ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Borina M. Y., Polezhaev A. A. Spatial-temporal patterns in a multidimensional active medium formed due to polymodal interaction near the wave bifurcation. Izvestiya VUZ. Applied Nonlinear Dynamics, 2012, vol. 20, iss. 6, pp. 15-24. DOI: 10.18500/0869-6632-2012-20-6-15-24

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 117)
Article type: 

Spatial-temporal patterns in a multidimensional active medium formed due to polymodal interaction near the wave bifurcation

Borina Marija Yurevna, P.N. Lebedev Physical Institute of the Russian Academy of Sciences
Polezhaev Andrej Aleksandrovich, P.N. Lebedev Physical Institute of the Russian Academy of Sciences

Investigation of a set of amplitude equations, describing interaction of several modes which became unstable due to the wave bifurcation, is carried out. It is shown that as a result of competition between modes depending on the value of the parameter defining the strength of interaction only two regimes are possible: either quasi one-dimensional travelling waves (there exists only one nonzero mode) or standing waves (al the modes are nonzero). This result is supported by numerical experiments for the Gierer-Mainhrdt model modified by addition of one more equation for the second fast diffusing inhibitor.

  1. Nicolis G, Prigogine I. Self-Organization in Nonequilibrium Systems. New York: John Wiley; 1977. 491 p.
  2. Prigogine I. From Being to Becoming. W H Freeman & Co; 1981. 272 p.
  3. Haken H. Synergetics. Berlin: Springer; 1983. 390 p. DOI: 10.1007/978-3-642-88338-5.
  4. Castets V, Dulos E, Boissonade J, Kepper PD. Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 1990;64(24):2953–2956. DOI: 10.1103/physrevlett.64.2953.
  5. Fields RJ, Burger M. Oscillations and Travelling Waves in Chemical Systems. New York: Wiley; 1985. 681 p.
  6. Kapral R, Showalter K. Chemical Waves and Patterns. Dordrecht: Kluwer; 1995. 524 p. DOI: 10.1007/978-94-011-1156-0.
  7. Zhabotinsky AM. A history of chemical oscillations and waves. Chaos. 1991;1(4):379–386. DOI: 10.1063/1.165848.
  8. Gong Y, Christini DJ. Antispiral waves in reaction-diffusion systems. Phys. Rev. Lett. 2003;90(8):088302. DOI: 10.1103/PhysRevLett.90.088302.
  9. Vanag VK, Epstein IR. Packet waves in a reaction-diffusion system. Phys. Rev. Lett. 2002;88(8):088303. DOI: 10.1103/physrevlett.88.088303.
  10. Vanag VK, Epstein IR. Dash waves in a reaction-diffusion system. Phys. Rev. Lett. 2003;90(9):098301. DOI: 10.1103/physrevlett.90.098301.
  11. Yang L, Berenstein I, Epstein IR. Segmented waves from a spatiotemporal transverse wave instability. Phys. Rev. Lett. 2005;95(3):038303. DOI: 10.1103/physrevlett.95.038303.
  12. Vanag VK, Epstein IR. Resonance-induced oscillons in a reaction-diffusion system. Phys. Rev. E. 2006;73(1):016201. DOI: 10.1103/PhysRevE.73.016201.
  13. Vanag VK. Waves and patterns in reaction-diffusion systems. Belousov-Zhabotinsky reaction in water-in-oil microemulsions. Phys. Usp. 2004;47(9):923–941. DOI: 10.1070/PU2004v047n09ABEH001742.
  14. Vanag VK, Epstein IR. Pattern formation in a tunable medium: the Belousov-Zhabotinsky reaction in an aerosol OT microemulsion. Phys. Rev. Lett. 2001;87(22):228301. DOI: 10.1103/physrevlett.87.228301.
  15. Turring AM. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B. 1952;237(641):37–72. DOI: 10.1098/rstb.1952.0012.
  16. Zhabotinsky AM, Dolnik M, Epstein IR, Rovinsky AB. Spatio-temporal patterns in a reaction-diffusion system with wave instability. J. Chem. Science. 2000;55(2):223–231. DOI: 10.1016/S0009-2509(99)00318-8.
  17. Kuramoto Y. Chemical Oscillations, Waves, and Turbulence. Berlin: Springer–Verlag; 1984. 156 p. DOI: 10.1007/978-3-642-69689-3.
  18. Nicolis G. Introduction to Nonlinear Science. Cambridge University Press; 1995. 254 p. DOI: 10.1017/CBO9781139170802.
  19. Gierer A, Meinhardt HA. Theory of biological pattern formation. Kibernetik. 1972;12(1):30–39. DOI: 10.1007/BF00289234.
  20. Borina MY, Polezhaev AA. Diffusion instability in a threevariable reaction–diffusion model. Computer Research and Modeling. 2011;3(2):135–146. DOI: 10.20537/2076-7633-2011-3-2-135-146.
  21. Lobanov AI, Petrov IB. Lectures on Computational Mathematics. Moscow: Binom; 2006. 523 p. (in Russian).
Short text (in English):
(downloads: 71)