Delay Embeddings for Forced Systems. I. Deterministic Forcing

Summary.

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 that gives 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. Unfortunately this is not the case for many real systems; in the laboratory we often force an experimental system in order for it to exhibit interesting behaviour, whilst in the case of naturally occurring systems it is very rare for us to be able to isolate the system to ensure that there are no external influences. In this paper we therefore prove two versions of Takens Theorem relevant to forced systems: one applicable to the case where the forcing is unknown, and the other to the situation where we are able to determine independently the state of the forcing system (usually because we are responsible for the forcing ourselves). In a subsequent paper we shall show how to extend these results to give an analogue of Takens Theorem for randomly forced systems, leading to a new framework for the analysis of time series arising from nonlinear stochastic systems.