July  2004, 10(3): 805-826. doi: 10.3934/dcds.2004.10.805

Asymptotic behaviors in a transiently chaotic neural network

1. 

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, Taiwan

Received  January 2002 Revised  May 2003 Published  January 2004

We are interested in the asymptotic behaviors of a discrete-time neural network. This network admits transiently chaotic behaviors which provide global searching ability in solving combinatorial optimization problems. As the system evolves, the variables corresponding to temperature in the annealing process decrease, and the chaotic behaviors vanish. We shall find sufficient conditions under which evolutions for the system converge to a fixed point of the system. Attracting sets and uniqueness of fixed point for the system are also addressed. Moreover, we extend the theory to the neural networks with cycle-symmetric coupling weights and other output functions. An application of this annealing process in solving travelling salesman problems is illustrated.
Citation: Shyan-Shiou Chen, Chih-Wen Shih. Asymptotic behaviors in a transiently chaotic neural network. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 805-826. doi: 10.3934/dcds.2004.10.805
[1]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Stability of the dynamics of an asymmetric neural network. Communications on Pure & Applied Analysis, 2009, 8 (2) : 655-671. doi: 10.3934/cpaa.2009.8.655

[2]

Yuantian Xia, Juxiang Zhou, Tianwei Xu, Wei Gao. An improved deep convolutional neural network model with kernel loss function in image classification. Mathematical Foundations of Computing, 2020, 3 (1) : 51-64. doi: 10.3934/mfc.2020005

[3]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077

[4]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025

[5]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2021, 8 (2) : 131-152. doi: 10.3934/jcd.2021006

[6]

Ndolane Sene. Fractional input stability and its application to neural network. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 853-865. doi: 10.3934/dcdss.2020049

[7]

Ying Sue Huang, Chai Wah Wu. Stability of cellular neural network with small delays. Conference Publications, 2005, 2005 (Special) : 420-426. doi: 10.3934/proc.2005.2005.420

[8]

King Hann Lim, Hong Hui Tan, Hendra G. Harno. Approximate greatest descent in neural network optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 327-336. doi: 10.3934/naco.2018021

[9]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619

[10]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[11]

Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645

[12]

Yixin Guo, Aijun Zhang. Existence and nonexistence of traveling pulses in a lateral inhibition neural network. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1729-1755. doi: 10.3934/dcdsb.2016020

[13]

Jianhong Wu, Ruyuan Zhang. A simple delayed neural network with large capacity for associative memory. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 851-863. doi: 10.3934/dcdsb.2004.4.851

[14]

Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062

[15]

Sanjay K. Mazumdar, Cheng-Chew Lim. A neural network based anti-skid brake system. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 321-338. doi: 10.3934/dcds.1999.5.321

[16]

Hiroaki Uchida, Yuya Oishi, Toshimichi Saito. A simple digital spiking neural network: Synchronization and spike-train approximation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1479-1494. doi: 10.3934/dcdss.2020374

[17]

Lidong Liu, Fajie Wei, Shenghan Zhou. Major project risk assessment method based on BP neural network. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1053-1064. doi: 10.3934/dcdss.2019072

[18]

K. L. Mak, J. G. Peng, Z. B. Xu, K. F. C. Yiu. A novel neural network for associative memory via dynamical systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 573-590. doi: 10.3934/dcdsb.2006.6.573

[19]

Hyeontae Jo, Hwijae Son, Hyung Ju Hwang, Eun Heui Kim. Deep neural network approach to forward-inverse problems. Networks & Heterogeneous Media, 2020, 15 (2) : 247-259. doi: 10.3934/nhm.2020011

[20]

Danilo Costarelli, Gianluca Vinti. Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators. Mathematical Foundations of Computing, 2020, 3 (1) : 41-50. doi: 10.3934/mfc.2020004

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]