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Chaos and asymptotical stability in discrete-time neural networks. (English) Zbl 0889.68122
Summary: By applying Marotto’s theorem this paper aims to prove that both transiently chaotic neural networks (TCNN) and discrete-time recurrent neural networks (DRNN) have chaotic structure. A significant property of 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 can actually evolve into a snap-back repeller which generates a 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 versions of both TCNN and DRNN, under which TCNN and DRNN asymptotically converge to a fixed point. Furthermore, related bifurcations are also considered in this paper. Since both TCNN and DRNN are not special but simple and general, the obtained theoretical results hold for a wide class of discrete-time neural networks. To demonstrate the theoretical results of this paper better, several numerical simulations are provided as illustrating examples.

68T05Learning and adaptive systems
37D45Strange attractors, chaotic dynamics