带有两个小世界联接的时滞环形神经网络系统的稳定性分析
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摘要
本文采用算子半群理论和谱分析方法研究一类带有两个小世界联接的时滞环形神经网络系统的适定性以及与时滞无关的稳定性,并根据Schur-Cohn准则讨论了小世界联接权值与系统稳定性之间的关系,建立了系统参数与小世界联接权值所满足的不等式条件.最后通过数值实例的仿真结果佐证了结论的有效性和可行性.
In this article, we adopt the semigroup theory and the spectral analysis method to study the well-posedness and delay-independent stability for a class of delayed ring neural network with two small world connections. Firstly, we formulate the system into an abstract evolution equation and prove the well-posedness by the semigroup approach. Next, we devote to the detailed spectral analysis. It is obtained that with some conditions required on the connection weights between neurons, all eigenvalues are located in the left half complex plane and their real parts go to —∞ as n →∞. The asymptotic spectral expression is also presented. Moreover, according to the Schur-Cohn criterion, we discuss the stability region of the connection weights, the two small world connections and time delay. The results show that the system remains stable when the weight of the small world connection belongs to a certain interval, for any positive value of the delay τ. Finally, the simulation results of numerical examples are presented to illustrate the validity and feasibility of the conclusion.
In this article, we adopt the semigroup theory and the spectral analysis method to study the well-posedness and delay-independent stability for a class of delayed ring neural network with two small world connections. Firstly, we formulate the system into an abstract evolution equation and prove the well-posedness by the semigroup approach. Next, we devote to the detailed spectral analysis. It is obtained that with some conditions required on the connection weights between neurons, all eigenvalues are located in the left half complex plane and their real parts go to —∞ as n →∞. The asymptotic spectral expression is also presented. Moreover, according to the Schur-Cohn criterion, we discuss the stability region of the connection weights, the two small world connections and time delay. The results show that the system remains stable when the weight of the small world connection belongs to a certain interval, for any positive value of the delay τ. Finally, the simulation results of numerical examples are presented to illustrate the validity and feasibility of the conclusion.
引文
[1] Hopfield J J. Neurons with Graded Response have Collective Computational Properties Like Those of Two-state Neurons. Proc. Natl. Acad. Sci. USA, 1984, 81:3088-3092.
[2] Pakdaman K, Grotta-Ragazzo C, Malta C P, Arino O, Vibert J F. Transient Oscillations in Continuoustime Excitatory Ring Neural Networks with Delay. Physical Review E, 1997, 55(3):3234-3248
[3]孙健,陈杰,刘国平.时滞系统稳定性分析与应用.北京:科学出版社,2012(Sun J, Chen J,Liu G P. Stability Analysis and Applications of Time-delay Systems. Beijing:Science Press, 2012)
[4]徐鉴,裴利军.时滞系统动力学近期研究进展与展望,力学进展,2006, 36(1):17-30(Xu J, Pei L J. Advances in dynamics for delayed systems. Advances in Mechanics, 2006, 36(1):17-30)
[5]邹劭芬,黄立宏.多时滞环状神经网络的稳定性.应用数学学报,2007, 30(4):707-718(Zou S F, Huang L H. Stability of a Ring Neural Network with Delays. Acta Mathematicae Applicatae Sinica, 2007, 30(4):707-718)
[6] Belair J. Stability in a model of a delayed neural network. Journal of Dynamics and Differential Equations, 1993, 5:607-623
[7]沈轶,廖晓昕. Hopfield型时滞神经网络的指数稳定性,数学物理学报,1999, 19(2):211-218(Shen Y, Liao X X. Exponential Stability of Hopfield-type Stochastic Delay Neural Networks. Acta Mathematica Scientia,1999, 19(2):211-218)
[8] Xu X. Complicated dynamics of a ring neural network with time delays. Journal of Physics A:Mathematical and Theoretical, 2008, 41:035102
[9]黄萍,张许杰,刘刚.小世界网络的研究现状与展望.情报杂志,2007, 4:66-68(Huang P, Zhang X J, Liu G. The Present Research Situation and Forecast of the Small World Network. Journal of Information, 2007, 4:66-68)
[10] Li C G, Chen G R. Stability of a neural network model with small-world connections. Physical Review E, 2003, 68:052901
[11] Xu X, Wang Z H. Effects of small world connection on the dynamics of a delayed ring network.Nonlinear Dynamic, 2009, 56:127-144
[12]徐旭,张洁,梁艳春.小世界联接对神经网络稳定性的影响,吉林大学学报(理学版),2008, 46(5):873-876(Xu X, Zhang J, Liang Y C. Effects of Small World Connection on the Stability of a Neural Network.Journal of Jilin University(Science Edition), 2008, 46(5):873-876)
[13] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York:Springer-Verlag,1983
[14] Jury E I. Inners and Stability of Dynamic Systems. New York:John Wiley&Sons, 1982
[2] Pakdaman K, Grotta-Ragazzo C, Malta C P, Arino O, Vibert J F. Transient Oscillations in Continuoustime Excitatory Ring Neural Networks with Delay. Physical Review E, 1997, 55(3):3234-3248
[3]孙健,陈杰,刘国平.时滞系统稳定性分析与应用.北京:科学出版社,2012(Sun J, Chen J,Liu G P. Stability Analysis and Applications of Time-delay Systems. Beijing:Science Press, 2012)
[4]徐鉴,裴利军.时滞系统动力学近期研究进展与展望,力学进展,2006, 36(1):17-30(Xu J, Pei L J. Advances in dynamics for delayed systems. Advances in Mechanics, 2006, 36(1):17-30)
[5]邹劭芬,黄立宏.多时滞环状神经网络的稳定性.应用数学学报,2007, 30(4):707-718(Zou S F, Huang L H. Stability of a Ring Neural Network with Delays. Acta Mathematicae Applicatae Sinica, 2007, 30(4):707-718)
[6] Belair J. Stability in a model of a delayed neural network. Journal of Dynamics and Differential Equations, 1993, 5:607-623
[7]沈轶,廖晓昕. Hopfield型时滞神经网络的指数稳定性,数学物理学报,1999, 19(2):211-218(Shen Y, Liao X X. Exponential Stability of Hopfield-type Stochastic Delay Neural Networks. Acta Mathematica Scientia,1999, 19(2):211-218)
[8] Xu X. Complicated dynamics of a ring neural network with time delays. Journal of Physics A:Mathematical and Theoretical, 2008, 41:035102
[9]黄萍,张许杰,刘刚.小世界网络的研究现状与展望.情报杂志,2007, 4:66-68(Huang P, Zhang X J, Liu G. The Present Research Situation and Forecast of the Small World Network. Journal of Information, 2007, 4:66-68)
[10] Li C G, Chen G R. Stability of a neural network model with small-world connections. Physical Review E, 2003, 68:052901
[11] Xu X, Wang Z H. Effects of small world connection on the dynamics of a delayed ring network.Nonlinear Dynamic, 2009, 56:127-144
[12]徐旭,张洁,梁艳春.小世界联接对神经网络稳定性的影响,吉林大学学报(理学版),2008, 46(5):873-876(Xu X, Zhang J, Liang Y C. Effects of Small World Connection on the Stability of a Neural Network.Journal of Jilin University(Science Edition), 2008, 46(5):873-876)
[13] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York:Springer-Verlag,1983
[14] Jury E I. Inners and Stability of Dynamic Systems. New York:John Wiley&Sons, 1982