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Frequency domain approach to the critical step size of discrete-time recurrent neural networks

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Abstract

It is usually essential to reveal the relationship between continuous-time systems and discrete-time ones. First, a discrete-time recurrent neural network is presented by the Euler scheme in this paper. Then, the time step size is set to a bifurcation parameter and frequency domain approach is adopted for Hopf bifurcation analysis. Moreover, the periodic solutions are obtained by the harmonic balance method; then the stability conditions are presented. The critical step size is determined with which the discrete-time recurrent neural network can inherit the stable state of the continuous-time one. Finally, one numerical example of the discrete-time recurrent neural network is given to support the theoretical analysis.

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The datasets analyzed during the current study are included in this article.

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Funding

This work was partially supported by National Natural Science Foundation of China (NSFC) (Grant No. 62076141), Natural Science Foundation of Chongqing Science and Technology Bureau (Grant No. cstc2019jcyj-msxmX0020), and Graduate Student Innovation Program of Chongqing University of Technology (Grant No. gzlcx20223305).

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Authors and Affiliations

  1. School of Science, Chongqing University of Technology, Chongqing, 400054, People’s Republic of China

    Hu-Shuang Hou & Hua Zhang

  2. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Cheng Luo

  3. Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, 641100, Sichuan Province, People’s Republic of China

    Guo-Cheng Wu

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  1. Hu-Shuang Hou
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  2. Cheng Luo
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Correspondence to Hua Zhang.

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Appendix

Appendix

The direction of half-line staring at \(-1+{\textbf {i}}0\) can be obtained as \(\varepsilon = l_{0}\Big (l_{1}{\textrm{e}^{-{\textbf {i}}\omega \tau _{1}}}+l_{2}{\textrm{e}^{-{\textbf {i}}\omega \tau _{2}}}+l_{3} {\textrm{e}^{-{\textbf {i}}\omega ( \tau _{1}-1 ) }}+ l_{4}{\textrm{e}^{-{\textbf {i}}\omega ( \tau _{2}-1 ) }}+l_{5} {\textrm{e}^{-{\textbf {i}}\omega ( \tau _{1}-2 ) }}+l_{6} {\textrm{e}^{-{\textbf {i}}\omega ( \tau _{2}-2 ) }}+ l_{7}{\textrm{e}^{-{\textbf {i}}\omega ( \tau _{1}-3 ) }}+ l_{8} {\textrm{e}^{-{\textbf {i}}\omega ( \tau _{2}-3 ) }} \Big ) \Big / ( (6h-5)(\cos \omega -1)+4h^{2} ) ( {\textrm{e}^{{\textbf {i}}\omega }}-1+h ) ^{2} ( {\textrm{e}^{{\textbf {i}}\omega }}h-{\textrm{e}^{{\textbf {i}}\omega }}+1) ( {\textrm{e}^{{\textbf {i}}\omega }}-1+2\,h ) \), where

$$\begin{aligned}{} & {} \varphi =\frac{h e^{-{\textbf {i}}\omega \tau _{1}}(e^{{\textbf {i}}\omega }-1+2h)^{2}(e^{-{\textbf {i}}\omega }-1+2h)}{4\eta ^{3}(e^{{\textbf {i}}\omega }-1+h)^{2}(e^{-{\textbf {i}}\omega }-1+h)} \\{} & {} \quad +\frac{4h e^{-{\textbf {i}}\omega \tau _{2}}}{\eta ^{3}},\\{} & {} l_{0}=h \left( \left( 2\,h-2 \right) \cos \omega +{h}^{2}-2\,h+2 \right) ,\\{} & {} \quad l_{1}=\frac{1}{2}\left( -\frac{1}{2}+h \right) ^{2},\\{} & {} \quad l_{2}=2\left( -1+h\right) ^{2}, \\{} & {} \quad l_{3}=\left( -\frac{1}{2}+h \right) \left( {h}^{2}-h+\frac{3}{4} \right) ,\\{} & {} \quad l_{4}=2\,{h}^{3}-6\,{h}^{2}+10\,h-6, \\{} & {} \quad l_{5}={h}^{2}-h+\frac{3}{8}, \\{} & {} \quad l_{6}= 4\,{h}^{2}-8\,h+6, \\{} & {} \quad l_{7}=\frac{h}{4}-\frac{1}{8}, \\{} & {} \quad l_{8}= 2\,h-2. \end{aligned}$$

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Hou, HS., Luo, C., Zhang, H. et al. Frequency domain approach to the critical step size of discrete-time recurrent neural networks. Nonlinear Dyn 111, 8467–8476 (2023). https://doi.org/10.1007/s11071-023-08278-0

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