Hopfield networks have been applied and their stable behavior studied. Nevertheless, neural networks as non-linear systems may exhibit very rich behaviors which can eventually be exploited to extend the range of their applications. In this paper we show different behaviors that are found when the condition of symmetry in the weight matrix is broken in discrete and continuous Hopfield networks, in both cases being discrete-time synchronous. Specifically, an outer product asymmetry (OPA) is defined which lends itself to experimentally find expressions for bounds of structural stability. It is found, for all the cases examined, that these formulas lead to rational numbers and depend only on the size of the network and the number of stored patterns. Also, another type of asymmetry, which we refer to as competitive asymmetry, is applied. This type of asymmetry produces quasi-periodic and chaotic behaviors. Lyapunov exponents and the fractal dimension of the attractors, which characterize these behaviors, are calculated by means of several computational methods.
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