CHAOS AND ASYMPTOTICAL STABILITY IN DISCRETE-TIME NEURAL NETWORKS

This paper aims to theoretically prove by applying Marotto's Theorem t hat both transiently chaotic neural networks (TCNN) and discrete-time recurrent neural networks (DRNN) have chaotic structure. A significant property TCNN and DRNN is that they have only one bounded fixed point , when absolute values of the self-feedback connection weights in TCNN and the difference time in DRNN are sufficiently large. We show that this unique fixed point tan actually evolve into a snap-back repeller which generates chaotic structure, if several conditions are satisfied . On the other hand, by using the Lyapunov functions, we also derive; sufficient conditions on asymptotical stability for symmetrical versio ns of both TCNN and DRNN, under which TCNN and DRNN asymptotically con verge to a fixed point. Furthermore, related bifurcations are also con sidered in this paper. Since both TCNN and DRNN are not special but si mple and general, the obtained theoretical results hold for a wide cla ss of discrete-time neural networks. To demonstrate the theoretical re sults of this paper better, several numerical simulations ale provided as illustrating examples.