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Abstract: The authors present their analysis of the differential equation dX(t)/dt=AX(t)-X^T(t)BX(t)X(t), where A is an unsymmetrical real matrix, B is a positive definite symmetric real matrix, X∈Rⁿ; showing that the equation characterizes a class of continuous type full-feedback artificial neural network; We give the analytic expression of the solution; discuss its asymptotic behavior; and finally present the result showing that, in almost all cases, one and only one of following cases is true. 1. For any initial value X₀∈Rⁿ, the solution approximates asymptotically to zero vector. In this case, the real part of each eigenvalue of A is non-positive. 2. For any initial value X₀ outside a proper subspace of Rⁿ, the solution approximates asymptotically to a nontrivial constant vector &Ytilde;(X₀). In this case, the eigenvalue of A with maximal real part is the positive number λ=‖(X₀)‖²_B and (X₀) is the corresponding eigenvector. 3. For any initial value X₀ outside a proper subspace of Rⁿ, the solution approximates asymptotically to a non-constant periodic function &Ytilde;(X₀,t). Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed.

[1] Bryan P. Rynne, Martin A. Youngson, 2000. Linear Functional Analysis. Springer.

[2] Chen Tianping, Shun Ichi Amari, Qin Lin, 1998. An unified algorithm for principal and minor components extraction. Neural Networks 11(3):385-390.

[3] David Betounes, 2001.Differential equations: theory and application. Maple, Springer-Verlag, New York, p.89.

[4] Lei Xu, 1993. Least MSE Reconstruction: A principle for self-organizing nets, Neural Networks, 6(6):627-648.

[5] Luo Falong, Li Yanda, Wang Zhe, 1994. Finding eigenvectors of a class of positive matrix by neural networks. Science in China, 24(1): 83-88 (in Chinese).

[6] Luo Falong, Rolf Unbehauen, 1999. Comments on: A unified algorithm for principal and minor components extraction, Neural Networks 12(2):393.

[7] Oja, E., 1982. A simplified neuron model as a principle component analyzer. J Math Biol, 15:267-273.

[8] Oja, E., 1989. Neural networks, principal components, and subspace, Int J Neural System, 1:61-68.

[9] Stephen H. Friedberg, Arndd J.Insel, Lawrence E. Spence, 1989. Linear Algebra. Prentice Hall, Inc.

[10] Wang Zhe, Li Yanda, Luo Falong, 1996, A neural network learning algorithm for PCA and MCA. Acta Electronica Sinica, 24:12-16(in Chinese).

[11] Yang Jiangang, Wang Kai, Yang huayong et al.,2000. A real-time adaptive control algorithm using neural nets with perturbation. Journal of Zhejiang University SCIENCE, 1(1):61-65.

[12] Yang Jiangang, Wang Ruming.,2000. A Rapid Fuzzy Rule Extraction Method For Fuzzy Controller. Journal of Zhejiang University SCIENCE, 1(3):311-376.

[13] Zhang Yi, Wang Pingan, Zhou Mingtian, 2000, Computing eigenvalues and eigen vectors of matrix by neural networks. CHINESE J.COMPUTERS, 23(1):71-76 (in Chinese).