Information Technology Reference
In-Depth Information
The mapping
f
defines the evolution of the dynamical system, by determining the
next state
x
n+1
=f(x
n
)
.
4
Te above definitions concerns the systems with discrete time
(cascades). For continuous time systems (flows) the evolution is given by a set of
k
differential equations
&
x
=
F
(
x
)
.
Def. 21.
A
trajectory
or an
orbit
is a series of consecutive states of a system.
For a class of systems, known as dissipative systems (see [26]), the trajectory usually
settles on a subset of state space known as attractor.
Def. 22.
An
attractor
A
of a dynamical system
DS=(X,f)
is a bounded closed subset of
system space
A
X
that is invariant
f(A)=A
and has such a neighborhood that every
trajectory from it settles on
A
.
From the invariant property of an attractor it follows that if the state of a system
converges to an attractor, then every consecutive states belongs to the attractor as
well.
In some special occasions a dissipative dynamical system can be sensitive to a
initial state. In these case even the smallest difference in the initial conditions gets
strengthen in time and two close trajectories disperse quickly. Such systems are called
chaotic [46]. The attractor of a chaotic dynamical system is usually a fractal set and
has an non-integer fractal dimension, and is being called a strange attractor.
An exemplary chaotic system is represented by the Mackey-Glass (MG) equation,
introduced in [45] as a model of blood cells production. Its dynamics is defined with a
following equation:
⊂
(
)
0
.
2
x
t
−
τ
&
MG
x
=
−
0
.
1
x
(1)
(
)
10
1
+
x
t
−
τ
MG
MG system belongs to the class of delayed feedback systems [32] that are common
for biological systems. Systems from this class have a infinite-dimensional state
space, because to establish its initial condition a generic function over a set
[-
, 0]
is
needed. For delayed feedback systems the attractor's dimension can by arbitrary high,
but if the delay is small system's dynamics is usually low-dimensional, e.g. for MG
with
τ
τ
=17
the dimension of attractor is about 2 [29].
3.1 State Space Reconstruction
One of the most widely used methods for dynamic systems analysis is the state space
reconstruction, proposed in [23] and justified on theoretical basis in [30] and [31]. It
allows for the reconstruction of system's underlying dynamics basing on the
univariate time series. There are three basic approaches to state space reconstruction
[24]: (1) the Method of Delays (MOD); (2) the derivatives method; and (3) the
principal components method. The simplest and most popular (although not
chronologically first) is MOD [20, 21, 22, 28, 32, 33, 34, 35]. In the method of delays
a reconstructed system space is represented by a delay vector, defined as follows:
4
Assuming an autonomous system, in which
f
does not depend on
n
.
