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Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay

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Abstract

In this paper, the dynamic behaviors of coupled reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay under the Neumann boundary conditions are investigated. By constructing a basis of phase space based on the eigenvectors of Laplace operator, the characteristic equation of this system is obtained. Then, the local stability of zero solution and the occurrence of Hopf bifurcation are established by regarding the time delay as the bifurcation parameter. In particular, by using the normal form theory and center manifold theorem of the partial differential equation, the normal forms are obtained, which determine the bifurcation direction and the stability of the periodic solutions. Finally, two examples are given to verify the theoretical results.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 61503310, in part by the supported by the Fundamental Research Funds for the Central Universities under Grant XDJK2016B018, XDJK2017D170 and XDJK2017D183, in part by China Postdoctoral Foundation 2016M600720, in part by Chongqing Postdoctoral Project under Grant Xm2016003 and in part by the Natural Science Foundation project of CQCSTC under Grant ctsc2014cyjA40053 and cstc2016jcyjA0559.

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  1. College of Electronics and Information Engineering, Southwest University, Chongqing, 400715, People’s Republic of China

    Tao Dong, Weiyi Xu & Xiaofeng Liao

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  1. Tao Dong
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  2. Weiyi Xu
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  3. Xiaofeng Liao
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Correspondence to Tao Dong.

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Dong, T., Xu, W. & Liao, X. Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay. Nonlinear Dyn 89, 2329–2345 (2017). https://doi.org/10.1007/s11071-017-3589-8

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  • DOI: https://doi.org/10.1007/s11071-017-3589-8

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