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Stability and Bifurcation Behavior of a Neuron System with Hyper-Strong Kernel

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Abstract

At present, there are few studies on the delayed kernel function of hyper-strong kernel. This paper attempts to analyze the stability and bifurcation of neural networks with distributed delayed hyper-strong kernels. Firstly, considering the average delay as a bifurcation parameter, the study discusses the characteristic equations of delayed kernels with weak kernel, strong kernel and hyper-strong kernel to provide sufficient conditions for the stability and generation of Hopf bifurcation. Secondly, it applies the normal theory and the center manifold theory to derive the formulas for determining the stability and direction of the bifurcating periodic solution. Finally, it verifies the correctness of the calculation results by numerical simulation with an example.

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Acknowledgements

The authors would like to express their sincere appreciation to the reviewers and editor for his/her helpful comments in improving the presentation and quality of the paper. This work was supported by National Natural Science Foundation of China (Grant Nos. 61374011, 62103215), and Natural Science Foundation of Shandong Province of China (Grant Nos. ZR2020MF080, ZR2020MF065.)

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, 266061, China

    Xinyu Li & Zunshui Cheng

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

  3. Yonsei Frontier Lab, Yonsei university, Seoul, 03722, South Korea

    Jinde Cao

  4. Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia

    Fawaz E. Alsaadi

Authors
  1. Xinyu Li
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  2. Zunshui Cheng
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  3. Jinde Cao
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  4. Fawaz E. Alsaadi
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Contributions

X. Li and Z. Cheng wrote the main manuscript text and prepared figures. Z. Cheng, J. Cao and F. E. Alsaadi reviewed the manuscript.

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Correspondence to Zunshui Cheng.

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Appendix

Appendix

Concrete expressions in system (37)

$$\begin{aligned} \left\{ \begin{aligned} f_1=&\,c_{11}^{(2)}[\phi _1^2(0)-b_1\int \limits ^{0}_{-\infty }T_0(-s)\phi _1^2(s)ds]+c_{12}^{(2)}[\phi _2^2(0)-b_1\int \limits ^{0}_{-\infty }T_0(-s)\phi _2^2(s)ds]\\&\,+c_{13}^{(2)}[\phi _3^2(0)-b_1\int \limits ^{0}_{-\infty }T_0(-s)\phi _3^2(s)ds]+c_{11}^{(3)}[\phi _1^3(0)-b_1\int \limits ^{0}_{-\infty }T_0(-s)\phi _1^3(s)ds]\\&\,+c_{12}^{(3)}[\phi _2^3(0)-b_1\int \limits ^{0}_{-\infty }T_0(-s)\phi _2^3(s)ds]+c_{13}^{(3)}[\phi _3^3(0)-b_1\int \limits ^{0}_{-\infty }T_0(-s)\phi _3^3(s)ds]+\cdot \cdot \cdot ,\\ f_2=&\,c_{21}^{(2)}[\phi _1^2(0)-b_2\int \limits ^{0}_{-\infty }T_0(-s)\phi _1^2(s)ds]+c_{22}^{(2)}[\phi _2^2(0)-b_2\int \limits ^{0}_{-\infty }T_0(-s)\phi _2^2(s)ds]\\&\,+c_{23}^{(2)}[\phi _3^2(0)-b_2\int \limits ^{0}_{-\infty }T_0(-s)\phi _3^2(s)ds]+c_{21}^{(3)}[\phi _1^3(0)-b_2\int \limits ^{0}_{-\infty }T_0(-s)\phi _1^3(s)ds]\\&\,+c_{22}^{(3)}[\phi _2^3(0)-b_2\int \limits ^{0}_{-\infty }T_0(-s)\phi _2^3(s)ds]+c_{23}^{(3)}[\phi _3^3(0)-b_2\int \limits ^{0}_{-\infty }T_0(-s)\phi _3^3(s)ds]+\cdot \cdot \cdot ,\\ f_3=&\,c_{31}^{(2)}[\phi _1^2(0)-b_3\int \limits ^{0}_{-\infty }T_0(-s)\phi _1^2(s)ds]+c_{32}^{(2)}[\phi _2^2(0)-b_3\int \limits ^{0}_{-\infty }T_0(-s)\phi _2^2(s)ds]\\&\,+c_{33}^{(2)}[\phi _3^2(0)-b_3\int \limits ^{0}_{-\infty }T_0(-s)\phi _3^2(s)ds]+c_{31}^{(3)}[\phi _1^3(0)-b_3\int \limits ^{0}_{-\infty }T_0(-s)\phi _1^3(s)ds]\\&\,+c_{32}^{(3)}[\phi _2^3(0)-b_3\int \limits ^{0}_{-\infty }T_0(-s)\phi _2^3(s)ds]+c_{33}^{(3)}[\phi _3^3(0)-b_3\int \limits ^{0}_{-\infty }T_0(-s)\phi _3^3(s)ds]+\cdot \cdot \cdot , \end{aligned} \right. \end{aligned}$$

where \(c^{(u)}_{ij}=\frac{1}{u!}a_{ij}f^{(u)}(0), i,j=1,2,3, u=2,3,\cdot \cdot \cdot \).

Concrete expressions in system (52)

$$\begin{aligned} \left\{ \begin{aligned} K_{11}&=(c_{11}^{(2)}+c^{(2)}_{12}B^2_{1}+c^{(2)}_{13}B^2_{2})\left[ 1-\frac{b_1\alpha }{(\alpha +2\textrm{i}\omega _0)}\right] ,\\ K_{12}&=2(c_{11}^{(2)}+c^{(2)}_{12}B_{1}\overline{B}_{1}+c^{(2)}_{13}B_{2}\overline{B}_{2})(1-b_1),\\ K_{13}&=(c_{11}^{(2)}+c^{(2)}_{12}\overline{B}^2_{1}+c^{(2)}_{13}\overline{B}^2_{2})\left[ 1-\frac{b_1\alpha }{(\alpha -2\textrm{i}\omega _0)}\right] ,\\ K_{14}&=c_{11}^{(2)}(W_{20}^{(1)}(0)+2W_{11}^{(1)}(0))\\&\quad +c_{12}^{(2)}(W^{(2)}_{20}(0)\overline{B}_{1}+2W_{11}^{(2)}(0)B_{1})+c_{13}^{(2)}(W^{(3)}_{20}(0)\overline{B}_{2}+2W_{11}^{(3)}(0)B_{2})-c_{11}^{(2)}b_1\\&\quad \int \limits _{-\infty }^0F(-s)(W_{20}^{(1)}(s)e^{-\textrm{i}\omega _0s}+2W_{11}^{(1)}(s)e^{\textrm{i}\omega _0s})ds-c_{12}^{(2)}b_1\int \limits _{-\infty }^0F(-s)(\overline{B}_1W_{20}^{(2)}(s)\\&e^{-\textrm{i}\omega _0s}+2B_{1}W_{11}^{(2)}(s)e^{\textrm{i}\omega _0s})ds-c_{13}^{(2)}b_1\int \limits _{-\infty }^0F(-s)(\overline{B}_2W_{20}^{(2)}(s)e^{-\textrm{i}\omega _0s}+2B_{2}W_{11}^{(2)}(s)e^{\textrm{i}\omega _0s})ds\\&+3(c_{11}^{(3)}+c^{(3)}_{12}B_{1}^2\overline{B}_1+c^{(3)}_{23}B_{2}^2\overline{B}_2)[1-\frac{b_1\alpha }{(\alpha +\textrm{i}\omega _0)}]\\ K_{21}&=(c_{21}^{(2)}+c^{(2)}_{22}B^2_{1}+c^{(2)}_{23}B^2_{2})\left[ 1-\frac{b_2\alpha }{(\alpha +2\textrm{i}\omega _0)}\right] ,\\ K_{22}&=2(c_{21}^{(2)}+c^{(2)}_{22}B_{1}\overline{B}_{1}+c^{(2)}_{23}B_{2}\overline{B}_{2})(1-b_2),\\ K_{23}&=(c_{21}^{(2)}+c^{(2)}_{22}\overline{B}^2_{1}+c^{(2)}_{23}\overline{B}^2_{2})\left[ 1-\frac{b_2\alpha }{(\alpha -2\textrm{i}\omega _0)}\right] ,\\ K_{24}&=c_{21}^{(2)}(W_{20}^{(1)}(0)+2W_{11}^{(1)}(0))+c_{22}^{(2)}(W^{(2)}_{20}(0)\overline{B}_{1}+2W_{11}^{(2)}(0)B_{1})\\&\quad +c_{23}^{(2)}(W^{(3)}_{20}(0)\overline{B}_{2}+2W_{11}^{(3)}(0)B_{2})-c_{21}^{(2)}b_2\int \limits _{-\infty }^0F(-s)(W_{20}^{(1)}(s)e^{-\textrm{i}\omega _0s}+2W_{11}^{(1)}(s)e^{\textrm{i}\omega _0s})ds\\&\quad -c_{22}^{(2)}b_2\int \limits _{-\infty }^0F(-s)(\overline{B}_1W_{20}^{(2)}(s)e^{-\textrm{i}\omega _0s}+2B_{1}W_{11}^{(2)}(s)e^{\textrm{i}\omega _0s})ds\\&\quad -c_{23}^{(2)}b_2\int \limits _{-\infty }^0F(-s)(\overline{B}_2W_{20}^{(2)}(s)e^{-\textrm{i}\omega _0s}+2B_{2}W_{11}^{(2)}(s)e^{\textrm{i}\omega _0s})ds\\&\quad +3(c_{21}^{(3)}+c^{(3)}_{22}B_{1}^2\overline{B}_1+c^{(3)}_{23}B_{2}^2\overline{B}_2)\left[ 1-\frac{b_2\alpha }{(\alpha +\textrm{i}\omega _0)}\right] \\ K_{31}&=(c_{31}^{(2)}+c^{(2)}_{32}B^2_{1}+c^{(2)}_{33}B^2_{2})[1-\frac{b_3\alpha }{(\alpha +2\textrm{i}\omega _0)}],\\ K_{32}&=2(c_{31}^{(2)}+c^{(2)}_{32}B_{1}\overline{B}_{1}+c^{(2)}_{33}B_{2}\overline{B}_{2})(1-b_3),\\ K_{33}&=(c_{31}^{(2)}+c^{(2)}_{32}\overline{B}^2_{1}+c^{(2)}_{33}\overline{B}^2_{2})\left[ 1-\frac{b_3\alpha }{(\alpha -2\textrm{i}\omega _0)}\right] ,\\ K_{34}&=c_{31}^{(2)}(W_{20}^{(1)}(0)+2W_{11}^{(1)}(0))+c_{32}^{(2)}(W^{(2)}_{20}(0)\overline{B}_{1}+2W_{11}^{(2)}(0)B_{1})\\&\quad +c_{33}^{(2)}(W^{(3)}_{20}(0)\overline{B}_{2}+2W_{31}^{(3)}(0)B_{2})-c_{31}^{(2)}b_3\int \limits _{-\infty }^0F(-s)(W_{20}^{(1)}(s)e^{-\textrm{i}\omega _0s}+2W_{11}^{(1)}(s)e^{\textrm{i}\omega _0s})ds\\&\quad -c_{32}^{(2)}b_3\int \limits _{-\infty }^0F(-s)(\overline{B}_1W_{20}^{(2)}(s)e^{-\textrm{i}\omega _0s}+2B_{1}W_{11}^{(2)}(s)e^{\textrm{i}\omega _0s})ds\\&\quad -c_{33}^{(2)}b_3\int \limits _{-\infty }^0F(-s)(\overline{B}_2W_{20}^{(2)}(s)e^{-\textrm{i}\omega _0s}+2B_{2}W_{11}^{(2)}(s)e^{\textrm{i}\omega _0s})ds\\&\quad +3(c_{31}^{(3)}+c^{(3)}_{32}B_{1}^2\overline{B}_1+c^{(3)}_{33}B_{2}^2\overline{B}_2)\left[ 1-\frac{b_3\alpha }{(\alpha +\textrm{i}\omega _0)}\right] \\ \end{aligned} \right. \end{aligned}$$

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Li, X., Cheng, Z., Cao, J. et al. Stability and Bifurcation Behavior of a Neuron System with Hyper-Strong Kernel. Neural Process Lett 55, 12143–12167 (2023). https://doi.org/10.1007/s11063-023-11413-y

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