Identification of the dynamic characteristics of nonlinear structures

 
 
(4-19)
Since and 
are incommensurable,
is not periodic and repeatedly
comes arbitrarily close to every point in [0, 
Therefore, in the 
coordinates,
the limit set of the Poincare map is the circle S. In the original Euclidean coordinates, the
limit set is a closed curve. To illustrate this point, the Poincare map of quasi-periodic
solution shown in figure 4.7 with sampling frequency equal to the the forcing frequency
is shown in Fig.4.12.
i
 
.
Fig. 4.12 The Poincare Map of a Quasi-periodic Solution of van der 
System
For chaotic trajectories, the steady-state Poincare maps are distinctive and often quite
beautiful. In order to illustrate this, the chaotic attractor 
map) of Duffing’s
equation with 
and 
is calculated and is shown in Fig.4.13. Looking at
these orbits, it becomes immediately clear that the steady-state orbits do not lie on a

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4
Identification of Chaotic Vibrational Systems
121
simple geometrical form as is the case with periodic and quasi-periodic behaviour. The
attractor has fine structure which is 
as will be discussed later on. Such fine
structure of 
map is typical of chaotic systems.
-
6
.
1
1
 
Strange Attractor of Duffing’s System
4.2.5 
STABILITY OF LIMIT SETS AND LYAPUNOV EXPONENTS
The study of the stability of limit sets is important because only the attracting limit sets
(structurally stable) can be physically observed. In this section, the conditions for a limit
set to be stable will be discussed both in the case of equilibrium points and periodic
solutions. The Lyapunov exponents which can be used to determine the stability of any
type of steady-state behaviour, including quasi-periodic and chaotic solutions, will be
introduced.
Consider an equilibrium point of equation (4-l). It is well-known that the local
behaviour (for small perturbations) of a nonlinear system near the equilibrium point is
determined by linearising at as
6x
(4-20)
where 
is the Jacobian matrix at point and 6x = (x
The thus derived linear
vector field (4-20) governs the time evolution of perturbations near of the original
nonlinear system. In particular, the stability of the flow near can be determined based
on the linear vector field by examining the real parts of the eigenvalues of the Jacobian
matrix 

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 Identification of Chaotic Vibrational Systems
122
Suppose [‘h.] and 
are the eigenvalue and eigenvector matrices of the Jacobian matrix,
then mathematically, the trajectory with initial condition 
is, to the first order,
6x(t) = 
+ 
= 
+
(4-2 1)
r = l
where 
contains scalar constants chosen to achieve the correct initial conditions.
From equation 
it can be seen that the real part of gives the rate of expansion (if
0) or contraction (if 
0) in the neighbourhood of the equilibrium point
along the direction of 
If 
0 for all 
then all sufficiently small perturbations will die out as t and
is 
 
If 
0 for some 
then 
is not stable. If one of the
eigenvalues has zero real part, then the stability cannot be determined from the linearised
vector field and higher terms need to be included in the expression of (4-20).
The stability of a periodic solution is determined by its