Embedding Theorems for Non-uniformly Sampled Dynamical Systems

Huke, J. P. and Broomhead, D. S. (2006) Embedding Theorems for Non-uniformly Sampled Dynamical Systems. [MIMS Preprint]

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Abstract

The embedding theorem of Takens, and its extensions, have provided the theoretical underpinning for a wide range of investigations of time series derived from nonlinear dynamical systems. The theorem applies when the dynamical system is sampled uniformly in time. There has, however, been increasing interest in situations where observations on the system are not uniform in time, and in particular where the consist of a series of inter-event (`interspike') intervals. Sauer has provided an embedding theorem for the case where these intervals are generated by an integrate-and-fire mechanism. Here we prove several embedding theorems pertaining to non-uniform sampling. We consider two situations: in the first, observations consist of the values of some function on the state space of the system, with the times between the successive observations being given by another function—the sampling interval function; in the second, sampling times are generated in the same way, but now the observations consist only of the intersample intervals. We prove embedding theorems both when the sampling interval function is allowed to be rather general, and when it corresponds to integrate-and-fire sampling. We point out that non-uniform sampling might lead to better reconstructions than uniform sampling, for certain kinds of time series.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	time series, nonlinear dynamics, embedding theorem
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory MSC 2010, the AMS's Mathematics Subject Classification > 53 Differential geometry
Depositing User:	Dr J.P. Huke
Date Deposited:	20 Sep 2006
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/599

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