| Abstrakt: |
The authors present their analysis of the differential equation d X( t/d t = AX( t) - X( t) BX( t) X( t), where A is an unsymmetrical real matrix, B is a positive definite symmetric real matrix, X ∈ R; 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 X∈ R, 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 X outside a proper subspace of R, the solution approximates asymptotically to a nontrivial constant vector Ỹ( X). In this case, the eigenvalue of A with maximal real part is the positive number λ = | Ỹ( X) | and B is the corresponding eigenvector. 3. For any initial value X outside a proper subspace of R, the solution approximates asymptotically to a non-constant periodic function Ỹ( X, t). Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed. [ABSTRACT FROM AUTHOR] |
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