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

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Kurushina S. E., Ivanov A. A. Dissipative structures of reaction–diffusion system simulation in multiplicative fluctuation phone. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 3, pp. 85-103. DOI: 10.18500/0869-6632-2010-18-3-85-103

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517.957, 519.21, 519.62

Dissipative structures of reaction–diffusion system simulation in multiplicative fluctuation phone

Kurushina Svetlana Evgenevna, Samara State University
Ivanov Andrej Aleksandrovich, Samara National Research University

The influence of multiplicative fluctuations of parameters of reaction­diffusion system on example of Gierer–Meinhardt model to formation of dissipative structures in soft mode instability regime was investigated. The system described interaction of non­decreased modes (order parameters) was received. It was shown that fluctuations of parameters are lead to changing of number of unstable modes, shifting of their eigenvalues and parametrical excitation of the system. The numerical simulation of described model evolution was received. The dependences of the dynamical variables fluctuation intensity in process of dissipative structures formation from the noise intensity were founded.

  1. Vasil'ev VA, Romanovsky YM, Yakhno VG. Autowave processes. Moscow: Nauka; 1987. (in Russian).
  2. Haken G. Synergetics. Moscow: Mir; 1980. 404 p. (in Russian).
  3. Landa PS. Self-oscillations in distributed systems. Moscow: Nauka, Fizmatlit; 1983. 320 p. (in Russian).
  4. Romanovsky YuM., Stepanova NV, Chernavsky DS. Mathematical modeling in biophysics (Introduction to theoretical biophysics). Moscow-Izhevsk: IKI; 2004. (in Russian).
  5. Ebeling V. Formation of structures at irreversible processes. Introduction intro theory of dissipative structures. Moscow-Izhevsk: IKI SIC RCD; 2004. (in Russian).
  6. Belintsev BN. Dissipative structures and the problem of biological pattern formation. Phys. Usp. 1983;26(9):775–800.
  7. Nicolis G,  Prigogine I. Self-organization in nonequilibrium systems. Moscow: Mir; 1979. (in Russian).
  8. Meinhardt H. The Algorithmic Beauty of Sea Shells. Heidelberg, New York. Berlin: Springer-Verlag; 1999.
  9. Horsthemke W, Lefever R. Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology. Moscow: Mir, 1987. (in Russian).
  10. Klyatskin VI. Stochastic Equations through the Eye of the Physicist: Basic Concepts, Exact Results, and Asymptotic Approximations. Amsterdam: Elsevier; 2005. 
  11. Stratonovich RL. Selected questions of the theory of fluctuations in radio engineering. Moscow: Sov. Radio; 1961. 558 p. (in Russian).
  12. Stratonovich RL, Romanovskii YuM. Parametric representation of linear and nonlinear oscillatory systems by a random force. Nauchn. Dokl. Vyssh. Shkoly Fiz.-Mat. Nauki. 1958;3:221–224. (in Russian).
  13. Svirezhev YuM, Logofet DO. Stability of biological communities. Moscow: Nauka, Fizmatlit; 1978. 352 p. (in Russian).
  14. Landa PS. Nonlinear oscillations and waves. Moscow: Nauka, Fizmatlit; 1997. 496 p. (in Russian).
  15. Mikhailov AS, Uporov IV. Critical phenomena in media with breeding, decay, and diffusion. Phys. Usp. 1984;27(9):695–714.
  16. Polak LS, Mikhailov AS. Processes of self-organization in physico-chemical systems. Moscow: Nauka; 1983. (in Russian).
  17. Kurushina SE, Maksimov VV. Noise-­induced phase transitions in competition processes in the external fluctuated media. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(1):88–100 (in Russian). DOI: 10.18500/0869-6632-2010-18-1-88-100.
  18. Belintsev BN. Dynamic collective properties of developed systems. Cand. Dissertation. Moscow: MIPT; 1979. (in Russian).
  19. Solyanik GI, Chernavsky DS. Mathematical models of morphogenesis. Preprint FIAN. 1980;8 (in Russian).
  20. Astashkina EV, Romanovsky YuM. Fluctuations in the process of self-organization. Mathematical models in ecology. Gorky: Gorky University Publ. 1980:74–82 (in Russian).
  21. Kurushina SE. Analytical research and numerical simulation of contrast dissipative structures in the field of fluctuations of dynamical variables. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(6):125–138 (in Russian). DOI: 10.18500/0869-6632-2009-17-6-125-138.
  22. Gromova LI, Ivanov AA, Kurushina SE. The dependence of the time of formation of contrast dissipative structures on the intensity and radius of correlation of the field of fluctuations of dynamic variables. Proceedings of the XVI International Conference of VMSPPS'2009 (May 25-31, 2009. Alushta, Crimea). Moscow: MAI-Print. 2009:245–247 (in Russian).
  23. Meinhardt H, Gierer A. Generation and regeneration of sequences of structures during morphogenesis. J. Theor. Biol. 1980;85:429–450. DOI: 10.1016/0022-5193(80)90318-5.
  24. Gierer A, Meinhardt H. Biological pattern formation involving lateral inhibition. Lectures on Mathematics in the Life Sciences. 1974;7:163–183.
  25. Meinhardt H, Gierer A. Applications of a theory of biological pattern formation based on lateral inhibition. Journ. Cell. Sci. 1974;15:321–346.
  26. Belintsev B.N. Physical foundations of biological shaping. Moscow: Nauka, Fizmatlit; 1991. (in Russian).
  27. Akhmanov SA, D'yakov YuE, Chirkin AS. Introduction to Statistical Radiophysics and Optics.  Moscow: Nauka, Fizmatlit; 1981. 640 p. (in Russian).
  28. Rytov SM. Introduction to statistical radiophysics. Moscow: Nauka; 1966. (in Russian).
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