Identification of the dynamic characteristics of nonlinear structures

periodic 
solution (often referred to as 
almost periodic 
solution) which is the sum of
periodic functions with their fundamental frequencies (the reciprocals of their minimal
periods) to be incommensurable. To see how quasi-periodic solutions arise in dynamic
systems, again consider the van der 
equation with external forcing as
(4-16)
The system has a limit cycle oscillation with fundamental frequency 
If the forcing
frequency is incommensurable with 
then a quasi-periodic solution occurs. The
quasi-periodic solution of equation 
16) with B=l .O and 
is shown in Fig.4.7.

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 Identification of Chaotic Vibrational Systems
116
time domain
state-space domain
frequency domain
Fig. 4.7 Quasi-periodic Solution of van der Pol’s Equation
Mathematically, a quasi-periodic trajectory which contains n different incommensurable
fundamental frequencies lies on an n-dimensional torus. Taking the two-periodic
trajectory (contains two incommensurable fundamental frequencies) as an example, the
trajectory lies on a two dimensional torus 
as shown in Fig.4.8 with each S
representing one of the base frequencies. Since a trajectory is a curve while 
is a
surface, not every point on the torus lies on the trajectory. However, it can be shown that
the trajectory repeatedly passes arbitrarily close to any point on the torus and, therefore,
the torus is the limit set of the quasi-periodic behaviour.
Fig. 4.8 Two Periodic Behaviour Lies on Two Dimensional Torus 
4.2.3 
CHAOTIC ATTRACTOR
There is no generally-accepted definition of a chaotic attractor. From a practical point of
view, chaotic solution can be defined as none of the above mentioned steady-state
solutions; that is, as bounded steady-state behaviour which is neither an equilibrium
point, nor periodic, and not a quasi-periodic limit set either. For this reason, chaotic
attractors are often referred to as “strange attractors”. Since the solution is nonperiodic
(which means that the solution contains some random components) while the system is
deterministic (there are no random parameters involved in describing the system), chaotic

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4 Identification of Chaotic Vibrational Systems
117
systems are very often described as “deterministic systems that exhibit random
behaviour”.
The chaotic behaviour of discrete-time dynamic systems such as the one-dimensional
logistic equation and the two-dimensional 
map has been briefly discussed and the
bifurcation diagram for the logistic equation and the strange attractor for the 
map
are shown in figures 
Now, a chaotic solution of Duffing’s equation (4-14) with
and 
is calculated and is shown in Fig.4.9. It is evident from this that
the trajectory is indeed bounded and nonperiodic. However, one should be careful to note
that boundedness and nonperiodicity do not necessarily mean that the solution is chaotic
because a quasi-periodic solution is bounded and nonperiodic as well. In order to
distinguish chaotic solutions from quasi-periodic ones, the frequency spectrum of the
signal needs to be calculated. For a quasi-periodic signal, the spectrum only contains
discrete frequency components while a chaotic solution has a spectrum with a continuous,
broad-band nature, as shown in figure 4.9. This noise-like spectrum is characteristic of
chaotic systems.
time domain
state-space domain
frequency domain
Fig.4.9 Chaotic Solution of Duffing’s Equation
Unlike the classical types of attractor that are associated with classical geometric objects
such as an equilibrium state with a point, the periodic motion or limit cycle with a closed
curve and a quasi-periodic motion with a surface in multi-dimensional space, the limit set
of chaotic behaviour is related to a new geometric object called a fractal set 
which
will be discussed later on.
Another property of chaotic systems is