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Dynamics and control in an \(({\varvec{n}}+{\varvec{2}})\)-neuron BAM network with multiple delays

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Abstract

The issues of the stability and bifurcation for a delayed BAM network involving two neurons in the I-layer and arbitrary neurons in the J-layer are concerned in the present paper. By adopting the sum of the delays as the bifurcation parameter, we discuss the distribution of the roots of the characteristic equation for high-dimension system in terms of stability switches theory and further present some sufficient conditions for the occurrence of Hopf bifurcations. Analysis reveals that Hopf bifurcation will emerge after the given system loses its stability. Moreover, we derive explicit general formulae to determine the properties of bifurcation via the normal theory and the center manifold theorem. It is demonstrated that the sum of the delays can effectively affect the dynamics of the proposed system. Finally, an illustrative example is employed to verify the validity of the theoretical results obtained.

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Acknowledgments

The authors would like to thank the anonymous reviewers and the handling editor for their helpful comments and constructive suggestions, which have been very useful in improving the quality of the manuscript. This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant No. (17-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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Authors and Affiliations

  1. Research Center for Complex Systems and Network Sciences, Department of Mathematics, Southeast University, Nanjing, 210096, China

    Chengdai Huang & Jinde Cao

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Jinde Cao, Abdulaziz Alofi, Abdullah AI-Mazrooei & Ahmed Elaiw

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  1. Chengdai Huang
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  2. Jinde Cao
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  3. Abdulaziz Alofi
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  4. Abdullah AI-Mazrooei
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Correspondence to Jinde Cao.

Appendices

Appendix 1

Derivation of the expressions of \(\varTheta _{ij}\), \(\varPsi _{ij}\) \((i=1,2;j=1,2,3)\) in Eqs. (10) and (11)

$$\begin{aligned} \varTheta _{11}=&\,(-1)^{k+1}\omega ^{n+2}+(-1)^{k}\omega ^{n}a_{2}\\&+\sum \limits _{i=1}^{(n-2)/2}(-1)^{k-i}\omega ^{n-2i}(c_{2i-1} +a_{2i+2}),\\ \varTheta _{12}=&\,-(-1)^{k}\omega ^{n+1}a_{1}-(-1)^{k-1}\omega ^{n-1}a_{3}\\&+\sum \limits _{i=1}^{(n-2)/2}(-1)^{k-(i+1)}\omega ^{n-(2i+1)}(c_{2i} -a_{2i+3}), \end{aligned}$$
$$\begin{aligned} \varTheta _{13}=&\,-\sum \limits _{i=1}^{(n+2)/2}(-1)^{k-(i-1)}\omega ^{n-2(i-1)}b_{2i-1},\\ \varTheta _{21}=&\,(-1)^{k}\omega ^{n+1}a_{1}+(-1)^{k-1}\omega ^{n-1}a_{3}\\&+\sum \limits _{i=1}^{(n-2)/2}(-1)^{k-(i+1)}\omega ^{n-(2i+1)}(a_{2i+3} -c_{2i}),\\ \varTheta _{22}=&\,(-1)^{k+1}\omega ^{n+2}+(-1)^{k}\omega ^{n}a_{2}\\&+\sum \limits _{i=1}^{(n-2)/2}(-1)^{k-i}\omega ^{n-2i}(a_{2i+2} -c_{2i+1}),\\ \varTheta _{23}=&\,-\sum \limits _{i=1}^{n/2}(-1)^{k-i}\omega ^{n-(2i-1)}b_{2i}.\\ \varPsi _{11}=&\,(-1)^{k+1}\omega ^{n+1}a_{1}+(-1)^{k}\omega ^{n-1}a_{3}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega ^{n-(2i+1)}(c_{2i} +a_{2i+3}),\\ \varPsi _{12}=&\,-(-1)^{k+1}\omega ^{n+2}-(-1)^{k}\omega ^{n}a_{2}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega ^{n-2i}(c_{2i-1} +a_{2i+2}),\\ \varPsi _{13}=&\,-\sum \limits _{i=1}^{(n+1)/2}(-1)^{k-(i+1)}\omega ^{n-(2i-1)}b_{2i},\\ \varPsi _{21}=&\,(-1)^{k+1}\omega ^{n+2}+(-1)^{k-1}\omega ^{n}a_{2}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega ^{n-2i}(a_{2i+2} -c_{2i+1}),\\ \varPsi _{22}=&\,(-1)^{k+1}\omega ^{n+1}a_{1}+(-1)^{k}\omega ^{n-1}a_{3}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega ^{n-(2i+1)}(a_{2i+3} -c_{2i}),\\ \varPsi _{23}=&\,-\sum \limits _{i=1}^{(n+1)/2}(-1)^{k-(i-1)}\omega ^{n-(2i-1)}b_{2i-1}. \end{aligned}$$

Appendix 2

Derivation of the expressions of \(M_{ij}(i=1,2;j=1,2)\) in Eq. (17)

$$\begin{aligned} M_{11}=&\,\bigg [(-1)^{k+1}\omega _{0}^{n+3}+(-1)^{k}\omega _{0}^{n+1}a_{2}\\&+\sum \limits _{i=1}^{(n-2)/2}(-1)^{k-i}\omega _{0}^{n-2i+1}(a_{2i+2} -c_{2i+1})\bigg ]\sin (\omega _{0}\tau _{0})\\&+\bigg [(-1)^{k}\omega _{0}^{n+2}a_{1}+(-1)^{k-1}\omega _{0}^{n}a_{3}\\&+\sum \limits _{i=1}^{(n-2)/2}(-1)^{k-(i+1)}\omega _{0}^{n-2i}(a_{2i+3} -c_{2i})\bigg ]\cos (\omega _{0}\tau _{0}),\\ \end{aligned}$$
$$\begin{aligned} M_{12}=&\,\bigg [(-1)^{k+1}\omega _{0}^{n+3}+(-1)^{k}\omega _{0}^{n+1}a_{2}\\&+\sum \limits _{i=1}^{(n-2)/2}(-1)^{k-i}\omega _{0}^{n-2i+1}(a_{2i+2} -c_{2i+2})\bigg ]\cos (\omega _{0}\tau _{0})\\&+\bigg [(-1)^{k}\omega _{0}^{n+2}a_{1}+(-1)^{k-1}\omega _{0}^{n}a_{3}\\&+\sum \limits _{i=1}^{(n-2)/2}(-1)^{k-(i+1)}\omega _{0}^{n-2i}(a_{2i+3} +c_{2i})\bigg ]\sin (\omega _{0}\tau _{0}),\\ M_{21}=&\,\tau _{0}\Bigg \{\bigg [(n+1)(-1)^{k}\omega _{0}^{n}a_{1}+(n-1)(-1) ^{k-1}\omega _{0}^{n-2}a_{3}\\&+\sum \limits _{i=1}^{(n-2)/2}[n-(2i+1)](-1)^{k-(i+1)}\omega _{0} ^{n-2(i+1)}\\&\times (a_{2i+3}+c_{2i})\bigg ]\cos (\omega _{0}\tau _{0})\\&-\bigg [(n+2)(-1)^{k}\omega _{0}^{n+1}+n(-1)^{k-1}\omega _{0}^{n-1}a_{2}\\&-\sum \limits _{i=1}^{(n-2)/2}(n-2i)(-1)^{k-(i+1)}\omega _{0}^{n-(2i+1)}(a_{2(i+1)} -c_{i})\bigg ]\\&\times \sin (\omega _{0}\tau _{0})\Bigg \}\\&+\bigg [(n+1)(-1)^{k}\omega _{0}^{n}a_{1}+(n-1)(-1)^{k-1}\omega _{0}^{n-2}a_{3}\\&+\sum \limits _{i=1}^{(n-2)/2}[n-(2i+1)](-1)^{k-(i+1)}\omega _{0}^{n-2(i+1)}\\&\times (a_{2i+3}-c_{2i})\bigg ]\cos (\omega _{0}\tau _{0})\\&+\bigg [(n+2)(-1)^{k}\omega _{0}^{n+1}+n(-1)^{k-1}\omega _{0}^{n-1}a_{2}\\&\quad +\sum \limits _{i=1}^{(n-2)/2}(n-2i)(-1)^{k-(i+1)}\\&\times \omega _{0}^{n-2(i+1)}\cdot (a_{2(i+1)}-c_{i})\bigg ]\sin (\omega _{0}\tau _{0})\\&\quad +\sum \limits _{i=1}^{n/2}(n-(2i+1))(-1)^{k-i}\omega _{0}^{n-2i}b_{2i}, \end{aligned}$$
$$\begin{aligned} M_{22}=&\,\tau _{0}\Bigg \{\bigg [(n+1)(-1)^{k}\omega _{0}^{n}a_{1}+(n-1)(-1) ^{k-1}\omega _{0}^{n-2}a_{3}\\&\quad +\sum \limits _{i=1}^{(n-2)/2}[n-(2i+1)](-1)^{k-(i+1)}\omega _{0} ^{n-2(i+1)}\\&\quad \times (a_{2i+3}-c_{2i})\bigg ]\sin (\omega _{0}\tau _{0})\\&\quad -\bigg [(n+2)(-1)^{k}\omega _{0}^{n+1}+n(-1)^{k-1}\omega _{0}^{n-1}a_{2}\\&\quad +\sum \limits _{i=1}^{(n-2)/2}(n-2i)(-1)^{k-(i+1)}\omega _{0}^{n-(2i+1)}(a_{2(i+1)}\\&\quad +c_{i})\bigg ]\sin (\omega _{0}\tau _{0})\Bigg \}\\&\quad +\bigg [(n+1)(-1)^{k}\omega _{0}^{n}a_{1}+(n-1)(-1)^{k-1}\omega _{0}^{n-2}a_{3}\\&\quad +\sum \limits _{i=1}^{(n-2)/2}[n-(2i+1)](-1)^{k-(i+1)}\omega _{0}^{n-2(i+1)}\\&\quad \times (a_{2i+3}-c_{2i})\bigg ]\sin (\omega _{0}\tau _{0})\\&\quad +\bigg [(n+2)(-1)^{k}\omega _{0}^{n+1}+n(-1)^{k-1}\omega _{0}^{n-1}a_{2}\\&\quad +\sum \limits _{i=1}^{(n-2)/2}(n-2i)(-1)^{k-(i+1)}\\&\quad \times \omega _{0}^{n-2(i+1)}\cdot (a_{2(i+1)}-c_{i})\bigg ]\cos (\omega _{0}\tau _{0})\\&\quad +\sum \limits _{i=1}^{n/2}[n-(2i+1)](-1)^{k-i}\omega _{0}^{n-2i}b_{2i+1}.\\ \end{aligned}$$

Appendix 3

Derivation of the expressions of \(N_{ij}(i=1,2;j=1,2)\) in Eq. (17)

$$\begin{aligned} N_{11}=&\,\bigg [(-1)^{k+1}\omega _{0}^{n+2}a_{1}+(-1)^{k}\omega _{0}^{n}a_{3}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega _{0}^{n-(2i-1)}(a_{2i+3} -c_{2i})\bigg ]\sin (\omega _{0}\tau _{0})\\&+\bigg [(-1)^{k+1}\omega _{0}^{n+3}+(-1)^{k-1}\omega _{0}^{n+1}a_{2}\\&\times \sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega _{0}^{n-2i+1}(a_{2i+2} -c_{2i+1})\bigg ]\cos (\omega _{0}\tau _{0}), \end{aligned}$$
$$\begin{aligned} N_{12}=&\,\bigg [(-1)^{k+1}\omega _{0}^{n+2}a_{1}+(-1)^{k}\omega _{0}^{n}a_{3}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega _{0}^{n-2i}(a_{2i+3} -c_{2i})\bigg ]\cos (\omega _{0}\tau _{0})\\&+\bigg [(-1)^{k+1}\omega _{0}^{n+3}+(-1)^{k}\omega _{0}^{n+1}a_{2}\\&-\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega _{0}^{n-2i+1}(a_{2i+2} -c_{2i-1})\bigg ]\\&\times \sin (\omega _{0}\tau _{0}),\\ N_{21}=&\,\tau _{0}\Bigg \{\bigg [(-1)^{k+1}\omega _{0}^{n+1}a_{1}+(-1)^{k}\omega _{0}^{n-1}a_{3}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega _{0}^{n-(2i+1)}(a_{2i+3} -c_{2i})\bigg ]\cos (\omega _{0}\tau _{0})\\&-\bigg [(-1)^{k+1}\omega _{0}^{n+2}+(-1)^{k}\omega _{0}^{n}a_{2}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega _{0}^{n-2i}(a_{2(i+1)} -c_{2i-1})\bigg ]\sin (\omega _{0}\tau _{0})\Bigg \}\\&+\bigg [(n+2)(-1)^{k+1}\omega _{0}^{n+1}+n(-1)^{k}\omega _{0}^{n-1}a_{2}\\&+\sum \limits _{i=1}^{(n-1)/2}(n-2i)(-1)^{k-i}\omega _{0}^{n-(2i+1)}(a_{2(i+1)} -c_{2i-1})\bigg ]\\&\times \cos (\omega _{0}\tau _{0})\\&+\bigg [(n+1)(-1)^{k}\omega _{0}^{n}a_{1}+(n-1)(-1)^{k-1}\omega _{0}^{n-2}a_{3}\\&+\sum \limits _{i=1}^{(n-3)/2}(n-(2i+1))(-1)^{k-(i+1)}\omega _{0}^{n-2(i+1)}\\&\times (a_{2i+3)}-c_{2i})\bigg ]\sin (\omega _{0}\tau _{0})\\&+\sum \limits _{i=1}^{(n+1)/2}(n-2(i-1))(-1)^{k-(i-1)}\omega _{0}^{n-(2i-1)} b_{2i-1}, \end{aligned}$$
$$\begin{aligned} N_{22}=&\,\tau _{0}\Bigg \{\bigg [(-1)^{k+1}\omega _{0}^{n+1}a_{1}+(-1)^{k}\omega _{0}^{n-1}a_{3}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega _{0}^{n-(2i+1)}(a_{2i+3} -c_{2i})\bigg ]\sin (\omega _{0}\tau _{0})\\&+\bigg [(-1)^{k+1}\omega _{0}^{n+2}+(-1)^{k}\omega _{0}^{n}a_{2}\\&+\sum \limits _{i=1}^{(n-1)/2}(-1)^{k-i}\omega _{0}^{n-2i}(a_{2(i+1)} -c_{2i-1})\bigg ]\cos (\omega _{0}\tau _{0})\Bigg \}\\&+\bigg [(n+2)(-1)^{k+1}\omega _{0}^{n+1}+n(-1)^{k}\omega _{0}^{n-1}a_{2}\\&+\sum \limits _{i=1}^{(n-1)/2}(n-2i)(-1)^{k-i}\omega _{0}^{n-(2i+1)}(a_{2(i+1)} -c_{2i-1})\bigg ]\\&\times \sin (\omega _{0}\tau _{0})\\&-\bigg [(n+1)(-1)^{k}\omega _{0}^{n}a_{1}+(n-1)(-1)^{k-1}\omega _{0}^{n-2}a_{3}\\&+\sum \limits _{i=1}^{(n-3)/2}(n-(2i+1))(-1)^{k-(i+1)}\omega _{0}^{n-2(i+1)}\\&\times (a_{2i+3)}+c_{2i})\bigg ]\cos (\omega _{0}\tau _{0})\\&+\sum \limits _{i=1}^{(n-1)/2}(n-2(i-1))(-1)^{k-i}\omega _{0}^{n-(2i-1)} b_{2i-1}. \end{aligned}$$

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Huang, C., Cao, J., Alofi, A. et al. Dynamics and control in an \(({\varvec{n}}+{\varvec{2}})\)-neuron BAM network with multiple delays. Nonlinear Dyn 87, 313–336 (2017). https://doi.org/10.1007/s11071-016-3045-1

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