Time series-based bifurcation diagram reconstruction

We consider the problem of reconstructing bifurcation diagrams (BDs) of map s using time series. This study goes along the same line of ideas presented by Tokunaga et al. [Physica D 79 (1994) 348] and Tokuda et al. [Physica D 95 (1996) 380]. The aim is to reconstruct the ED of a dynamical system with out the knowledge of its functional form and its dependence on the paramete rs. Instead, time series at different parameter values, assumed to be avail able, are used. A three-layer fully-connected neural network is employed in the approximation of the map. The task of the network is to learn the dyna mics of the system as function of the parameters from the available time se ries. We determine a class of maps for which one can always find a linear s ubspace in the weight space of the network where the network's bifurcation structure is qualitatively the same as the bifurcation structure of the map . We discuss a scheme in locating this subspace using the time series. We f urther discuss how to recognize time series generated by this class of maps . Finally, we propose an algorithm in reconstructing the BDs of this class of maps using predictor functions obtained by neural network. This algorith m is flexible so that other classes of predictors, apart from neural networ ks, can be used in the reconstruction. (C)1999 Elsevier Science B.V. All ri ghts reserved.