Skip to main content
SpringerLink
Log in

Boundary Effects on the Structural Stability of Stationary Patterns in a Bistable Reaction-Diffusion System

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We study a piecewise linear version of a one-component, two-dimensional bistable reaction-diffusion system subjected to partially reflecting boundary conditions, with the aim of analyzing the structural stability of its stationary patterns. Dirichlet and Neumann boundary conditions are included as limiting cases. We find a critical line in the space of the parameters which divides different dynamical behaviors. That critical line merges as the locus of the coalescence of metastable and unstable nonuniform structures.

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. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1976).

    Google Scholar 

  2. P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems (Springer-Verlag, Berlin, 1979).

    Google Scholar 

  3. P. C. Fife, Current topics in reaction-diffusion systems, in Nonequilibrium Cooperative Phenomena in Physics and Related Fields, M. G. Velarde, ed. (Plenum Press, New York, 1984).

    Google Scholar 

  4. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65:851 (1993).

    Google Scholar 

  5. E. Meron, Phys. Rep. 218:1 (1992).

    Google Scholar 

  6. G. Nicolis, T. Erneux, and M. Herschkowitz-Kaufman, Pattern formation in reacting and diffusing systems, in Advances in Chemical Physics, Vol. 38, I. Prigogine and S. Rice, eds. (Wiley, New York, 1978).

    Google Scholar 

  7. A. S. Mikhailov, Foundations of Synergetics 1 (Springer-Verlag, Berlin, 1990).

    Google Scholar 

  8. J. D. Murray, Mathematical Biology (Springer-Verlag, Berlin, 1985).

    Google Scholar 

  9. H. S. Wio, An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics (World Scientific, Singapore, 1994).

    Google Scholar 

  10. C. Schat and H. S. Wio, Physica A 180:295 (1992).

    Google Scholar 

  11. S. A. Hassan, M. N. Kuperman, H. S. Wio, and D. H. Zanette, Physica A 206:380 (1994).

    Google Scholar 

  12. S. A. Hassan, D. H. Zanette, and H. S. Wio, J. Phys. A 27:5129 (1994).

    Google Scholar 

  13. D. Zanette, H. Wio, and R. Deza, Phys. Rev. E 52:129 (1995).

    Google Scholar 

  14. H. S. Wio, G. Izús, O. Ramírez, R. Deza, and C. Borzi, J. Phys. A: Math. Gen. 26:4281 (1993).

    Google Scholar 

  15. G. Izús, R. Deza, O. Ramírez, H. Wio, D. Zanette, and C. Borzi, Phys. Rev. E 52:129 (1995).

    Google Scholar 

  16. G. Izús, H. Wio, J. Reyes de Rueda, O. Ramírez, and R. Deza, Int. J. Mod. Phys. B 10:1273 (1996).

    Google Scholar 

  17. G. Izús, R. Deza, H. Wio, and C. Borzi, Phys. Rev. E 55:4005 (1997).

    Google Scholar 

  18. G. Izús, R. Deza, C. Borzi, and H. Wio, Physica A 237:135 (1997).

    Google Scholar 

  19. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1984).

    Google Scholar 

  20. T. Boddington, P. Gray, and C. Wake, Proc. R. Soc. Lond. A 357:403 (1977).

    Google Scholar 

  21. S. P. Fedotov, Phys. Lett. A 176:220 (1993).

    Google Scholar 

  22. V. Castest, E. Dulos, J. Boissonade, and P. de Keeper, Phys. Rev. Lett. 64:2953 (1990).

    Google Scholar 

  23. Q. Ouyang and H. Swinney, Nature 352 (1991).

  24. G. Nicolis, Introduction to Nonlinear Science (Cambridge University Press, Cambridge, 1995).

    Google Scholar 

  25. G. Izús, O. Ramírez, R. Deza, and H. Wio, J. Chem. Phys. 105:10424 (1996).

    Google Scholar 

  26. B. Ross and J. D. Lister, Phys. Rev. A 15:1246 (1977).

    Google Scholar 

  27. R. Landauer, Phys. Rev. A 15:2117 (1977).

    Google Scholar 

  28. D. Bedeaux, P. Mazur, and R. A. Pasmanter, Physica A 86:355 (1977).

    Google Scholar 

  29. D. Bedeaux and P. Mazur, Physica A 105:1 (1981).

    Google Scholar 

  30. W. J. Skoepol, M. R. Beasley, and M. Tinkham, J. Appl. Phys. 45:4054 (1974).

    Google Scholar 

  31. H. L. Frisch, Some recent exact solutions of the Fokker-Planck equation, in Non-Equilibrium Statistical Mechanics in One Dimension, V. Privman, ed. (Cambridge University Press, Cambridge, 1996), Chapter XVII.

    Google Scholar 

  32. H. L. Frisch, V. Privman, C. Nicolis, and G. Nicolis, J. Phys. A 23:1147 (1990).

    Google Scholar 

  33. V. Privman and H. L. Frisch, J. Chem. Phys. 94:8216 (1991).

    Google Scholar 

  34. B. Von Haeften, G. Izús, R. Deza, and C. Borzi, Phys. Lett. A, in press.

  35. A. C. Scott, Rev. Mod. Phys. 47:47 (1975).

    Google Scholar 

  36. R. Graham, Lecture Notes in Physics, Vol. 84 (Springer-Verlag, Berlin, 1978).

  37. R. Graham and T. Tel, Phys. Rev. A 42:4661 (1990).

    Google Scholar 

  38. R. Graham, Weak noise limit and nonequilibrium potentials of dissipative dynamical systems, in Instabilities and Nonequilibrium Structures, E. Tirapegui and D. Villarroel, eds. (Reidel, Dordrecht, 1987).

    Google Scholar 

  39. M. Kerszberg, Phys. Rev. A 28:1198 (1983).

    Google Scholar 

  40. M. de la Torre and I. Rehberg, Phys. Rev. A 42:2096, 5998 (1990).

    Google Scholar 

  41. J. Viñals, E. Hernandez-Garcia, M. San Miguel, and R. Toral, Phys. Rev. A 44:1123 (1991).

    Google Scholar 

  42. E. Hernandez-Garcia, J. Viñals, R. Toral, and M. San Miguel, Phys. Rev. Lett. 70:3576 (1993).

    Google Scholar 

  43. P. Jung, Phys. Rep. 234:175 (1993).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Deán Funes 3350, (, 7600), Mar del Plata, Argentina

    G. G. Izús & J. Reyes de Rueda

  2. MONDITEC S.A., Olazabal 1927, (, 1428), Buenos Aires, Argentina

    C. H. Borzi

Authors
  1. G. G. Izús
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Reyes de Rueda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. H. Borzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Izús, G.G., de Rueda, J.R. & Borzi, C.H. Boundary Effects on the Structural Stability of Stationary Patterns in a Bistable Reaction-Diffusion System. Journal of Statistical Physics 90, 103–117 (1998). https://doi.org/10.1023/A:1023295416567

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023295416567

Search

Navigation