Advances in Computational Mathematics

, Volume 39, Issue 1, pp 175–192 | Cite as

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

  • Jonathan A. SherrattEmail author


A variety of numerical methods are available for determining the stability of a given solution of a partial differential equation. However for a family of solutions, calculation of boundaries in parameter space between stable and unstable solutions remains a major challenge. This paper describes an algorithm for the calculation of such stability boundaries, for the case of periodic travelling wave solutions of spatially extended local dynamical systems. The algorithm is based on numerical continuation of the spectrum. It is implemented in a fully automated way by the software package wavetrain, and two examples of its use are presented. One example is the Klausmeier model for banded vegetation in semi-arid environments, for which the change in stability is of Eckhaus (sideband) type; the other is the two-component Oregonator model for the photosensitive Belousov–Zhabotinskii reaction, for which the change in stability is of Hopf type.


Numerical continuation Periodic traveling wave Wavetrain Auto Eckhaus Hopf Spectral stability 

Mathematics Subject Classifications (2010)

65P99 35P05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kopell, N., Howard, L.N.: Plane wave solutions to reaction–diffusion equations. Stud. Appl. Math. 52, 291–328 (1973)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aranson, I.S., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Modern Phys. 74, 99–143 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aparicio, N.D., Malham, S.J.A., Oliver, M.: Numerical evaluation of the Evans function by Magnus integration. BIT 45, 219–258 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ledoux, V., Malham, S.J.A., Niesen, J., Thümmler, V.: Computing stability of multi-dimensional travelling waves. SIAM J. Appl. Dyn. Syst. 8, 480–507 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ledoux, V., Malham, S.J.A., Thümmler, V.: Grassmannian spectral shooting. Math. Comput. 79, 1585–1619 (2010)zbMATHCrossRefGoogle Scholar
  8. 8.
    Evans, J.W.: Nerve axon equations: IV The stable and unstable pulse. Indiana Univ. Math. J. 24, 1169–1190 (1975)zbMATHCrossRefGoogle Scholar
  9. 9.
    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)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gardner, R.A.: On the structure of the spectra of periodic travelling waves. J. Math. Pures Appl. 72, 415–439 (1993)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Deconinck, B., Kutz, J.N.: Computing spectra of linear operators using the Floquet–Fourier–Hill method. J. Comput. Phys. 219, 296–321 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Rademacher, J.D.M., Sandstede, B., Scheel, A.: Computing absolute and essential spectra using continuation. Physica D 229, 166–183 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bordiougov, G., Engel, H.: From trigger to phase waves and back again. Physica D 215, 25–37 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    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)MathSciNetGoogle Scholar
  16. 16.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Sherratt, J.A.: Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations. Appl. Math. Comput. 218, 4684–4694 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Trefethen, L.N., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)Google Scholar
  20. 20.
    Janiaud, B., Pumir, A., Bensimon, D., Croquette, V., Richter, H., Kramer, L.: The Eckhaus instability for traveling waves. Physica D 55, 269–286 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    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)zbMATHCrossRefGoogle Scholar
  23. 23.
    Sandstede, B.: Stability of travelling waves. In: Fiedler, B. (ed.) Handbook of Dynamical Systems II, pp. 983–1055. North-Holland, Amsterdam (2002)CrossRefGoogle Scholar
  24. 24.
    Rademacher, J.D.M., Scheel, A.: Instabilities of wave trains and Turing patterns in large domains. Int. J. Bifur. Chaos 17, 2679–2691 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Doedel, E.J., Kernevez, J.P.: A numerical analysis of wave phenomena in a reaction diffusion model. In: Othmer, H.G. (ed.) Nonlinear Oscillations in Biology and Chemistry (Lecture Notes in Biomathematics 66), pp. 261–273. Springer, Berlin (1986)CrossRefGoogle Scholar
  26. 26.
    Doedel, E.J., Kernevez, J.P.: Auto: software for continuation and bifurcation problems in ordinary differential equations. Applied Mathematics Report, California Institute of Technology, Pasadena (1986). See also pp. 374–388 of
  27. 27.
    Atkinson, F.V.: Discrete and Continuous Boundary Problems. Academic, New York (1964)zbMATHGoogle Scholar
  28. 28.
    Kreiss, H.O.: Difference approximation for boundary and eigenvalue problems for ordinary differential equations. Math. Comput. 26, 605–624 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    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)zbMATHCrossRefGoogle Scholar
  30. 30.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Merchant, S.M.: Spatiotemporal patterns in mathematical models for predator invasions. PhD thesis, University of British Columbia (2009).
  32. 32.
    Doedel, E.J.: Auto, a program for the automatic bifurcation analysis of autonomous systems. Cong. Numer. 30, 265–384 (1981)MathSciNetGoogle Scholar
  33. 33.
    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)zbMATHCrossRefGoogle Scholar
  34. 34.
    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)Google Scholar
  35. 35.
    Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du, J., Croz, Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: Lapack Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999)Google Scholar
  36. 36.
    Fornberg, B.: Calculation of weights in finite difference formulas. SIAM Rev. 40, 685–691 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Champneys, A.R., Kuznetsov, Y.A., Sandstede, B.: A numerical toolbox for homoclinic bifurcation analysis. Int. J. Bifur. Chaos 6, 867–887 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Dawes, J.H.P.: Localized pattern formation with a large-scale mode: slanted snaking. SIAM J. Appl. Dyn. Syst. 7, 186–206 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Dawes, J.H.P.: Modulated and localized states in a finite domain. SIAM J. Appl. Dyn. Syst. 8, 909–930 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    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)zbMATHCrossRefGoogle Scholar
  41. 41.
    Klausmeier, C.A.: Regular and irregular patterns in semiarid vegetation. Science 284, 1826–1828 (1999)CrossRefGoogle Scholar
  42. 42.
    Callaway, R.M.: Positive interactions among plants. Bot. Rev. 61, 306–349 (1995)CrossRefGoogle Scholar
  43. 43.
    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)CrossRefGoogle Scholar
  44. 44.
    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)CrossRefGoogle Scholar
  45. 45.
    Valentin, C., d’Herbès, J.M., Poesen, J.: Soil and water components of banded vegetation patterns. Catena 37, 1–24 (1999)CrossRefGoogle Scholar
  46. 46.
    Deblauwe, V.: Modulation des structures de végétation auto-organisées en milieu aride [trans: self-organized vegetation pattern modulation in arid climates]. PhD thesis, Université Libre de Bruxelles. (2010)
  47. 47.
    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)CrossRefGoogle Scholar
  48. 48.
    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)CrossRefGoogle Scholar
  49. 49.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Sherratt, J.A.: An analysis of vegetation stripe formation in semi-arid landscapes. J. Math. Biol. 51, 183–197 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments I. Nonlinearity 23, 2657–2675 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments III: the transition between homoclinic solutions (2012, submitted)Google Scholar
  53. 53.
    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
  54. 54.
    Doelman, A., Rademacher, J.D.M., van der Stelt, S.: Hopf dances near the tips of Busse balloons. Discrete Continuous Dyn. Syst., Ser. S 5, 61–92 (2012)zbMATHCrossRefGoogle Scholar
  55. 55.
    Lefever, R., Lejeune, O.: On the origin of tiger bush. Bull. Math. Biol. 59, 263–294 (1997)zbMATHCrossRefGoogle Scholar
  56. 56.
    HilleRisLambers, R., Rietkerk, M., van de Bosch, F., Prins, H.H.T., de Kroon, H.: Vegetation pattern formation in semi-arid grazing systems. Ecology 82, 50–61 (2001)CrossRefGoogle Scholar
  57. 57.
    von Hardenberg, J., Meron, E., Shachak, M., Zarmi, Y.: Diversity of vegetation patterns and desertification. Phys. Rev. Lett. 87(19), 198101 (2001)CrossRefGoogle Scholar
  58. 58.
    Rietkerk, M., Boerlijst, M.C., van Langevelde, F., HilleRisLambers, R., van de Koppel, J., Prins, H.H.T., de Roos, A.: Self-organisation of vegetation in arid ecosystems. Am. Nat. 160, 524–530 (2002)CrossRefGoogle Scholar
  59. 59.
    Gilad, E., von Hardenberg, J., Provenzale, A., Shachak, M., Meron, E.: A mathematical model of plants as ecosystem engineers. J. Theor. Biol. 244, 680–691 (2007)CrossRefGoogle Scholar
  60. 60.
    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)zbMATHCrossRefGoogle Scholar
  61. 61.
    Epstein, I.R., Showalter, K.: Nonlinear chemical dynamics: oscillations, patterns and chaos. J. Phys. Chem. 100, 13132–13147 (1996)CrossRefGoogle Scholar
  62. 62.
    Vanag, V.K., Epstein, I.R.: Design and control of patterns in reaction–diffusion systems. Chaos 18(2), 026107 (2008)CrossRefGoogle Scholar
  63. 63.
    Kapral, R., Showalter, K. (ed.): Chemical Waves and Patterns. Springer, New York (1995)Google Scholar
  64. 64.
    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)zbMATHCrossRefGoogle Scholar
  65. 65.
    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)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Maxwell Institute for Mathematical SciencesHeriot-Watt UniversityEdinburghUK

Personalised recommendations