Identification of the dynamic characteristics of nonlinear structures

are 
chaotic. Furthermore, it is worth mentioning that, in addition to the
quantification of the complexity of the chaotic motion, the calculation of fractal dimension
is very important for the modelling of chaotic systems because it is from this value that
the number of degrees of freedom necessary to model a practical chaotic system can be
determined so that all the topological nature of the attractor can be preserved.
l
l l
l
a 
--
8 I O 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 8 0 0 1 0 3 0
Values of 
l 
1 
Fig.4.24 Capacity Dimension Versus the Size of 
4.3.5 
SENSITIVITY TO INITIAL CONDITIONS AND
LYAPUNOV EXPONENTS
As discussed in section 
chaos in dynamics implies a sensitivity in the outcome of
a dynamic process to small changes in the initial conditions. When a system becomes
chaotic, the accurate prediction of long-term response becomes impossible because, in
this case, a small initial condition uncertainty will be magnified exponentially as time goes
on and, as a result, two originally indistinguishable initial conditions will lead to
completely different long-term solutions. This sensitivity to initial conditions for case 1
with 
is illustrated in Fig.4.25 (the time interval between two
successive points for these two trajectories is a forcing period).

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 Identification of Chaotic Vibrational Systems
134
T I M E
Fig.4.25 Sensitivity on Initial Conditions of Chaotic Solutions
In order to quantify this sensitivity to the initial conditions, the Lyapunov exponent of the
motion needs to be calculated. Imagine a set of initial conditions within a sphere of radius
in phase space, then the chaotic motion trajectories originating in the sphere will map the
sphere into an ellipsoid whose major axis grows as 
where is known as a
Lyapunov exponent. As mentioned in 
for regular motions, 
while for chaotic
motions, 
Thus, the sign of is a criterion of chaos. The numerical method for
calculating the Lyapunov exponent was well explained in 
Suppose we have two
chaotic trajectories, and 
starting with very close initial conditions as shown in
(4-33)
Fig.4.26, then can be calculated as:
N
k 1
where is the separation of