Skip to main content
SpringerLink
Log in

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

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kopell, N., Howard, L.N.: Plane wave solutions to reaction–diffusion equations. Stud. Appl. Math. 52, 291–328 (1973)

    MathSciNet  MATH  Google Scholar 

  2. Aranson, I.S., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Modern Phys. 74, 99–143 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  5. Aparicio, N.D., Malham, S.J.A., Oliver, M.: Numerical evaluation of the Evans function by Magnus integration. BIT 45, 219–258 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ledoux, V., Malham, S.J.A., Thümmler, V.: Grassmannian spectral shooting. Math. Comput. 79, 1585–1619 (2010)

    Article  MATH  Google Scholar 

  8. Evans, J.W.: Nerve axon equations: IV The stable and unstable pulse. Indiana Univ. Math. J. 24, 1169–1190 (1975)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  10. Gardner, R.A.: On the structure of the spectra of periodic travelling waves. J. Math. Pures Appl. 72, 415–439 (1993)

    MathSciNet  MATH  Google Scholar 

  11. Deconinck, B., Kutz, J.N.: Computing spectra of linear operators using the Floquet–Fourier–Hill method. J. Comput. Phys. 219, 296–321 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rademacher, J.D.M., Sandstede, B., Scheel, A.: Computing absolute and essential spectra using continuation. Physica D 229, 166–183 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bordiougov, G., Engel, H.: From trigger to phase waves and back again. Physica D 215, 25–37 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Trefethen, L.N., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  23. Sandstede, B.: Stability of travelling waves. In: Fiedler, B. (ed.) Handbook of Dynamical Systems II, pp. 983–1055. North-Holland, Amsterdam (2002)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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 http://cmvl.cs.concordia.ca/courses/comp-6361/fall-2011/notes.pdf

  27. Atkinson, F.V.: Discrete and Continuous Boundary Problems. Academic, New York (1964)

    MATH  Google Scholar 

  28. Kreiss, H.O.: Difference approximation for boundary and eigenvalue problems for ordinary differential equations. Math. Comput. 26, 605–624 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  31. Merchant, S.M.: Spatiotemporal patterns in mathematical models for predator invasions. PhD thesis, University of British Columbia (2009). http://www.iam.ubc.ca/theses/SandraMerchant/SMerchant_PhD_Thesis.pdf

  32. Doedel, E.J.: Auto, a program for the automatic bifurcation analysis of autonomous systems. Cong. Numer. 30, 265–384 (1981)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. 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. Fornberg, B.: Calculation of weights in finite difference formulas. SIAM Rev. 40, 685–691 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  37. Champneys, A.R., Kuznetsov, Y.A., Sandstede, B.: A numerical toolbox for homoclinic bifurcation analysis. Int. J. Bifur. Chaos 6, 867–887 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. Dawes, J.H.P.: Localized pattern formation with a large-scale mode: slanted snaking. SIAM J. Appl. Dyn. Syst. 7, 186–206 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Dawes, J.H.P.: Modulated and localized states in a finite domain. SIAM J. Appl. Dyn. Syst. 8, 909–930 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  41. Klausmeier, C.A.: Regular and irregular patterns in semiarid vegetation. Science 284, 1826–1828 (1999)

    Article  Google Scholar 

  42. Callaway, R.M.: Positive interactions among plants. Bot. Rev. 61, 306–349 (1995)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  45. Valentin, C., d’Herbès, J.M., Poesen, J.: Soil and water components of banded vegetation patterns. Catena 37, 1–24 (1999)

    Article  Google Scholar 

  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. http://theses.ulb.ac.be/ETD-db/collection/available/ULBetd-04122010-093151/ (2010)

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  50. Sherratt, J.A.: An analysis of vegetation stripe formation in semi-arid landscapes. J. Math. Biol. 51, 183–197 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  51. Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments I. Nonlinearity 23, 2657–2675 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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. 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)

    Article  MATH  Google Scholar 

  55. Lefever, R., Lejeune, O.: On the origin of tiger bush. Bull. Math. Biol. 59, 263–294 (1997)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  57. von Hardenberg, J., Meron, E., Shachak, M., Zarmi, Y.: Diversity of vegetation patterns and desertification. Phys. Rev. Lett. 87(19), 198101 (2001)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  61. Epstein, I.R., Showalter, K.: Nonlinear chemical dynamics: oscillations, patterns and chaos. J. Phys. Chem. 100, 13132–13147 (1996)

    Article  Google Scholar 

  62. Vanag, V.K., Epstein, I.R.: Design and control of patterns in reaction–diffusion systems. Chaos 18(2), 026107 (2008)

    Article  Google Scholar 

  63. Kapral, R., Showalter, K. (ed.): Chemical Waves and Patterns. Springer, New York (1995)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Jonathan A. Sherratt

Authors
  1. Jonathan A. Sherratt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan A. Sherratt.

Additional information

Communicated by: A. Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sherratt, J.A. Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations. Adv Comput Math 39, 175–192 (2013). https://doi.org/10.1007/s10444-012-9273-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-012-9273-0

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation