摘要
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 E 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 eases is true. 1. For any initial value X0∈R^n, the solution approximates asymptotically to zero vector. In thin cane, the real part of each eigenvalue of A is non-positive. 2. For any initial value X0 outside a proper subspace of R^n,the solution approximates asymptoticaUy to a nontrivial constant vector Y( X0 ). In this cane, the eigenvalue of A with maximal real part is the positive number λ=Ⅱ Y (X0)ⅡB^2 and Y (X0) is the corre-sponding eigenvector. 3. For any initial value X0 outsidea proper subspace of R^n, the solution approximates asymptotically to a non-constant periodic function Y( X0 , t ). Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed.
The authors present their analysis of the differential equation d X(t)/ d t = 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 ∈R n; 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 0∈R n, 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 0 outside a proper subspace of R n, the solution approximates asymptotically to a nontrivial constant vector (X 0) . In this case, the eigenvalue of A with maximal real part is the positive number λ=‖(X 0)‖ 2 B and (X 0) is the corresponding eigenvector. 3. For any initial value X 0 outside a proper subspace of R n, the solution approximates asymptotically to a non constant periodic function (X 0,t) . Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed.
