurcation and singularly perturbed boundary value problems. J. Differential Equations 84 (1990) 319-382
    • (1990) J. Differential Equations , vol.84 , pp. 319-382
    • Lin, X.-B.1
  • 27
    • 0013331861 scopus 로고    scopus 로고
    • Exchange lemmas for singular perturbation problems with certain turning points
    • Liu W. Exchange lemmas for singular perturbation problems with certain turning points. J. Differential Equations 167 (2000) 134-180
    • (2000) J. Differential Equations , vol.167 , pp. 134-180
    • Liu, W.1
  • 28
    • 33646345647 scopus 로고    scopus 로고
    • W. Liu, Geometric singular perturbations for problems with multiple turning points: Invariant manifolds and exchange lemmas, in preparation
  • 29
    • 0022146559 scopus 로고
    • Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction-diffusion systems
    • Maginu K. Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction-diffusion systems. SIAM J. Appl. Math. 45 (1985) 750-774
    • (1985) SIAM J. Appl. Math. , vol.45 , pp. 750-774
    • Maginu, K.1
  • 30
    • 0001834849 scopus 로고
    • On the stability of the center for time periodic perturbations
    • Melnikov V.K. On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12 (1963) 1-57
    • (1963) Trans. Moscow Math. Soc. , vol.12 , pp. 1-57
    • Melnikov, V.K.1
  • 32
    • 6344238035 scopus 로고
    • An active pulse transmission line simulating nerve axon
    • Nagumo J., Arimoto S., and Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc. Inst. Radio Eng. 50 (1964) 2061-2070
    • (1964) Proc. Inst. Radio Eng. , vol.50 , pp. 2061-2070
    • Nagumo, J.1    Arimoto, S.2    Yoshizawa, S.3
  • 33
    • 21944451412 scopus 로고    scopus 로고
    • Stability of multiple-pulse solutions
    • Sandstede B. Stability of multiple-pulse solutions. Trans. Amer. Math. Soc. 350 (1998) 429-472
    • (1998) Trans. Amer. Math. Soc. , vol.350 , pp. 429-472
    • Sandstede, B.1
  • 34
    • 0032338132 scopus 로고    scopus 로고
    • Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation
    • Sandstede B. Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation. SIAM J. Math. Anal. 29 (1998) 183-207
    • (1998) SIAM J. Math. Anal. , vol.29 , pp. 183-207
    • Sandstede, B.1
  • 35
    • 67649363651 scopus 로고    scopus 로고
    • Stability of travelling waves
    • North-Holland, Amsterdam
    • Sandstede B. Stability of travelling waves. Handbook of Dynamical Systems, vol. 2 (2002), North-Holland, Amsterdam 983-1055
    • (2002) Handbook of Dynamical Systems, vol. 2 , pp. 983-1055
    • Sandstede, B.1
  • 36
    • 0033589602 scopus 로고    scopus 로고
    • Travelling waves in time almost periodic structures governed by bistable nonlinearities I. Stability and uniqueness
    • Shen W. Travelling waves in time almost periodic structures governed by bistable nonlinearities I. Stability and uniqueness. J. Differential Equations 159 (1999) 1-54
    • (1999) J. Differential Equations , vol.159 , pp. 1-54
    • Shen, W.1
  • 37
    • 0033589576 scopus 로고    scopus 로고
    • Travelling waves in time almost periodic structures governed by bistable nonlinearities II. Existence
    • Shen W. Travelling waves in time almost periodic structures governed by bistable nonlinearities II. Existence. J. Differential Equations 159 (1999) 55-101
    • (1999) J. Differential Equations , vol.159 , pp. 55-101
    • Shen, W.1
  • 38
    • 0028743559 scopus 로고
    • Invariant manifolds and singularly perturbed boundary value problems
    • Tin S.-K., Kopell N., and Jones C. Invariant manifolds and singularly perturbed boundary value problems. SIAM J. Numer. Anal. 31 (1994) 1558-1576
    • (1994) SIAM J. Numer. Anal. , vol.31 , pp. 1558-1576
    • Tin, S.-K.1    Kopell, N.2    Jones, C.3
  • 39
    • 36749117168 scopus 로고
    • Target patterns in a realistic model of the Belousov-Zhabotinskii reaction
    • Tyson J.J., and Fife P.C. Target patterns in a realistic model of the Belousov-Zhabotinskii reaction. J. Chem. Phys. 73 (1980) 2224-2237
    • (1980) J. Chem. Phys. , vol.73 , pp. 2224-2237
    • Tyson, J.J.1    Fife, P.C.2
  • 40
    • 4544321870 scopus 로고    scopus 로고
    • Parameter estimation of the Hodgkin-Huxley gating model: An inversion procedure
    • Wang G.J., and Beaumont J. Parameter estimation of the Hodgkin-Huxley gating model: An inversion procedure. SIAM J. Appl. Math. 64 (2004) 1249-1267
    • (2004) SIAM J. Appl. Math. , vol.64 , pp. 1249-1267
    • Wang, G.J.1    Beaumont, J.2
  • 41
    • 0021853023 scopus 로고
    • Stability of Fast Travelling Pulse Solutions of the FitzHugh-Nagumo Equation
    • Yanagida E. Stability of Fast Travelling Pulse Solutions of the FitzHugh-Nagumo Equation. J. Math. Biol. 22 (1985) 81-104
    • (1985) J. Math. Biol. , vol.22 , pp. 81-104
    • Yanagida, E.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.