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


For citation:

Kurushina S. E., Gromova L. I., Shapovalova E. A. Nonlinear multivariate self­consistent fokker–planck equation for multicomponent reaction­diffusion systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 5, pp. 27-42. DOI: 10.18500/0869-6632-2014-22-5-27-42

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Language: 
Russian
Article type: 
Article
UDC: 
519.21, 517.957, 519.62, 536.75

Nonlinear multivariate self­consistent fokker–planck equation for multicomponent reaction­diffusion systems

Autors: 
Kurushina Svetlana Evgenevna, Samara State University
Gromova Lidija Ivanovna, Samara National Research University
Shapovalova Evgenija Aleksandrovna, Samara National Research University
Abstract: 

Mean field approximation is extended to multicomponent stochastic reaction­diffusion systems. A multivariate nonlinear self­consistent Fokker–Planck equation defining the probability density of the state of the system, which describes a well­known model of autocatalytic chemical reaction (Brusselator) with spatially correlated multiplicative noise, is obtained. The evolution of probability density and statistical characteristics of the system in the region of Turing bifurcation are studied. Numerical study of the equation solutions for a stochastic brusselator shows that in the region of Turing bifurcation several types of solutions exist if noise intensity increases: unimodal solution, transient bimodality, and an interesting solution which involves multiple «repumping» of probability density through bimodality.

Reference: 
  1. Lindnera B, Garcia-Ojalvo J, Neimand A, Schimansky-Geier L. Effects of noise in excitable systems. Physics Reports. 2004;392:321.
  2. Ibanes M. Garc Mean-field results. Phys. Rev. E. 1999;60:3597.
  3. Buceta J, Ibanes M, Sancho JM, Lindenberg K. Noise-driven mechanism for pattern formation. Phys. Rev. E. 2003;67:021113.
  4. Carrillo O, Ibanes M. Garc noise-induced phase transitions: Beyond the noise interpretation. Phys. Rev. E. 2003;67:046110.
  5. Zaikin AA, Garcia-Ojalvo J, Schimansky-Geier L. Nonequilibrium first-order phase transition induced by additive noise. Phys. Rev. E. 1999;60:R6275.
  6. Muller R, Lippert K, Kuhnel A, Behn U. First-order nonequilibrium phase transition in a spatially extended system. Phys. Rev. E. 1997;56:2658.
  7. Carrillo O, Ibanes M, Sancho JM. Noise induced phase transitions by nonlinear instability mechanism. Fluct. Noise Lett. 2002. Vol. 2. L1.
  8. Landa PS, Zaikin AA, Schimansky-Geier L. Influence of additive noise on noise-induced phase transitions in nonlinear chains. Chaos, Solitons and Fractals. 1998;9:1367.
  9. Van den Broeck C, Parrondo JMR, Toral R, Kawai R. Nonequilibrium phase transitions induced by multiplicative noise. Phys. Rev. E. 1997;55:4084.
  10. Buceta J, Parrondo JMR, and de la Rubia FJ. Random Ginzburg–Landau model revisited: Reentrant phase transitions. Phys. Rev. E. 2001;63:031103.
  11. Prigogine I, Lefever R. Symmetry breaking instabilities in dissipative systems. J. Chem. Phys. 1968;48:1695.
  12. Kurushinа SE, Maximov VV, Romanovskii YM. Spatial pattern formation in external noise: Theory and simulation. Phys. Rev. E. 2012;86:011124.
  13. Horsthemke W, Lefever M. Noise-Induced Transition. Berlin, Springer, 1984.
  14. Garcia-Ojalvo J, Lacasta AM, Sancho JM, Toral R. Phase separation driven by external fluctuations. Europhys. Lett. 1998;42:125.
  15. Stratonovich RL. Topics in the Theory of Random Noise. New York, Gordon and Breach. 1963, Vol. 1; 1967, Vol. 2.
  16. Karetkina NV. An unconditionally stable difference scheme for parabolic equations containing first derivatives. USSR Computational Mathematics and Mathematical Physics. 1980;20:257.
  17. Samarskii AA. On an economical difference method for the solution of a multi-dimensional parabolic equation in an arbitrary region. USSR Computational Mathematics and Mathematical Physics. 1963; 2:894.
  18. Samarskii AA. Local one dimensional difference schemes on non-uniform nets. USSR Computational Mathematics and Mathematical Physics. 1963;3:572.
  19. Samarskii AA. Homogeneous difference schemes on non-uniform nets for equations of parabolic type. USSR Computational Mathematics and Mathematical Physics. 1963;3:351.
  20. Van den Broeck C, Parrondo JMR, Toral R. Noise-induced nonequilibrium phase transition. Phys. Rev. Lett. 1994;73:3395.
Received: 
15.06.2014
Accepted: 
06.11.2014
Published: 
31.03.2015
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