Turing instability and pattern formation of a fractional Hopfield reaction–diffusion neural network with transmission delay
Articles
Jiazhe Lin
China Aerodynamics Research and Development Center
Jiapeng Li
China Aerodynamics Research and Development Center
Rui Xu
Shanxi University
https://orcid.org/0000-0001-6367-7631
Published 2022-05-05
https://doi.org/10.15388/namc.2022.27.27473
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Keywords

fractional derivative
neural network
Turing instability
pattern formation
reaction–diffusion

How to Cite

Lin, J., Li, J. and Xu, R. (2022) “Turing instability and pattern formation of a fractional Hopfield reaction–diffusion neural network with transmission delay”, Nonlinear Analysis: Modelling and Control, 27(5), pp. 823–840. doi:10.15388/namc.2022.27.27473.

Abstract

It is well known that integer-order neural networks with diffusion have rich spatial and temporal dynamical behaviors, including Turing pattern and Hopf bifurcation. Recently, some studies indicate that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we mainly investigate the Turing pattern in a delayed reaction–diffusion neural network with Caputo-type fractional derivative. In particular, we find that this fractional neural network can form steadily spatial patterns even if its first-derivative counterpart cannot develop any steady pattern, which implies that temporal fractional derivative contributes to pattern formation. Numerical simulations show that both fractional derivative and time delay have influence on the shape of Turing patterns.

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