跳到主要內容

臺灣博碩士論文加值系統

(44.220.251.236) 您好!臺灣時間:2024/10/11 04:01
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果 :::

詳目顯示

: 
twitterline
研究生:鄭昌源
論文名稱:延遲型神經網路之多重穩定性及收斂性
論文名稱(外文):Multistability and convergence in delayed neural networks
指導教授:石至文
學位類別:博士
校院名稱:國立交通大學
系所名稱:應用數學系所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2006
畢業學年度:94
語文別:英文
論文頁數:65
中文關鍵詞:神經網路延遲多重穩定性
外文關鍵詞:neural networksdelayedmultistability
相關次數:
  • 被引用被引用:0
  • 點閱點閱:256
  • 評分評分:
  • 下載下載:11
  • 收藏至我的研究室書目清單書目收藏:0
本論文主要在於研究延遲型神經網路系統具有多平衡點時的動態收斂性及多重穩定性。此篇論文首先討論了延遲型微分方程的基礎理論及單調性動態系統之收斂性質,繼而研究高維度延遲型神經網路系統中的多重穩定性和擬收斂性。我們在具有飽和或非飽和S型活化函數的延遲型神經系統中,藉由幾何方法設定參數條件以證明多平衡點的存在性,並在擁有多平衡點的系統中建立正向不變區域以及穩定性平衡點的吸引盆。當限制抑制性延遲回饋時間夠小時,可以更進一步探討此系統的強保序性質,並得知一般解存有擬收斂性。因此、本文在高維度延遲型神經網路系統中同時建立了多平衡點的存在性及一般解的擬收斂性。我們也在文中描敘幾個數值模擬,以佐證所獲得之理論。
1 Introduction 1
2 Basic Notions of Delayed Di®erential Equations 6
2.1 Fundamental Theorems in Delayed Equations . . . . . . . . . . 6
2.2 Fundamental Theorem for Delayed Neural Networks . . . . . . . . 8
2.3 Lyapunov Functional and Lyapunov-Razumikhin Theorem . . . . . . . 8
3 Monotone Dynamical Systems 12
3.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 The Convergence Criterion . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Generic Quasiconvergence . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Global Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Neural Networks with Delays 21
4.1 Global Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Comparison of Neural Networks with and without Delays . . . . . . . . 23
4.2.1 Characteristic Equations . . . . . . . . . . . . . . . . . . . . . . 23
4.2.2 Lyapunov Functionals and Lyapunov Functions . . . . . . . . . 24
4.3 Activation Functions and Multiple Equilibria . . . . . . . . . . . . . . . 26
4.4 Stability of Equilibria and Basins of Attraction . . . . . . . . . . . . . 32
4.5 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.6 Extending Basins of Attraction . . . . . . . . . . . . . . . . . . . . . . 42
4.7 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Monotonicity, Convergence and Quasiconvergence in Delayed Neural
Networks 50
5.1 Quasiconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
\bibitem{Amann76}
H. Amann, {\em Fixed point equations and nonlinear eigenvalue
problems in ordered Banach spaces}, SIAM Rev., 18 (1976), pp.
620--709.

\bibitem{Baldi-Atiya94}
P. Baldi and A. F. Atiya, {\em How delays affect neural dynamics
and learning}, IEEE Trans. Neural Networks, 5 (1994), pp.
612--621.

\bibitem{Belair-Campbell-Driessche96}
J. B\'{e}lair, S. A. Campbell and P. van den Driessche, {\em
Frustration, stability, and delay-induced oscillations in a neural
network model}, SIAM J. Appl. Math., 56 (1996), pp. 245--255.

\bibitem{Campbell-Edwards-Driessche04}
S. A. Campbell, R. Edwards and P. van den Driessche, {\em Delayed
coupling between two neural networks loops}, SIAM J. Appl. Math.,
65 (2004), pp. 316--335.

\bibitem{Cao99}
J. Cao, {\em Global stability analysis in delayed cellular neural
networks}, Phys. Rev. E, 59 (1999), pp. 5940--5944.

\bibitem{Cao03}
J. Cao, {\em New results concerning exponential stability and
periodic solutions of delayed cellular neural networks}, Phys.
Lett. A, 307 (2003), pp. 136--147.

\bibitem{Cao-Li00}
J. Cao and Q. Li, {\em On the exponential stability and periodic
solutions of delayed cellular neural networks}, J. Math. Anal.
Appl., 252 (2000), pp. 50--64.

\bibitem{Civalleri-Gilli94}
P. P. Civalleri and M. Gilli, {\em On stability of cellular neural
networks with delay}, IEEE Trans. Circuits Syst., 40 (1993), pp.
157--165.

\bibitem{Cheng-Lin-Shih06}
C. Y. Cheng, K. H. Lin and C. W. Shih, {\em Multistability in
recurrent neural networks}, SIAM Appl. Math., 66 (2006), pp.
1301-1320.

\bibitem{Cheng-Lin-Shih}
C. Y. Cheng, K. H. Lin and C. W. Shih, {\em Multistability and
convergence in delayed neural networks}, submitted, 2005.

\bibitem{Cheng-Shih}
C. Y. Cheng and C. W. Shih, {\em Global dynamics for multi-stable
delayed neural networks}, preprint, 2006.

\bibitem{Chu-Zhang-Wang03}
T. Chu, Z. Zhang and Z. Wang, {\em A decomposition approach to
analysis of competitive-cooperative neural networks with delay},
Phys. Lett. A, 312 (2003), pp. 339--347.

\bibitem{Chua}
L. O. Chua, {\em CNN: A paradigm for complexity}, World
Scientific, 1998.

\bibitem{Chua-Yang88}
L. O. Chua and L. Yang, {\em Cellular neural networks: Theory},
IEEE Trans. Circuits Syst., 35 (1988), pp. 1257--1272.

\bibitem{Cohen-Grossberg83}
M. A. Cohen and S. Grossberg, {\em Absolute stability of global
pattern formation and parallel memory storage by competitive
neural networks}, IEEE Trans. Syst. Man Cybern, 13 (1983), pp.
815--826.

\bibitem{Dieudonne}
J. Dieudonn\'{e}, {\em Foundations of Modern Analysis}, Academic
Press, New York, 1969.

\bibitem{Feng-Plamondon01}
C. Feng and R. Plamondon, {\em On the stability analysis of
delayed neural network systems}, Neural Networks, 14 (2001), pp.
1181--1188.

\bibitem{Foss}
J. Foss, A. Longtin, B. Mensour and J. Milton, {\em Multistability
and delayed recurrent loops}, Phys. Rev. Lett., 76 (1996), pp.
708--711.

\bibitem{Gilli94}
M. Gilli, {\em Stability of cellular neural networks and delayed
cellular neural networks with nonpositive templates and
nonmonotonic output functions}, IEEE Trans. Circuits Syst. I, 41
(1994), pp. 518--528.

\bibitem{Gyori-Hartung03}
I. Gy\H{o}ri and F. Hartung, {\em Stability analysis of a single
neuron model with delay}, J. Comp. Appl. Math., 157 (2003), pp.
73--92.

\bibitem{Haddock-Terjeki83}
J. R. Haddock and J. Terj\'{e}ki, {\em Liapunov-Razumikhin
functions and an invariance principle for functional differential
equations}, J. Diff. Eq., 48 (1983), pp. 95--122.

\bibitem{Hahnloser}
R. L. T. Hahnloser, {\em On the piecewise analysis of networks of
linear threshold neurons}, Neural Networks, 11 (1998), pp.
691--697.

\bibitem{Hale80}
J. Hale, {\em Ordinary differential equations}, Second edition,
Robert E. Krieger Publishing Co., 1980.

\bibitem{Hale88}
J. Hale, {\em Asymptotic behavior of dissipative systems}, AMS,
1988.

\bibitem{Hale}
J. Hale and S. V. Lunel, {\em Introduction to functional
differential equations}, Springer-Verlag, 1993.

\bibitem{Hirsch87}
M. W. Hirsch, {\em Convergence in neural networks}, In: Proc. of
the 1st Int. Conf. on Neueal Networks. San Diego, IEEE Service
Center, 1987, pp. 115--126.

\bibitem{Hirsch88}
M. W. Hirsch, {\em Stability and convergence in strongly monotone
dynamical systems}, J. Reine Angew. Math., 383 (1988), pp. 1--53.

\bibitem{Hirsch89}
M. W. Hirsch, {\em Convergence activation dynamics in continuous
time networks}, Neueal Networks, 2 (1989), pp. 331--349.

\bibitem{Hopfield84}
J. Hopfield, {\em Neurons with graded response have collective
computational properties like those of two sate neurons}, Proc.
Natl. Acad. Sci. USA, 81 (1984), pp. 3088--3092.

\bibitem{Joy}
M. Joy, {\em Results concerning the absolute stability of delayed
neural network}, Neural Networks, 13 (2000), pp. 613--616.

\bibitem{Krisztin-Walther-Wu99}
T. Krisztin, H. O. Walther and J. Wu, {\em Shape, smoothness and
invariant stratification of an attracting set for delayed positive
feedback}, Fiel. Inst. Mono. Seri., vol. 11, Amer. Math. Soc.,
Providence, RI, 1999.

\bibitem{Kolmanovskii}
V. Kolmanovskii and V. Nosov, {\em Stability of functional
differential equations}, Academic Press, London, 1986.

\bibitem{Liao-Chen-Sanchez02}
X. Liao, G. Chen and E. N. Sanchez, {\em Delay-dependent
exponential stability analysis of delayed neural networks: an LMI
appraoch}, Neural Networks, 15 (2002), pp. 855--866.

\bibitem{Liao-Li05}
X. Liao and C. Li, {\em An LMI approach to asymptotical stability
of multi-delayed neural network}, Phys. D, 200 (2005), pp.
139--155.

\bibitem{Liao-Wang03}
X. Liao and J. Wang, {\em Global dissipativity of continuous-time
recurrentneural networks with time delay}, Phys. Rev. E, 68
(2003), 016118.

\bibitem{Lin-Shih99}
S. S. Lin and C. W. Shih, {\em Complete stability for standard
cellular neural network}, Int. J. Bifurcation and Chaos, 9 (1999),
pp. 909--918.

\bibitem{Marcus-Westervelt89}
C. M. Marcus and R. M. Westervelt, {\em Stability analog neural
networks with delay}, Phys. Rev. A, 39 (1989), pp. 347--359.

\bibitem{Matano84}
H. Matano, {\em Existence of nontrival unstable sets for
equilibria of strongly order preserving systems}, J. Fac. Sci.
Univ. Tokyo, 30 (1984),pp. 645--673.

\bibitem{Mohamad-Gopalsamy03}
S. Mohamad and K. Gopalsamy, {\em Exponential stability of
continuous-time and discrete-time cellular neural networks with
delays}, Appl. Math. Comp., 135 (2003), pp. 17--38.

\bibitem{Morita}
M. Morita, {\em Associative memory with non-monotone dynamics},
Neural Networks, 6 (1993), pp. 115-126.

\bibitem{Olien-Belair97}
L. Olien and J. B\'{e}lair, {\em Bifurcations, stability, and
monotonicity properties of a delayed neural network model}, Phys.
D, 102 (1997), pp. 349--363.

\bibitem{Pituk03}
M. Pituk, {\em Convergence to equilibria in scalar
nonquasimonotone functional differential equations}, J. Diff. Eq.,
193 (2003), pp. 95--130.

\bibitem{Roska-Chua92}
T. Roska and L. O. Chua, {\em Cellular neural networks with
nonlinear and delay-type template elements and non-uniform grids},
Int. J. Circuit Theory Appl., 20 (1992), pp. 469--481.

\bibitem{Shayer-Campbell00}
L. P. Shayer and S. A. Campbell, {\em Stability, bifurcation, and
multistability in a system of two coupled neurons with multiple
time delays}, SIAM J. Appl. Math., 61 (2000), pp. 673--700.

\bibitem{Shih-Weng00}
C. W. Shih and C. W. Weng, {\em Cycle-symmetric matrices and
convergence neural networks}, Physica D, 146 (2000), pp. 213--220.

\bibitem{Shih01}
C. W. Shih, {\em Complete stability for a class of cellular neural
networks}, Int. J. Bifurcation and Chaos, 11 (2001), pp. 169--177.

\bibitem{Smith95}
H. L. Smith, {\em Monotone dynamical systems: an introduction to
the theory of competitive and cooperative systems}, Math. Surveys
Monographs 41, AMS, Providence, RI, 1995.

\bibitem{Smith-Thieme91}
H. L. Smith and H. R. Thieme, {\em Strongly order preserving
semiflows generated by functional differential equations}, J.
Diff. Eq., 93 (1991), pp. 332--363.

\bibitem{Stepan}
G. St\'{e}p\'{a}n, {\em Retarded dynamical systems}, Pitman
Research Notes in Mathematics, vol. 210, Longman Group, Essex,
1989.

\bibitem{Takahashi00}
N. Takahashi, {\em A new sufficient condition for complete
stability of cellular neural networks with delay}, IEEE Tran.
Circ. Syst. I, 47 (2000), pp. 793--799.

\bibitem{Takahashi-Chua98}
N. Takahashi and L. O. Chua, {\em On the complete stability of
nonsymmetric cellular neural networks}, IEEE Tran. Circ. Syst. I,
45 (1998), pp. 754--758.

\bibitem{Driessche98}
P. van den Driessche and X. Zou, {\em Global attractivity in
delayed Hopfield neural network models}, SIAM J. Appl. Math., 58
(1998), pp. 1878--1890.

\bibitem{Driessche-Wu-Zou01}
P. van den Driessche, J. Wu and X. Zou, {\em Stabilization role of
inhibitory self-connections in a delayed neural network}, Phys. D,
150 (2001), pp. 84--90.

\bibitem{Wu-Chua97}
C. W. Wu and L. O. Chua, {\em A more rigious proof of complete
stability of cellular neural networks}, IEEE Trams. Circuits Syst.
I,44 (1997), pp. 370--371.

\bibitem{Zhang-Wei-Xu03}
Q. Zhang, X. Wei and J. Xu, {\em Global exponential convergence
analysis of delayed neural networks with time-varying delays},
Phys. Lett. A, 318 (2003), pp. 537--544.

\bibitem{Zeng-Huang-Wang05}
Z. Zeng, D. S. Huang and Z. Wang, {\em Memory pattern analysis of
cellular neural networks}, Phys. Lett. A, 342 (2005), pp. 114-128.
QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top