Research article

Random attractors of the stochastic extended Brusselator system with a multiplicative noise

  • Received: 13 January 2020 Accepted: 07 April 2020 Published: 13 April 2020
  • MSC : 35B40, 35B41, 37L05

  • In this paper, we are devoted to study asymptotic dynamics of the stochastic extended Brusselator system with a multiplicative noise. The stochastic extended Brusselator system is composed of three pairs of symmetrical coupling components. We firstly study the pullback absorbing property for the stochastic extended Brusselator system with a multiplicative noise. But coupling terms bring great difficulty on this problem, we use the scaling method and estimate groups to overcome this difficulty. Then, we apply the bootstrap pullback estimations to prove the pullback asymptotic compactness for the stochastic extended Brusselator system with a multiplicative noise. Finally, we show the existence of random attractors. In the study of the existence of random attractors for stochastic dynamics, we use the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms. Based on this, we have to estimate groups to overcome the difficulties of coupling structure and make more complex estimates.

    Citation: Chunting Ji, Hui Liu, Jie Xin. Random attractors of the stochastic extended Brusselator system with a multiplicative noise[J]. AIMS Mathematics, 2020, 5(4): 3584-3611. doi: 10.3934/math.2020233

    Related Papers:

  • In this paper, we are devoted to study asymptotic dynamics of the stochastic extended Brusselator system with a multiplicative noise. The stochastic extended Brusselator system is composed of three pairs of symmetrical coupling components. We firstly study the pullback absorbing property for the stochastic extended Brusselator system with a multiplicative noise. But coupling terms bring great difficulty on this problem, we use the scaling method and estimate groups to overcome this difficulty. Then, we apply the bootstrap pullback estimations to prove the pullback asymptotic compactness for the stochastic extended Brusselator system with a multiplicative noise. Finally, we show the existence of random attractors. In the study of the existence of random attractors for stochastic dynamics, we use the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms. Based on this, we have to estimate groups to overcome the difficulties of coupling structure and make more complex estimates.


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    [1] I. Prigogine, R. Lefever, Symmetry-breaking instabilities in dissipative systems, J. Chem. Phys., 48 (1968), 1695-1700. doi: 10.1063/1.1668896
    [2] K. Brown, F. Davidson, Global bifurcation in the Brusselator system, Nonlinear Anal., 24 (1995), 1713-1725. doi: 10.1016/0362-546X(94)00218-7
    [3] J. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. doi: 10.1126/science.261.5118.189
    [4] K. J. Lee, W. D. Mccormick, Q. Ouyang, et al. Pattern formation by interacting chemical fronts, Science, 261 (1993), 192-194. doi: 10.1126/science.261.5118.192
    [5] P. Gray, S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability, Chem. Eng. Sci., 38 (1983), 29-43. doi: 10.1016/0009-2509(83)80132-8
    [6] P. Gray, S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; BC, Chem. Eng. Sci., 39 (1984), 1087-1097. doi: 10.1016/0009-2509(84)87017-7
    [7] B. Guo, Y. Han, Attractor and spatial chaos for the Brusselator in RN, Nonlinear Anal. Theor., 70 (2009), 3917-3931. doi: 10.1016/j.na.2008.08.002
    [8] Y. You, Global dynamics of the Brusselator equations, Dynam. Part. Differ. Eq., 4 (2007), 167-196. doi: 10.4310/DPDE.2007.v4.n2.a4
    [9] Y. You, Global attractor of a coupled two-cell Brusselator model, In: Infinite Dimensional Dynamical Systems, Springer, New York, 2013, 319-352.
    [10] Q. Bie, Pattern formation in a general two-cell Brusselator model, J. Math. Anal. Appl., 376 (2011), 551-564. doi: 10.1016/j.jmaa.2010.10.066
    [11] R. D. Parshad, S. Kouachi, N. Kumari, et al. Global existence and long time dynamics of a four compartment Brusselator type system, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 24 (2017), 79-120.
    [12] C. Florinda, D. L. Roberta, T. Isabella, Influence of diffusion on the stability of a full Brusselator model, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 29 (2018), 661-679. doi: 10.4171/RLM/827
    [13] H. Kitano, Systems biology: A brief overview, Science, 295 (2002), 1662-1664. doi: 10.1126/science.1069492
    [14] P. Gormley, K. Li, G. W. Irwin, Modeling molecular interaction pathways using a two-stage identification algorithm, Syst. Synth. Biol., 1 (2007), 145-160. doi: 10.1007/s11693-008-9012-5
    [15] Y. You, S. Zhou, Global dissipative dynamics of the extended Brusselator system, Nonlinear Anal. Real, 13 (2012), 2767-2789. doi: 10.1016/j.nonrwa.2012.04.005
    [16] J. Tu, Y. You, Random attractor of stochastic Brusselator system with multiplicative noise, Discrete Contin. Dyn. Syst., 36 (2016), 2757-2779. doi: 10.3934/dcds.2016.36.2757
    [17] H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Theory. Rel., 100 (1994), 365-393. doi: 10.1007/BF01193705
    [18] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225
    [19] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
    [20] P. W. Bates, K. Lu, B. Wang, Random attractors for stochastic reaction-diffusion equation on unbounded domains, J. Differ. Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017
    [21] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269
    [22] X. Jia, J. Gao, X. Ding, Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise, Open Math., 14 (2016), 586-602. doi: 10.1515/math-2016-0052
    [23] N. Liu, J. Xin, Existence of random attractor for stochastic fractional long-short wave equations with periodic boundary condition, Discrete Dyn. Nat. Soc., 2017 (2017), 1-11.
    [24] X. J. Li, X. L. Li, K. Lu, Random attractors for stochastic parabolic equations with additive noise in weighted spaces, Commun. Pure Appl. Anal., 17 (2018), 729-749. doi: 10.3934/cpaa.2018038
    [25] F. Wang, J. Li, Y. Li, Random attractors for Ginzburg-Landau equations driven by difference noise of a Wiener-like process, Adv. Differ. Equ., 2019 (2019), 1-18. doi: 10.1186/s13662-018-1939-6
    [26] R. Sell, Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
    [27] V. V. Chepyzhov, M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002.
    [28] I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, Heidelberg, 2002.
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