Delay Embeddings for Forced Systems. II. Stochastic Forcing

Abstract

Takens’ Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It typically allows us to reconstruct an unknown dynamical system which gave rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal separation, and most recently for control and targeting. Current versions of Takens’ Theorem assume that the underlying system is autonomous (and noise-free). Unfortunately this is not the case for many real systems. In a previous paper, one of us showed how to extend Takens’ Theorem to deterministically forced systems. Here, we use similar techniques to prove a number of delay embedding theorems for arbitrarily and stochastically forced systems. As a special case, we obtain embedding results for Iterated Functions Systems, and we also briefly consider noisy observations.