Identification of the dynamic characteristics of nonlinear structures

limit set 
of 
X
. Limit sets are closed and
invariant under the flow 
(a set L is invariant under 
if, for all 
L and all t,
e.g., the 1imit cycle of a nonlinear system.
A limit set L is attracting if there exists an open neighbourhood U of L such that 
for all 
U. The limit cycle of a nonlinear system is an attracting limit set.
The 
basin of attraction 
B(L) of an attracting set L is defined as the union of all such
neighbourhoods U. Every trajectory starting in B(L) tends towards L as t 
These
definitions are briefly illustrated in Fig.4.4.

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4
Identification of Chaotic Vibrational Systems
113
attracting limit set 
0
X
0
x 0
X
limit point
limit set
basin of attraction
Fig 
Illustration of Limit Point, Limit Set and Basin of Attraction
In a stable linear system, there is only one limit set corresponding to specific input and
therefore the steady-state behaviour is independent of initial conditions. In a typical
nonlinear system however, there can be several attracting limit sets, each with a different
basin of attraction. In this case, the initial condition determines in which limit set the
system eventually settles.
The concept of limit sets is very useful in understanding different classical types of
steady-state behaviour such as equilibrium points, limit cycles and quasi-periodic
solutions. However, as will be shown, it is far too simple to describe the complex 
state behaviour found in chaotic systems and some new mathematical concepts such as
fractal dimension and Lyapunov exponent need to be introduced when steady-state
chaotic behaviour (strange attractor) is considered. In what follows, different types of
steady-state behaviour are discussed based on the well-known Duffing’s and van der
systems. Each state will be described from three different points of view: in the time
domain, in the frequency domain and as a limit set in state-space domain.
An equilibrium point 
is related with an autonomous system (a nonautonomous system
does not have equilibrium points because the vector field
varies with time)
and is the constant solution of equation 
for all time t. In general, 
implies that x is an equilibrium point of the system. A simple example is the damped free
vibration system given by
(4-10)
It is well known that the system possesses an equilibrium point which is 
This equilibrium point can be obtained by solving 
as follows:
Rewrite 
10) into its state-space form as

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4 Identification of Chaotic Vibrational Systems
114
Therefore, 
means that 
and y x = 0 
Both autonomous and nonautonomous systems can have periodic solutions
initial and excitation conditions. A solution 
is a periodic solution if
= 
(4-13)
(4-12)
under certain
for all time t and some minimal period 
0. In general, a periodic solution of a
dynamic system has a Fourier transform consisting of a fundamental frequency
component at 
and evenly spaced harmonics at 
The amplitudes of
some of these spectral components may be zero. For a nonautonomous system, 
is
typically some multiple of forcing period 
-kT and the periodic solution is usually
referred to as a 
subharmonic. To illustrate this point, a periodic (periodic 3) solution
of the well-known Duffing’s equation with 
and 
(all units appear in
this chapter are supposed to be 
except where physical units are given).
(4-14)
was calculated and is shown in Fig.45
time domain
state-space domain
frequency domain
Fig. 4.5 Period 3 Solution of Duffing’s System
Also, periodic solutions exist in autonomous systems and in this case, the periodic
solution is called a 
limit cycle. 
A limit cycle is a self-sustained oscillation and cannot
occur in a linear system. One classical example of limit cycle is found in van der 
equation

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Identification of Chaotic Vibrational Systems
115
15)
The existence of a stable van der Pol limit cycle is shown in Fig.4.6 and can be physically
explained in terms of the damping mechanism of the system. When 1, the damping
of the system is negative and therefore, the solution is expanding. While on the other
hand, when the solution becomes 
1, it is contracting. As a result, the solution will
eventually settle down a limit cycle.
time domain
state-space domain
frequency domain
Fig. 4.6 Limit Cycle of the van der Polk System
Another type of steady-state solution which exists in some nonlinear systems is the