轮形符号图时滞神经网络分岔动力学分析
Bifurcation Dynamics Analysis of Wheel-Signed Graph Neural Networks with Time Delay
摘要: 本文研究了在轮形拓扑结构中符号图时滞神经网络的分岔动力学问题。首先，我们采用非线性耦合方式，通过符号连接权重来描述网络的节点间合作与竞争并存的现象。接着，我们以时滞为自由参数，通过分析特征根的分布，分别推导出网络系统在平衡点处局部渐进稳定与Hopf分岔发生的充分条件。最后使用DDE-BIFTOOL软件包进行数值仿真，计算了分岔点处的第一李雅普诺夫系数来判断Hopf分岔的方向和稳定性，以验证理论结果。
Abstract: This paper investigates the bifurcation dynamics of wheel-signed graph neural networks with time delay. First, we adopt a nonlinear coupling method and use signed connection weights to describe the coexistence of cooperation and competition among nodes. Then, taking time delay as the bifurcation parameter, we analyze the distribution of characteristic roots and establish sufficient conditions for both local asymptotic stability and the occurrence of Hopf bifurcation at the equilibrium point. Finally, numerical simulations are carried out by using the DDE-BIFTOOL package, and the first Lyapunov coefficient at the bifurcation point is computed to determine the direction and stability of the Hopf bifurcation, thereby validating the theoretical results.
文章引用：郑晨阳, 孙文. 轮形符号图时滞神经网络分岔动力学分析[J]. 应用数学进展, 2026, 15(2): 257-269. https://doi.org/10.12677/aam.2026.152067

参考文献

[1] Guckenheimer, J. and Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 150-154.
[2] Wei, J.J. and Zhang, C.R. (2008) Bifurcation Analysis of a Class of Neural Networks with Delays. Nonlinear Analysis: Real World Applications, 9, 2234-2252. [Google Scholar] [CrossRef] 
[3] Tao, B.B., Xiao, M., Zheng, W.X., Cao, J.D. and Tang, J.W. (2021) Dynamics Analysis and Design for a Bidirectional Super-Ring-Shaped Neural Network with n Neurons and Multiple Delays. IEEE Transactions on Neural Networks and Learning Systems, 32, 2978-2992. [Google Scholar] [CrossRef] [PubMed]
[4] Xiao, M., Zheng, W.X. and Cao, J.D. (2013) Hopf Bifurcation of an (n + 1)-Neuron Bidirectional Associative Memory Neural Network Model with Delays. IEEE Transactions on Neural Networks and Learning Systems, 24, 118-132. [Google Scholar] [CrossRef] [PubMed]
[5] Xu, W.Y., Cao, J.D., Xiao, M., et al. (2015) A New Framework for Analysis on Stability and Bifurcation in a Class of Neural Networks with Discrete and Distributed Delays. IEEE Transactions on Cybernetics, 45, 2224-2236. [Google Scholar] [CrossRef] [PubMed]
[6] Chen, J., Xiao, M., He, J. and Wang, F. (2023) Bifurcation Analysis of a Ring-Hub-Shaped Neural Network with (n + 1) Neurons and Multiple Delays. IEEE/CAA Journal of Automatica Sinica, 10, 1304-1315.
[7] Zhang, Y.Z., Xiao, M., Cao, J.D. and Zheng, W.X. (2022) Dynamical Bifurcation of Large-Scale-Delayed Fractional-Order Neural Networks with Hub Structure and Multiple Rings. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 52, 1731-1743. [Google Scholar] [CrossRef] 
[8] Sun, W., Li, B., Guo, W., Wen, S. and Wu, X. (2022) Interval Bipartite Synchronization of Multiple Neural Networks in Signed Graphs. IEEE Transactions on Neural Networks and Learning Systems, 34, 10970-10979. [Google Scholar] [CrossRef] [PubMed]
[9] Sun, W., Guo, W., Li, B., Wen, S. and Wu, X. (2022) Interval Bipartite Synchronization of Delayed Nonlinear Neural Networks with Signed Graphs. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 53, 1723-1733. [Google Scholar] [CrossRef] 
[10] Bizyaeva, A., Franci, A. and Leonard, N.E. (2023) Sustained Oscillations in Multi-Topic Belief Dynamics over Signed Networks. 2023 American Control Conference (ACC), San Diego, 31 May-2 June 2023, 4296-4301. [Google Scholar] [CrossRef] 
[11] Sieber, J., Engelborghs, K., Luzyanina, T., Samaey, G. and Roose, D. (2014) DDE-BIFTOOL v.3.1 Manual-Bifurcation Analysis of Delay Differential Equations.
https://arxiv.org/abs/1406.7144
[12] Ruan, S. and Wei, J. (2003) On the Zeros of Transcendental Functions with Applications to Stability of Delay Differential Equations with Two Delays. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10, 863-874.
[13] 魏俊杰, 王洪滨, 蒋卫华. 时滞微分方程的分支理论及应用[M]. 北京: 科学出版社, 2012.

为你推荐