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)
Language: 
Russian
Article type: 
Article
UDC: 
519.8

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

Autors: 
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
Abstract: 

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.

Reference: 
  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).
Received: 
15.02.2012
Accepted: 
11.05.2012
Published: 
29.03.2013
Short text (in English):
(downloads: 71)