臺灣博碩士論文加值系統

這篇論文主要研究散佈在一維整數網格上的遲滯型細胞神經網路似駝峰行進波的種類。細胞元之間的動態行為除了有瞬時的自身回饋外，由於信號傳播轉換速度的關係，會與左邊最鄰近的m個細胞元產生遲滯相互作用。在本文中，我們使用階梯法直接勾勒出解析解的形式，並進一步證明除了單調行進波的存在性外，在某些適當條件下，亦存在似駝峰的非單調行進波。最後我們也搭配一些數值結果來驗證理論分析。

In this thesis, we study the camel-like traveling wave solutions for a class of delayed cellular neural networks distributed in the one-dimensional integer lattice Z. The dynamics of a given cell is characterized by instantaneous
self-feedback and neighborhood interaction with its nearest m left neighbors with distributive delay due to, for example, finite switching speed and finite velocity of signal transmission.
Using the method of step, we can directly figure out the analytic solution and then prove that,
in addition to the existence of monotonic
traveling wave solutions, for certain templates there exist non-monotonic traveling wave solutions such as camel-like waves with many critical points. Some numerical results are also given.

Contents

˙Abstract ………………………………………………………………………1
˙Introduction …………………………………………………………………2
˙Construction of the solution formula …………………………………6
˙Proof of the main theorem ………………………………………………14
---3.1. Profiles for αj>0, 1≦j≦m ……………………………………15
---3.2. Profiles for αj<0, 1≦j≦m ……………………………………16
---3.3. Profiles for alternating αj with |αj|≧|αj+1| …………17
˙References …………………………………………………………………30

References
[1] P. W. Bates, X. Chen, and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), pp. 520-546.
[2] J. W. Cahn, J. Mallet-Paret, and E. S. Van Vleck, Traveling wave solutions for systems of ODE's on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), pp. 455-493.
[3] S.-N. Chow, J. Mallet-Paret, and W. Shen, Traveling waves in lattice dynamical systems, J. Diff. Eqns., 149 (1998), pp. 248-291.
[4] L. O. Chua, CNN: A Paradigm for Complexity, World Scientific Series on Nonlinear Science, Series A, Vol. 31, World Scientific, Singapore, 1998.
[5] L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), pp. 147-156.
[6] L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35 (1988), pp. 1257-1272.
[7] L. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits Syst., 35 (1988), pp. 1273-1290.
[8] T. Erneux and G. Nicolis, Propagation waves in discrete bistable reaction-diffusion systems, Physica D, 67 (1993), pp. 237-244.
[9] I. Gyori and G. Ladas, Oscillating Theory of Delay Differential Equations with
Applications, Oxford University Press, Oxford, 1991.
[10] C.-H. Hsu and S.-S. Lin, Existence and multiplicity of traveling waves in
a lattice dynamical system, J. Diff. Eqns., 164 (2000), pp. 431-450.
[11] C.-H. Hsu, S.-S. Lin, and W. Shen, Traveling waves in cellular neural networks,
Internat. J. Bifur. and Chaos, 9 (1999), pp. 1307-1319.
[12] C.-H. Hsu and S.-Y. Yang, On camel-like traveling wave solutions in cellular neural networks, J. Diff. Eqns., 196 (2004), pp. 481-514.
[13] C.-H. Hsu and S.-Y. Yang, Wave propagation in RTD-based cellular neural
networks, to appear in J. Diff. Eqns.
[14] C.-H. Hsu and S.-Y. Yang, Structure of a class of traveling waves in delayed
cellular neural networks, submitted for publication, 2002.
[15] J. Juang and S.-S. Lin, Cellular neural networks: mosaic pattern and spatial chaos, SIAM J. Appl. Math., 60 (2000), pp. 891-915.
[16] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), pp. 556-572.
[17] S.-S. Lin and C.-W. Shih, Complete stability for standard cellular neural networks, Internat. J. Bifur. and Chaos, 9 (1999), pp. 909-918.
[18] S. Ma, X. Liao, and J. Wu, Traveling wave solutions for planar lattice differential systems with applications to neural networks, J. Diff. Eqns., 182 (2002), pp. 269-297.
[19] J. Mallet-Paret, The global structure of traveling waves in spatial
discrete dynamical systems, J. Dynam. Diff. Eqns., 11 (1999), pp. 49-127.
[20] W. Shen, Traveling waves in time almost periodic structure governed by bistable nonlinearities I. stability and uniqueness, J. Diff. Eqns., 159 (1999), pp. 1-54.
[21] W. Shen, Traveling waves in time almost periodic structure governed by bistable nonlinearities II. existence, J. Diff. Eqns., 159 (1999), pp. 55-101.
[22] C.-W. Shih, Complete stability for a class of cellular neural
networks, {it Internat. J. Bifur. and Chaos},
9 (2001), pp. 169-177.
[23] P. Thiran, Dynamics and Self-Organization of Locally Coupled
Neural Networks, Presses Polytechniques et Universitaires Romandes, Lausanne,
Switzerland, 1997.
[24] P. Thiran, K. R. Crounse, L. O. Chua, and M. Hasler, Pattern formation properties of autonomous cellular neural networks, IEEE Trans. Circuit Syst., 42 (1995), pp. 757-774.
[25] F. Werblin, T. Roska, and L. O. Chua, The analogic cellular neural network as a bionic eye, Internat. J. Circuit Theory Appl., 23 (1994), pp. 541-569.
[26] P. Weng and J. Wu, Deformation of traveling waves in delayed cellular neural
networks, Internat. J. Bifur. and Chaos, 13 (2003), pp. 797-813.
[27] J. Wu and X. Zou, Asymptotical and periodic boundary value problems of mixed
FDEs and wave solutions of lattice differential equations, J. Diff. Eqns., 135 (1997), pp. 315-357.
[28] B. Zinner, Existence of traveling wavefront solutions for discrete
Nagumo equation, {it J. Diff. Eqns.}, 96 (1992), pp. 1-27.
[29] B. Zinner, G. Harris, and W. Hudson, Traveling wavefronts for the discrete
Fisher's equation, J. Diff. Eqns., 105 (1993), pp. 46-62.