Oscillations, Waves, and Interactions - GWDG

422 U. Parlitz
current state future state
similar states in the past
whose temporal evolution
is known
?
flow
Figure 12. Local modelling using nearest neighbours in (reconstructed) state space.
known) are some neighbouring (reconstructed) states of this reference state that occurred
(in the given time series) in the past such that their evolution over a period of
time T is already known (as illustrated in Fig. 12). If the dynamical flow in (reconstructed)
state space is continuous then the future values of the neighbouring states
provide good approximations of the future evolution of the reference state. This is
the main idea of the local approach and there are several options for further improving
its performance [54]. Local modelling can also be applied to complex extendend
system if a suitable state space reconstruction method is used [55].
An alternative to local modelling are global models, for example given as a superposition
of nonlinear basis functions. Such models have been employed to describe
not only the dynamics of a given system but also its parameter dependence [56]. If
models with good generalisation capabilities are required (i. e., models with good
performance on data not seen during the learning process) it is often advantageous
to use not a single type of model but an ensemble of different models. Averaging
their individual forecasts provides in most cases better results (on average) than any
single model [56,57]. A Matlab T M toolbox ENTOOL for such ensemble modelling
was developed by former DPI students Christian Merkwirth and Jörg Wichard. 7
5 Synchronisation of chaotic dynamics
Synchronisation of periodic signals is a well-known phenomenon in physics, engineering
and many other scientific disciplines. It was first investigated in 1665 by
Christiaan Huygens who observed that two pendulum clocks hanging at the same
beam of his room oscillated in exact synchrony [58,59]. Huygens made experiments
with his clocks and found that the synchronisation originates from invisible vibrations
of the beam enabling some interaction between both oscillators. He reported
his findings on the “sympathy of two clocks” (as he called it) at the Royal Society
of London but it took more than 200 years before research on synchronisation was
continued. J. W. Strutt (Lord Rayleigh) described in the middle of the 19th century
that two (similar) organ pipes sound unisono if placed close together so that they
can interact acoustically [60].
7 It is available at http://zti.if.uj.edu.pl/~merkwirth/entool.htm.