Affiliation:
1. School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China
2. School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, P. R. China
Abstract
The purpose of the present paper is to investigate the pattern formation and secondary instabilities, including Eckhaus instability and zigzag instability, of an activator–inhibitor system, known as the Gierer–Meinhardt model. Conditions on the Hopf bifurcation and the Turing instability are obtained through linear stability analysis at the unique positive equilibrium. Then, the method of weakly nonlinear analysis is used to derive the amplitude equations. Especially, by adding a small disturbance to the Turing instability critical wave number, the spatiotemporal Newell–Whitehead–Segel equation of the stripe pattern is established. It is found that Eckhaus instability and zigzag instability may occur under certain conditions. Finally, Turing and non-Turing patterns are obtained via numerical simulations, including spotted patterns, mixed patterns, Eckhaus patterns, spatiotemporal chaos, nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous solutions in two-dimensional or one-dimensional space. Theoretical analysis and numerical results are in good agreement for this diffusive Gierer–Meinhardt model.
Funder
National Natural Science Foundation of China
Publisher
World Scientific Pub Co Pte Lt
Subject
Applied Mathematics,Modeling and Simulation,Engineering (miscellaneous)
Cited by
15 articles.
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