Complex Neuro-Cognitive Systems - Foundations on Natural and Artificial Computation - page 5

pendulum is suited for illustrating the principles behind complex dynamics and
chaotic attractors (Fig. 2). The dimensionless motion equation of the pendulum
is:

d 2 θ
dt 2

+ 1
q

dθ
dt

+sin θ = g cos( ω D t ) (1)

with θ the angular position in radians, q the damping parameter, g the ampli-
tude of the driving force, and ω D the frequency of that force. For small angle θ
the equation can be integrated, i.e. the pendulum either undergoes regular os-
cillations or, without a driving term, eventually stop swinging. For larger angles
θ , however, this approximation is invalid and hence the equation can no longer
be solved analytically.
The dynamical variables of the system, i.e. the angular position θ and velocity
ω = dθ/dt , are the coordinates defining the system's phase space .Inthetwo-
dimensional case, the variables can be plotted to display a phase portrait of the
dynamical behaviour.
By varying the parameters q , g and ω D of the motion equation (1) and then
plotting the resulting phase portrait a wide range of behaviour can be observed.
In the case of a non-zero damping parameter and no driving force to replace the
energy loss, the pendulum is a dissipative system, i.e. it comes to the resting
point (0,0), a fixpoint attractor. Using a non-zero driving force g , the attractor
is no longer a single point at (0,0) but now a closed orbit, that is, the pendulum
undergoes regular motion. A bifurcation has occurred, changing the fixpoint into
a limit cycle attractor. Increasing the driving force g further, a sequence of period
doublings occurs which continues as g is increased until a point is reached where
the motion of the pendulum ceases to be regular and becomes chaotic, i.e. a
chaotic attractor occurred.
This example nicely shows that even in a compositionally very simple system
the dynamics can be chaotic. In this case, the analysis could be made because
the dynamical system model, i.e. the equation of motion (1), is known.

4.3 Nonlinear Time Series Analysis

We have discussed complex systems with nonlinear dynamics so far using the
bottom-up, deductive approach: given the system equations, the behaviour of
the system can be predicted. However, the situation in cognitive neuroscience is
totally different: instead of the system equations, a set of observations is given,
say in the form of an EEG record. The problem then is to find a way to get from
the observations of a system with unknown properties to an understanding of
the dynamics of the underlying system. This can be achieved by nonlinear time
series analysis , a systemic approach starting with the output of the system, and
working towards the state space, attractors and their properties.
Nonlinear time series analysis proceeds as follows: (i) reconstruction of the
systems dynamics in state space; (ii) characterization of the reconstructed at-
tractor; (iii) checking the validity of the procedure [17].