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Turing Bifurcation and Pattern Formation of Stochastic Reaction‐Diffusion System

Author

Listed:
  • Qianiqian Zheng
  • Zhijie Wang
  • Jianwei Shen
  • Hussain Muhammad Ather Iqbal

Abstract

Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction‐Diffusion System (RDS), Stochastic Reaction‐Diffusion System (SRDS) is more complex and it is very difficult to deal with the noise function. In this paper, we have presented a method to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation.

Suggested Citation

  • Qianiqian Zheng & Zhijie Wang & Jianwei Shen & Hussain Muhammad Ather Iqbal, 2017. "Turing Bifurcation and Pattern Formation of Stochastic Reaction‐Diffusion System," Advances in Mathematical Physics, John Wiley & Sons, vol. 2017(1).
    • Handle: RePEc:wly:jnlamp:v:2017:y:2017:i:1:n:9648538
    DOI: 10.1155/2017/9648538
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    References listed on IDEAS

    as
    1. Ya-nan Zhu & Jianwei Shen & Yong Xu, 2013. "Mechanism of Stochastic Resonance in a Quorum Sensing Network Regulated by Small RNAs," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. Li, Meifeng & Han, Bo & Xu, Li & Zhang, Guang, 2013. "Spiral patterns near Turing instability in a discrete reaction diffusion system," Chaos, Solitons & Fractals, Elsevier, vol. 49(C), pages 1-6.
    3. E. Ahmed & H. A. Abdusalam & E. S. Fahmy, 2001. "On Telegraph Coupled Map Lattice And Its Applications," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 12(10), pages 1525-1535.
    4. Duccio Fanelli & Claudia Cianci & Francesca Patti, 2013. "Turing instabilities in reaction-diffusion systems with cross diffusion," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 86(4), pages 1-8, April.
    5. Ya-nan Zhu & Jianwei Shen & Yong Xu, 2013. "Mechanism of Stochastic Resonance in a Quorum Sensing Network Regulated by Small RNAs," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-6, November.
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