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Journal of Vibration Testing and System Dynamics 8(4) (2024) 379--390 | DOI:10.5890/JVTSD.2024.12.001

Cheng Luo$^{1}$, Guo-Cheng Wu$^2$, Hu-Shuang Hou$^3$

$^1$ School of Mathematics and Statistics, Southwest University, Chongqing 400715, PR China

$^2$ Key Laboratory of Intelligent Analysis and Decision on Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing, 400065, PR China

$^3$ School of Mathematical Science, Sichuan Normal University, Chengdu 610066, Sichuan Province, PR China

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Abstract

Many neural networks can be modeled by differential equations. They hold rich dynamics and numerical methods are generally used for numerical simulations. It becomes challenging to determine an appropriate time step--size for long time numerical simulations. This paper suggests a new application of Hopf bifurcation analysis for numerical optimization. First, a discrete--time neural network is obtained by Euler scheme. Then the time step--size is set as a bifurcation parameter and the frequency domain analysis approach is used to determine the critical value. Periodic solutions are obtained by the harmonic balance method. The direction of Hopf bifurcation and sufficient stability conditions are given. Finally, it can be concluded that the method is efficient to find the critical time step--size such that a continuous--time system can be approximated stably.

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