Oscillations, Waves, and Interactions - GWDG

420 U. Parlitz
series” where for the first time state space reconstruction methods were applied to
scalar time series. A mathematical justification of this approach was given by Takens
[43] at about the same time. He proved that it is possible to (re)construct, from
a scalar time series only, a new state space that is diffeomorphically equivalent to
the (in general unknown) original state space of the experimental system. Based
on these reconstructed states the time series can then be analysed from the point
view of (deterministic) nonlinear dynamics. It is possible to model and predict the
underlying dynamics and to characterize the dynamics in terms of dimensions and
Lyapunov exponents, for example. In the following we will address some of these
issues. For further reading we refer to other review articles [35–39] and implementations
of many time series algorithms can be found in the TSTOOL box already
mentioned in Sect. 3.1.
4.1 State space reconstruction
Essentially two methods for reconstructing the state space from scalar time series are
available: delay coordinates and derivative coordinates. Derivative coordinates were
used by Packard et al. [42] and consist of higher-order derivatives of the measured
time series. Since derivatives are susceptible to noise, derivative coordinates usually
are not very useful for experimental data. Therefore, we will discuss the method of
delay coordinates only.
Let M be a smooth (C 2 ) m–dimensional manifold in the state space in which
the dynamics of interest takes place and let φ t : M → M be the corresponding flow
describing the temporal evolution of states in M. Suppose that we can measure some
scalar quantity s(t) = h(x(t)) that is given by the measurement function h : M → IR,
where x(t) = φ t (x(0)). Then one may construct a delay coordinates map
F : M → IR d
x ↦→ y = F (x) = (s(t), s(t − tl), ..., s(t − (d − 1)tl)
that maps a state x from the original state space M to a point y in a reconstructed
state space IR d , where d is the embedding dimension and tl gives the delay time (or
lag) used. Figure 11 shows a visualisation of this construction. Takens [43] proved
that for d ≥ 2m + 1 it is a generic property of F to be an embedding of M in
IR d , i. e., F : M → F (M) ⊂ IR d is a (C 2 –) diffeomorphism. Generic means that
the subset of pairs (h, tl) which yield an embedding is an open and dense subset
in the set of all pairs (h, tl). This theorem was generalised by Sauer, Yorke and
Casdagli [44,45] who replaced the condition d ≥ 2m + 1 by d > 2d0(A) where d0(A)
denotes the capacity (or: box–counting) dimension of the attractor A ⊂ M. This is
a great progress for experimental systems that possess a low–dimensional attractor
(e. g., d0(A) < 5) in a very high–dimensional space (e.g., m = 100). In this case,
the theorem of Takens guarantees only for very large embedding dimensions d (e. g.,
d ≥ 201) the existence of a diffeomorphic equivalence, whereas with the condition of
Sauer et al. a much smaller d will suffice (e. g., d > 10). Furthermore, Sauer et al.
showed that for dimension estimation an embedding dimension d > d0(A) suffices. In
this case the delay coordinates map F is, in general, not one-to-one, but the points
where trajectories intersect are negligible for dimension calculations. More details