A nonlinear system with singular vector field near equilibria

Abstract

The system Nω=(N-α)ω+y, N= bN+aωω^T, N(t)∊R^m×m, ω(t)∊R^m which originally arose from a model for the pathological behavior of neural networks, is studied. Similar equations can arise in a variety of applications. It is shown that if N(0) is positive definite, then solutions exist for all time. Equilibrium points are determined. N is found to be singular at the equilibrium points, making the analysis of the asymptotic properties of the system non-trivial. The asymptotic behavior when y = 0 is completely described. Some results are proven on the asymptotic behavior of N and ω when y≠0

^†Supported in part by National Science Foundation Grant MCS75-15153-A01 and the Weizmann Institute.

^†Supported in part by National Science Foundation Grant MCS75-15153-A01 and the Weizmann Institute.

Notes

^†Supported in part by National Science Foundation Grant MCS75-15153-A01 and the Weizmann Institute.