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Volumn 39, Issue 1, 2013, Pages 175-192

Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations

Author keywords

Auto; Eckhaus; Hopf; Numerical continuation; Periodic traveling wave; Spectral stability; Wavetrain

Indexed keywords


EID: 84878397207     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-012-9273-0     Document Type: Article
Times cited : (30)

References (65)
  • 1
    • 0000588222 scopus 로고
    • Plane wave solutions to reaction-diffusion equations
    • Kopell, N., Howard, L. N.: Plane wave solutions to reaction-diffusion equations. Stud. Appl. Math. 52, 291-328 (1973).
    • (1973) Stud. Appl. Math. , vol.52 , pp. 291-328
    • Kopell, N.1    Howard, L.N.2
  • 2
    • 0036013603 scopus 로고    scopus 로고
    • The world of the complex Ginzburg-Landau equation
    • Aranson, I. S., Kramer, L.: The world of the complex Ginzburg-Landau equation. Rev. Modern Phys. 74, 99-143 (2002).
    • (2002) Rev. Modern Phys. , vol.74 , pp. 99-143
    • Aranson, I.S.1    Kramer, L.2
  • 3
    • 0037112236 scopus 로고    scopus 로고
    • Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework
    • Bridges, T. J., Derks, G., Gottwald, G.: Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Physica D 172, 190-216 (2002).
    • (2002) Physica D , vol.172 , pp. 190-216
    • Bridges, T.J.1    Derks, G.2    Gottwald, G.3
  • 4
    • 17744400704 scopus 로고    scopus 로고
    • Evans functions for integral neural field equations with Heaviside firing rate function
    • Coombes, S., Owen, M. R.: Evans functions for integral neural field equations with Heaviside firing rate function. SIAM J. Appl. Dyn. Syst. 34, 574-600 (2004).
    • (2004) SIAM J. Appl. Dyn. Syst. , vol.34 , pp. 574-600
    • Coombes, S.1    Owen, M.R.2
  • 5
    • 27144551687 scopus 로고    scopus 로고
    • Numerical evaluation of the Evans function by Magnus integration
    • Aparicio, N. D., Malham, S. J. A., Oliver, M.: Numerical evaluation of the Evans function by Magnus integration. BIT 45, 219-258 (2005).
    • (2005) Bit , vol.45 , pp. 219-258
    • Aparicio, N.D.1    Malham, S.J.A.2    Oliver, M.3
  • 8
    • 0016578299 scopus 로고
    • Nerve axon equations: IV The stable and unstable pulse
    • Evans, J. W.: Nerve axon equations: IV The stable and unstable pulse. Indiana Univ. Math. J. 24, 1169-1190 (1975).
    • (1975) Indiana Univ. Math. J. , vol.24 , pp. 1169-1190
    • Evans, J.W.1
  • 9
    • 0005235678 scopus 로고
    • A topological invariant arising in the stability analysis of travelling waves
    • Alexander, J., Gardner, R., Jones, C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167-212 (1990).
    • (1990) J. Reine Angew. Math. , vol.410 , pp. 167-212
    • Alexander, J.1    Gardner, R.2    Jones, C.3
  • 10
    • 0001307276 scopus 로고
    • On the structure of the spectra of periodic travelling waves
    • Gardner, R. A.: On the structure of the spectra of periodic travelling waves. J. Math. Pures Appl. 72, 415-439 (1993).
    • (1993) J. Math. Pures Appl. , vol.72 , pp. 415-439
    • Gardner, R.A.1
  • 11
    • 33750370251 scopus 로고    scopus 로고
    • Computing spectra of linear operators using the Floquet-Fourier-Hill method
    • Deconinck, B., Kutz, J. N.: Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219, 296-321 (2006).
    • (2006) J. Comput. Phys. , vol.219 , pp. 296-321
    • Deconinck, B.1    Kutz, J.N.2
  • 12
    • 33847169349 scopus 로고    scopus 로고
    • SpectrUW: a laboratory for the numerical exploration of spectra of linear operators
    • Deconinck, B., Kiyak, F., Carter, J. D., Kutz, J. N.: SpectrUW: a laboratory for the numerical exploration of spectra of linear operators. Math. Comput. Simul. 74, 370-379 (2007).
    • (2007) Math. Comput. Simul. , vol.74 , pp. 370-379
    • Deconinck, B.1    Kiyak, F.2    Carter, J.D.3    Kutz, J.N.4
  • 13
    • 34248583204 scopus 로고    scopus 로고
    • Computing absolute and essential spectra using continuation
    • Rademacher, J. D. M., Sandstede, B., Scheel, A.: Computing absolute and essential spectra using continuation. Physica D 229, 166-183 (2007).
    • (2007) Physica D , vol.229 , pp. 166-183
    • Rademacher, J.D.M.1    Sandstede, B.2    Scheel, A.3
  • 14
    • 33644903758 scopus 로고    scopus 로고
    • From trigger to phase waves and back again
    • Bordiougov, G., Engel, H.: From trigger to phase waves and back again. Physica D 215, 25-37 (2006).
    • (2006) Physica D , vol.215 , pp. 25-37
    • Bordiougov, G.1    Engel, H.2
  • 15
    • 33847621215 scopus 로고    scopus 로고
    • Wave trains in an excitable FitzHugh-Nagumo model: bistable dispersion relation and formation of isolas
    • Röder, G., Bordyugov, G., Engel, H., Falcke, M.: Wave trains in an excitable FitzHugh-Nagumo model: bistable dispersion relation and formation of isolas. Phys. Rev. E 75(3), 036202 (2007).
    • (2007) Phys.Rev. , vol.75 , Issue.3 , pp. 036202
    • Röder, G.1    Bordyugov, G.2    Engel, H.3    Falcke, M.4
  • 16
    • 35649003882 scopus 로고    scopus 로고
    • The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems
    • Smith, M. J., Sherratt, J. A.: The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems. Physica D 236, 90-103 (2007).
    • (2007) Physica D , vol.236 , pp. 90-103
    • Smith, M.J.1    Sherratt, J.A.2
  • 17
    • 40749098198 scopus 로고    scopus 로고
    • Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models
    • Sherratt, J. A., Smith, M. J.: Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models. J. R. Soc. Interface 5, 483-505 (2008).
    • (2008) J. R. Soc. Interface , vol.5 , pp. 483-505
    • Sherratt, J.A.1    Smith, M.J.2
  • 18
    • 83555172511 scopus 로고    scopus 로고
    • Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations
    • Sherratt, J. A.: Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations. Appl. Math. Comput. 218, 4684-4694 (2012).
    • (2012) Appl. Math. Comput. , vol.218 , pp. 4684-4694
    • Sherratt, J.A.1
  • 21
    • 0035892848 scopus 로고    scopus 로고
    • Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation
    • Brusch, L., Torcini, A., van Hecke, M., Zimmermann, M. G., Bär, M.: Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation. Physica D 160, 127-148 (2001).
    • (2001) Physica D , vol.160 , pp. 127-148
    • Brusch, L.1    Torcini, A.2    van Hecke, M.3    Zimmermann, M.G.4    Bär, M.5
  • 22
    • 67650468398 scopus 로고    scopus 로고
    • Locating the transition from periodic oscillations to spatiotemporal chaos in the wake of invasion
    • Sherratt, J. A., Smith, M. J., Rademacher, J. D. M.: Locating the transition from periodic oscillations to spatiotemporal chaos in the wake of invasion. Proc. Natl. Acad. Sci. USA 106, 10890-10895 (2009).
    • (2009) Proc. Natl. Acad. Sci. USA , vol.106 , pp. 10890-10895
    • Sherratt, J.A.1    Smith, M.J.2    Rademacher, J.D.M.3
  • 23
    • 67649363651 scopus 로고    scopus 로고
    • Stability of travelling waves
    • B. Fiedler (Ed.), Amsterdam: North-Holland
    • Sandstede, B.: Stability of travelling waves. In: Fiedler, B. (ed.) Handbook of Dynamical Systems II, pp. 983-1055. North-Holland, Amsterdam (2002).
    • (2002) Handbook of Dynamical Systems II , pp. 983-1055
    • Sandstede, B.1
  • 24
    • 34748878233 scopus 로고    scopus 로고
    • Instabilities of wave trains and Turing patterns in large domains
    • Rademacher, J. D. M., Scheel, A.: Instabilities of wave trains and Turing patterns in large domains. Int. J. Bifur. Chaos 17, 2679-2691 (2007).
    • (2007) Int. J. Bifur. Chaos , vol.17 , pp. 2679-2691
    • Rademacher, J.D.M.1    Scheel, A.2
  • 28
    • 84968486225 scopus 로고
    • Difference approximation for boundary and eigenvalue problems for ordinary differential equations
    • Kreiss, H. O.: Difference approximation for boundary and eigenvalue problems for ordinary differential equations. Math. Comput. 26, 605-624 (1972).
    • (1972) Math. Comput. , vol.26 , pp. 605-624
    • Kreiss, H.O.1
  • 29
    • 84966234694 scopus 로고
    • Collocation approximation to eigenvalues of an ordinary differential equation: the principle of the thing
    • de Boor, C., Swartz, B.: Collocation approximation to eigenvalues of an ordinary differential equation: the principle of the thing. Math. Comput. 35, 679-694 (1980).
    • (1980) Math. Comput. , vol.35 , pp. 679-694
    • de Boor, C.1    Swartz, B.2
  • 30
    • 0001309976 scopus 로고
    • The spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators
    • Chatelin, F.: The spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators. SIAM Rev. 23, 495-522 (1981).
    • (1981) SIAM Rev. , vol.23 , pp. 495-522
    • Chatelin, F.1
  • 32
    • 0001755071 scopus 로고
    • Auto, a program for the automatic bifurcation analysis of autonomous systems
    • Doedel, E. J.: Auto, a program for the automatic bifurcation analysis of autonomous systems. Cong. Numer. 30, 265-384 (1981).
    • (1981) Cong. Numer. , vol.30 , pp. 265-384
    • Doedel, E.J.1
  • 33
    • 0000378937 scopus 로고
    • Numerical analysis and control of bifurcation problems: (I) bifurcation in finite dimensions
    • Doedel, E. J., Keller, H. B., Kernévez, J. P.: Numerical analysis and control of bifurcation problems: (I) bifurcation in finite dimensions. Int. J. Bifurc. Chaos 1, 493-520 (1991).
    • (1991) Int. J. Bifurc. Chaos , vol.1 , pp. 493-520
    • Doedel, E.J.1    Keller, H.B.2    Kernévez, J.P.3
  • 34
    • 77958055560 scopus 로고    scopus 로고
    • Numerical continuation of branch points of equilibria and periodic orbits
    • E. J. Doedel, G. Domokos, and I. G. Kevrekidis (Eds.), Singapore: World Scientific
    • Doedel, E. J., Govaerts, W., Kuznetsov, Y. A., Dhooge, A.: Numerical continuation of branch points of equilibria and periodic orbits. In: Doedel, E. J., Domokos, G., Kevrekidis, I. G. (eds.) Modelling and Computations in Dynamical Systems, pp. 145-164. World Scientific, Singapore (2006).
    • (2006) Modelling and Computations in Dynamical Systems , pp. 145-164
    • Doedel, E.J.1    Govaerts, W.2    Kuznetsov, Y.A.3    Dhooge, A.4
  • 36
    • 0032166374 scopus 로고    scopus 로고
    • Calculation of weights in finite difference formulas
    • Fornberg, B.: Calculation of weights in finite difference formulas. SIAM Rev. 40, 685-691 (1998).
    • (1998) SIAM Rev. , vol.40 , pp. 685-691
    • Fornberg, B.1
  • 38
    • 43049104801 scopus 로고    scopus 로고
    • Localized pattern formation with a large-scale mode: slanted snaking
    • Dawes, J. H. P.: Localized pattern formation with a large-scale mode: slanted snaking. SIAM J. Appl. Dyn. Syst. 7, 186-206 (2008).
    • (2008) SIAM J. Appl. Dyn. Syst. , vol.7 , pp. 186-206
    • Dawes, J.H.P.1
  • 39
    • 68649095203 scopus 로고    scopus 로고
    • Modulated and localized states in a finite domain
    • Dawes, J. H. P.: Modulated and localized states in a finite domain. SIAM J. Appl. Dyn. Syst. 8, 909-930 (2009).
    • (2009) SIAM J. Appl. Dyn. Syst. , vol.8 , pp. 909-930
    • Dawes, J.H.P.1
  • 40
    • 65249145224 scopus 로고    scopus 로고
    • Continuation of connecting orbits in 3D-ODEs: (II) cycle-to-cycle connections
    • Doedel, E. J., Kooi, B. W., van Voorn, G. A. K., Kuznetsov, Y. A.: Continuation of connecting orbits in 3D-ODEs: (II) cycle-to-cycle connections. Int. J. Bifurc. Chaos 19, 159-169 (2009).
    • (2009) Int. J. Bifurc. Chaos , vol.19 , pp. 159-169
    • Doedel, E.J.1    Kooi, B.W.2    van Voorn, G.A.K.3    Kuznetsov, Y.A.4
  • 41
    • 0033546065 scopus 로고    scopus 로고
    • Regular and irregular patterns in semiarid vegetation
    • Klausmeier, C. A.: Regular and irregular patterns in semiarid vegetation. Science 284, 1826-1828 (1999).
    • (1999) Science , vol.284 , pp. 1826-1828
    • Klausmeier, C.A.1
  • 42
    • 0028994383 scopus 로고
    • Positive interactions among plants
    • Callaway, R. M.: Positive interactions among plants. Bot. Rev. 61, 306-349 (1995).
    • (1995) Bot. Rev. , vol.61 , pp. 306-349
    • Callaway, R.M.1
  • 43
    • 0007600349 scopus 로고
    • The influence of land management and soil characteristics on infiltration and the occurrence of overland flow
    • Hills, R. C.: The influence of land management and soil characteristics on infiltration and the occurrence of overland flow. J. Hydrol. 13, 163-181 (1971).
    • (1971) J. Hydrol. , vol.13 , pp. 163-181
    • Hills, R.C.1
  • 44
    • 0033910820 scopus 로고    scopus 로고
    • Multiscale soil and vegetation patchiness along a gradient of herbivore impact in a semi-arid grazing system in West Africa
    • Rietkerk, M., Ketner, P., Burger, J., Hoorens, B., Olff, H.: Multiscale soil and vegetation patchiness along a gradient of herbivore impact in a semi-arid grazing system in West Africa. Plant Ecol. 148, 207-224 (2000).
    • (2000) Plant Ecol. , vol.148 , pp. 207-224
    • Rietkerk, M.1    Ketner, P.2    Burger, J.3    Hoorens, B.4    Olff, H.5
  • 45
    • 0032796361 scopus 로고    scopus 로고
    • Soil and water components of banded vegetation patterns
    • Valentin, C., d'Herbès, J. M., Poesen, J.: Soil and water components of banded vegetation patterns. Catena 37, 1-24 (1999).
    • (1999) Catena , vol.37 , pp. 1-24
    • Valentin, C.1    d'Herbès, J.M.2    Poesen, J.3
  • 47
    • 0001507360 scopus 로고    scopus 로고
    • Vegetation dynamics: recruitment and regeneration in two-phase mosaics
    • D. J. Tongway, C. Valentin, and J. Seghieri (Eds.), New York: Springer
    • Montaña, C., Seghieri, J., Cornet, A.: Vegetation dynamics: recruitment and regeneration in two-phase mosaics. In: Tongway, D. J., Valentin, C., Seghieri, J. (eds.) Banded Vegetation Patterning in Arid and Semi-Arid Environments, pp. 132-145. Springer, New York (2001).
    • (2001) Banded Vegetation Patterning in Arid and Semi-Arid Environments , pp. 132-145
    • Montana, C.1    Seghieri, J.2    Cornet, A.3
  • 48
    • 0001896948 scopus 로고    scopus 로고
    • Theories on the origins, maintainance, dynamics, and functioning of banded landscapes
    • D. J. Tongway, C. Valentin, and J. Seghieri (Eds.), New York: Springer
    • Tongway, D. J., Ludwig, J. A.: Theories on the origins, maintainance, dynamics, and functioning of banded landscapes. In: Tongway, D. J., Valentin, C., Seghieri, J. (eds.) Banded Vegetation Patterning in Arid and Semi-Arid Environments, pp. 20-31. Springer, New York (2001).
    • (2001) Banded Vegetation Patterning in Arid and Semi-Arid Environments , pp. 20-31
    • Tongway, D.J.1    Ludwig, J.A.2
  • 49
    • 80755130540 scopus 로고    scopus 로고
    • Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments II: patterns with the largest possible propagation speeds
    • Sherratt, J. A.: Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments II: patterns with the largest possible propagation speeds. Proc. R. Soc. Lond. A 467, 3272-3294 (2011).
    • (2011) Proc. R. Soc. Lond. A , vol.467 , pp. 3272-3294
    • Sherratt, J.A.1
  • 50
    • 22844447274 scopus 로고    scopus 로고
    • An analysis of vegetation stripe formation in semi-arid landscapes
    • Sherratt, J. A.: An analysis of vegetation stripe formation in semi-arid landscapes. J. Math. Biol. 51, 183-197 (2005).
    • (2005) J. Math. Biol. , vol.51 , pp. 183-197
    • Sherratt, J.A.1
  • 51
    • 78149344290 scopus 로고    scopus 로고
    • Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments I
    • Sherratt, J. A.: Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments I. Nonlinearity 23, 2657-2675 (2010).
    • (2010) Nonlinearity , vol.23 , pp. 2657-2675
    • Sherratt, J.A.1
  • 53
    • 84873408969 scopus 로고    scopus 로고
    • Rise and fall of periodic patterns for a generalized Klausmeier-Gray-Scott model
    • doi: 10. 1007/s00332-012-9139-0
    • van der Stelt, S., Doelman, A., Hek, G., Rademacher, J. D. M.: Rise and fall of periodic patterns for a generalized Klausmeier-Gray-Scott model. J. Nonlinear Sci. (2012). doi: 10. 1007/s00332-012-9139-0.
    • (2012) J. Nonlinear Sci
    • van der Stelt, S.1    Doelman, A.2    Hek, G.3    Rademacher, J.D.M.4
  • 55
    • 0030947102 scopus 로고    scopus 로고
    • On the origin of tiger bush
    • Lefever, R., Lejeune, O.: On the origin of tiger bush. Bull. Math. Biol. 59, 263-294 (1997).
    • (1997) Bull. Math. Biol. , vol.59 , pp. 263-294
    • Lefever, R.1    Lejeune, O.2
  • 57
    • 34548443143 scopus 로고    scopus 로고
    • Diversity of vegetation patterns and desertification
    • von Hardenberg, J., Meron, E., Shachak, M., Zarmi, Y.: Diversity of vegetation patterns and desertification. Phys. Rev. Lett. 87(19), 198101 (2001).
    • (2001) Phys. Rev. Lett. , vol.87 , Issue.19 , pp. 198101
    • von Hardenberg, J.1    Meron, E.2    Shachak, M.3    Zarmi, Y.4
  • 60
    • 33750998491 scopus 로고    scopus 로고
    • Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments
    • Sherratt, J. A., Lord, G. J.: Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments. Theor. Popul. Biol. 71, 1-11 (2007).
    • (2007) Theor. Popul. Biol. , vol.71 , pp. 1-11
    • Sherratt, J.A.1    Lord, G.J.2
  • 61
    • 0030219251 scopus 로고    scopus 로고
    • Nonlinear chemical dynamics: oscillations, patterns and chaos
    • Epstein, I. R., Showalter, K.: Nonlinear chemical dynamics: oscillations, patterns and chaos. J. Phys. Chem. 100, 13132-13147 (1996).
    • (1996) J. Phys. Chem. , vol.100 , pp. 13132-13147
    • Epstein, I.R.1    Showalter, K.2
  • 62
    • 46449086521 scopus 로고    scopus 로고
    • Design and control of patterns in reaction-diffusion systems
    • Vanag, V. K., Epstein, I. R.: Design and control of patterns in reaction-diffusion systems. Chaos 18(2), 026107 (2008).
    • (2008) Chaos , vol.18 , Issue.2 , pp. 026107
    • Vanag, V.K.1    Epstein, I.R.2
  • 64
    • 77949491358 scopus 로고    scopus 로고
    • Anomalous dispersion in the Belousov-Zhabotinsky reaction: experiments and modeling
    • Bordyugov, G., Fischer, N., Engel, H., Manz, N., Steinbock, O.: Anomalous dispersion in the Belousov-Zhabotinsky reaction: experiments and modeling. Physica D 239, 766-775 (2010).
    • (2010) Physica D , vol.239 , pp. 766-775
    • Bordyugov, G.1    Fischer, N.2    Engel, H.3    Manz, N.4    Steinbock, O.5
  • 65
    • 33751553761 scopus 로고
    • Analysis of the modified complete Oregonator accounting for oxygen sensitivity and photosensitivity of Belousov-Zhabotinsky reaction
    • Krug, H.-J., Pohlmann, L., Kuhnert, L.: Analysis of the modified complete Oregonator accounting for oxygen sensitivity and photosensitivity of Belousov-Zhabotinsky reaction. J. Phys. Chem. 94, 4862-4865 (1990).
    • (1990) J. Phys. Chem. , vol.94 , pp. 4862-4865
    • Krug, H.-J.1    Pohlmann, L.2    Kuhnert, L.3


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