JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (8): 92-103.doi: 10.6040/j.issn.1671-9352.0.2022.539

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Hopf bifurcation analysis in the Lengyel-Epstein reaction diffusion system with time delay

Yadi WANG(),Hailong YUAN*()   

  1. School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi'an 710021, Shaanxi, China
  • Received:2022-10-19 Online:2023-08-20 Published:2023-07-28
  • Contact: Hailong YUAN E-mail:842219671@qq.com;yuanhailong@sust.edu.cn

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Abstract:

The Lengyel-Epstein reaction diffusion system with time delay subject to Neumann boundary conditions is considered. By choosing the time delay as the bifurcation parameter, the stability/instability of the unique positive constant equilibrium and the existence of Hopf bifurcation are investigated. In addition, the formulae to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by applying the normal form theory and center manifold theorem for partial differential equation are derived. Finally, some numerical simulations are carried out to support the analytical results.

Key words: Hopf bifurcation, time delay, Lengyel-Epstein system, stability, numerical simulations

CLC Number: 

  • O175.12

Fig.1

Parameter is τ=0.030 5 and the initial value is (u0, v0)=(1, 1)"

Fig.2

Parameter is τ=0.033 4 and the initial value is (u0, v0)=(2.36, 6.07)"

Fig.3

Parameter is τ=0.030 4 and the initial value is u(x, t)=α+0.01 cos(2.5x), v(x, t)=vα+0.01 cos(2.5x)"

Fig.4

Parameter is τ=0.038 5 and the initial value is u(x, t)=α+0.45t cos(0.05x), v(x, t)=vα+0.45t cos (0.05x)"

Fig.5

Parameter is τ=0.038 5 and the initial value is u(x, t)=α+1.65 cos(4.21x), v(x, t)=vα+1.65 cos (4.21x)"

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