Reconstructing Dynamical Systems Driven by External Events

Jaroslav Stark,
Centre for Nonlinear Dynamics and its Applications,
University College London, Gower Street, London, WC1E 6BT.

Historically, the vast majority of the theory of nonlinear dynamical
systems was developed under the assumption that one knew the state space
and evolution equations of the system under consideration. In principle the
application of these results to real problems thus required the
simultaneous observation of all the state space variables. This is
obviously not possible in many practical applications, where all that one
can manage to do is to make a sequence of repeated measurements of one or
more observables, whose relationship to the state variables is at best
uncertain. It is then of fundamental importance to understand how much
information about the original dynamical system can be extracted from such
a time series.

In the case of autonomous systems there exists a well developed theoretical
framework for addressing this question. This is based on Takens Embedding
Theorem which typically allows us to reconstruct the dynamics of an unknown
system from a scalar time series generated by that system. Over the last
two decades, this has led to a wide variety of new techniques for the
analysis and manipulation of such time series, including algorithms 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.

Unfortunately, few systems in the real world are genuinely autonomous, but
are rather subject to a variety of external influences including both
regular and irregular forcing, stochastic effects, input of external data
(e.g. communications systems), irregular sampling and so on. In such a
situation, Takens Theorem does not apply and hence although many of the
usual algorithms are still employed, they lack the theoretical
justification which they possess in the autonomous case.

In this talk, we outline a new framework for the study of such
non-autonomous systems, state appropriate generalizations of Takens Theorem
and discuss their relevance to practical signal processing methods.