Prediction and system identification in chaotic nonlinear systems: Time series with broadband spectra

Abstract

We consider the problem of prediction and system identification for time series having broadband power spectra which arise from the intrinsic nonlinear dynamics of the system. We view the motion of the system in a reconstructed phase space which captures the attractor (usually strange) on which the system evolves, and give a procedure for constructing parameterized maps which evolve points in the phase space into the future. The predictor of future points in the phase space is a combination of operation on past points by the map and its iterates. Thus the map is regarded as a dynamical system, not just a fit to the data. The invariants of the dynamical system — the Lyapunov exponents and aspects of the invariant density on the attractor — are used as constraints on the choice of mapping parameters. The parameter values are chosen through a least-squares optimization procedure. The method is applied to “data” from the Hénon map and shown to be feasible. It is found that the parameter values which minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system. The maps which do reproduce the values of the invariants are not optimum in the least-squares sense, yet still are excellent predictors. We discuss several technical and general problems associated with prediction and system identification on strange attractors. In particular, we consider the matter of the evolution of points that are off the attractor (where little or no data is available), onto the attractor, where long-term motion takes place.