Delay Embeddings for Forced Systems. II. Stochastic Forcing |
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Authors: | Stark J Broomhead DS Davies ME Huke J |
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Institution: | (1) Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom;(2) Department of Mathematics, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, M60 1QD, United Kingdom;(3) Department of Electronic Engineering, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom |
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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. |
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