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

Chunting Ji; Hui Liu; Jie Xin; Chunting Ji; Hui Liu; Jie Xin

doi:10.3934/math.2020233

AIMS Mathematics

2020, Volume 5, Issue 4: 3584-3611. doi: 10.3934/math.2020233

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Research article

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

1.
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, P. R. China
2.
College of Information Science and Engineering, Shandong Agricultural University, Tai An, Shandong, 271018, P. R. China
3.
School of Mathematics and Statistics, Ludong University, Yantai, Shandong, 264025, P. R. China

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.
- random attractors,
- stochastic extended Brusselator system,
- multiplicative noise,
- pullback asymptotic compactness
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

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Abstract

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.

References

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