Quantitative analysis of EEG signals: Time-frequency methods and ...

the position vs. the linear momentum. In general, the phase space is identied with a
topological manifold. An n-dimensional phase space is spanned by a set of n-dimensional
\embedding vectors", each one dening a point in the phase space, thus representing
the instantaneous state of the system. The sequence of such states over the time scale
denes a curve in the phase space, called trajectory.
Attractor: In some cases, trajectories of dissipative dynamical systems (systems with
a volume contraction in the phase space) converge with increasing time to a bounded
subset of the phase space. This bounded region to which all suciently close trajectories
(trajectories lying in the basin of attraction) converges asymptotically is called the
attractor (Shuster, 1988). According to their topology, several types of attractors can
be distinguished (see Abraham and Shaw 1983 Shuster, 1988 Ott et al, 1994 Basar
and Quian Quiroga, 1998):
1. Fixed Point: Trajectories in phase space tend to a pointatrest. Atypical example
is a damped pendulum that has come to rest after some time.
2. Limit cycle: The attractor is a closed (one dimensional) curve in the phase space
representing a periodic motion. The standard example is a damped periodically
driven oscillator. circumstances
3. Torus: The attractor is a two-dimensional toroidal surface. This type of attractor
represent a quasiperiodic motion where two incommensurable frequencies correspond
to the movement around and along the torus.
4. Strange attractor: The main property of strange attractors is their sensitive dependence
on initial conditions. Points that are initially close in the phase space,
become exponentially separated after some time. All known strange attractors
have a non-integer dimension. Signals corresponding to strange attractors have a
random appearance.
5.2.2 Correlation Dimension
The Correlation Dimension (D 2 ) has become the most widely used quantitative parameter
to describe attractors. It is a measure of complexity ofthe system related with its
number of degrees of freedom, or in a more intuitive way with its topological dimension.
It is already a common practice to calculate D 2 and to investigate how itchanges upon
dierent . Furthermore, since in principle D 2 converges to nites values for deterministic
systems and do not converge in the case of a random signal, D 2 isagoodparameterfor
evaluating the deterministic or noisy inherent nature of a system. As we will see later
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