Turing-Hopf bifurcation of reaction-diffusion neural networks with leakage delay. (English) Zbl 1450.35049
Summary: In artificial neural networks, the diffusion phenomenon of electrons exists inevitably, due to the electromagnetic field of neural networks is heterogeneous. In this paper, we study the spatio-temporal dynamical behaviors of a reaction-diffusion neural network with leakage delay. By analyzing the corresponding characteristic equation, the sufficient and necessary conditions of Turing instability are obtained and the existence of Turing, Hopf, and Turing-Hopf bifurcations is also established. Furthermore, the truncated normal form up to third order is derived to understand and classify the spatio-temporal dynamics close to the Turing-Hopf bifurcation point. By numerical simulations, we find a pair of spatially inhomogeneous periodic solutions and illustrate the effects of time delays and spatial diffusion on the spatio-temporal dynamics of the model.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
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