Critical dynamical analysis for α-UAM RNNs without diagonal nonlinear requirements

Critical dynamics research of recurrent neural networks (RNNs) is very meaningful in both theoretical importance and practical significance. Recently, because of the application requirements, the study on the critical dynamics behaviors of RNNs has drawn special attention. The critical condition is that a discriminant matrix M1 (Γ) related with an RNN is nonnegative definite. Due to the essential difficulty in analysis, there were only a few critical results up to now. Further, nearly all of the existing dynamic results are with diagonally nonlinear requirements on the activation mappings, i.e., the activation mapping G should satisfy the strict necessary condition that G (x) = (g1 (x1) , g2 (x2) , ⋯ , gN (xN)) T. This is because of the essential difficulty on the analysis of the energy function. The requirement is so strict and it limits the applications of RNNs. In this paper, under the critical conditions, some new global asymptotically stable conclusions are presented for RNNs without the diagonally nonlinear requirement on the activation mappings. The results present here not only improve substantially upon the existing relevant critical stability results, but also provide some further cognizance on the essentially dynamical behavior of RNNs, and further, enlarge the application fields of them.