Identification of the dynamic characteristics of nonlinear structures

THE DIMENSION OF AN ATTRACTOR
As discussed above, Lyapunov exponents can be used to 
different types of
limit set and here in this section another important concept the concept of dimension of
an attractor which serves to quantify the complexity of a given attractor is introduced.
An attractor could be defined to be n-dimensional if, in the neighbourhood of every point
on the attractor, it looks like an open subset of 
This is how the dimension of a
manifold is defined in differential topology. For example, a limit cycle is one-dimensional

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4 Identification of Chaotic Vibrational Systems
125
because it looks locally like an interval. A two-dimensional torus has a dimension of 2
because for every point, locally, it resembles an open set of 
An equilibrium point is
considered to have zero dimension. However, as shown in figure 4.13, the
neighbourhood of any point of a strange attractor has a very finely defined structure and
does not resemble any Euclidean space. Therefore, strange attractors are not manifolds
and do not have integer dimension. There are several ways to generalise the dimension to
the general fractional case and in this section, only the capacity dimension is presented.
The simplest dimension is the capacity dimension. To illustrate how the capacity
dimension can be calculated, let us consider a long time trajectory in phase space as
shown in Fig.4.15. First, time sample the trajectory so that a large number of points on
the trajectory are obtained. Then place a sphere (or cube) of radius (or length) at some
point of the orbit and count the number of points within the sphere N(
E
). The probability
of finding a point in this sphere is then defined as
(4-26)
sampled data point
trajectory
Time Sampled Data points of a Trajectory
where 
is the total number of sampled time data points. For a one-dimensional orbit,
such as a closed periodic orbit, P(
E
) will be linear in 
E 
as 0 and 
P(
E
)
(where is a constant). If the orbit is quasi-periodic, two-periodic for example, then the
probability P(
E
), as 
E 
0 and 
will be P(
E
)
(where yis a constant). These
observations lead one to define the capacity dimension of an orbit at point 
by
measuring the relative percentage of time that the orbit spends in the small sphere; that is,