Europhysics Letters Robust Chaos Generation by a Perceptron

{ The properties of time series generated by a perceptron with monotonic and non-monotonic transfer function, where the next input vector is determined from past output values, are examined. Analysis of the parameter space reveals the following main nding: a perceptron with a monotonic function can produce fragile chaos only whereas a non-monotonic function can generate robust chaos as well. For non-monotonic functions, the dimension of the attractor can be controlled monotonically by tuning a natural parameter in the model. Traditionally, most neural networks with a non-local transfer function use monotonic functions , e.g., sigmoid or hyperbolic-tangent. On the other hand, networks based on local units, e.g., radial basis functions, use non-monotonic functions. In this letter, we focus on the interplay between the types of transfer functions used and some quantitative measures of the time series generated by the networks. Previous analytical studies concentrated on the stable regime of the parameter space of feed-forward networks with feedback that generate time series 1, 2, 3, 4, 5]. The model (previously termed \sequence generator"-SGen) is described mathematically as follows. For a given input vector at time step t, ~ S t (S t j ; j = 1; : : : ; N), the network's output S t out is given by S t out = f 0 @ N X j=1 W j S t j 1 A (1) where ~ W are the weights, is a gain parameter and f is a transfer function. The input vector at time t + 1 is given by the previous output values S t+1 1 = S t out ; S t+1 j = S t j?1 j = 2; : : :; N : (2) This model exhibits (quasi) periodic attractors in the stable regime, regardless the complexity of the weights, both in the case of a perceptron as well as multilayer SGen's 3, 5].