Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra.

TLDR: This work considers the problem of prediction and system identification for time series having broadband power spectra that arise from the intrinsic nonlinear dynamics of the system and finds that the parameter values that minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system.

Abstract: We consider the problem of prediction and system identification for time series having broadband power spectra that arise from the intrinsic nonlinear dynamics of the system. We view the motion of the system in a reconstructed phase space that captures the attractor (usually strange) on which the system evolves and give a procedure for constructing parametrized maps that 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 and not just a fit to the data. The invariants of the dynamical system, the Lyapunov exponents and optimum moments of the invariant density on the attractor, are used as constraints on the choice of mapping parameters. The parameter values are chosen through a constrained least-squares optimization procedure, constrained by the values of these invariants. We give a detailed discussion of methods to extract the Lyapunov exponents and optimum moments from data and show how to equate them to the values for the parametric map in the constrained optimization. We also discuss the motivation and methods we utilize for choosing the form of our parametric maps. Their form has a strong similarity to the work in statistics on kernel density estimation, but the goals and techniques differ in detail. Our methodology is applied to "data" from the H\'enon map and the Lorenz system of differential equations and shown to be feasible. We find that the parameter values that minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system. The maps that 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 few or no data are available), onto the attractor where long-term motion takes place. We find that we are able to realize maps that give a least-squares approximation to the data with rms variation over the attractor of 0.5% or less and still reproduce the dynamical invariants to 5% or better. The dynamical invariants are the classifiers of the dynamical system producing the broadband time series in the first place, so this quality of the maps is essential in representing the correct dynamics.