Identification of the dynamic characteristics of nonlinear structures

sensitive dependence on initial conditions: 
given
two initial conditions arbitrarily close to one another, the trajectories emanating from these
initial conditions diverge at an exponential rate (which is some kind of characteristic value
of the system) until for all practical purposes, they are uncorrelated. This sensitive
dependence on initial condition for Duffing’s equation is illustrated in 
(the two
initial conditions differ by only 1%). In practice, the initial state of the system can never

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4
Identification of 
 Vibrational Systems
118
be specified exactly, but only within some tolerance and therefore, if two initial
conditions and lie within of one another, they cannot be distinguished. However,
after a finite amount of time, flows 
and 
will diverge and become
uncorrelated. As a result, no matter how precisely the initial condition is known, the 
term behaviour of a chaotic system can never be predicted (of course, the more accurate
the initial conditions are, the longer the prediction can be, but since the divergence is
exponential with time, unless the initial condition could be specified to infinite precision,
accurate long-term prediction becomes impossible).
 
Fig.4.10 Illustration of Sensitive Dependence on Initial Conditions
4.2.4 THE 
MAPS
A very useful classical technique for analysing dynamical systems was developed by
Poincare. The technique replaces the flow of a continuous-time dynamical system with a
discrete map called Poincare map. For autonomous and nonautonomous systems, the
definitions of the Poincare map are slightly different and two cases are treated separately.
Consider an 
autonomous system with a limit cycle as shown in 
Let
be a point on the limit cycle and let T be the minimal period of the limit cycle. Take
an (n-l)-dimensional hyper-plane (a plane has a dimension more than two) transverse
to 
at 
The trajectory emanating from will hit at in T seconds. Due to the
continuity of with respect to the initial conditions, trajectories starting on in a
sufficiently small neighbourhood of will, in approximately T seconds, intersect in
the vicinity of 
Therefore, vector field and hyper-plane define a mapping P of
some neighbourhood 
of onto another neighbourhood 
of 
The 
defined P is called the Poincare map of an autonomous system.

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 Identification of Chaotic Vibrational 
119
For nonautonomous systems, as shown in section 4.2.1, an 
nonautonomous
system in 
Euclidean space with period T may be transformed into an 
dimensional autonomous system in cylindrical state-space 
Consider the 
dimensional hyper-plane in 
defined by
(4-17)
Every T seconds, the trajectory intersects as shown in 
Thus a map P: 
is defined by 
where P is called the Poincare map. Such a Poincare map can
be thought of in following two ways:
(i) P(x) indicates where the flow takes x after a T seconds and this is called T advance
mapping; or
(ii) the orbit (sequence of points) 
is a sampling of a single trajectory every T
seconds; that is 
= 
for 2, . . . .
map of
autonomous system
The 
map of a first order
nonautonomous system
Poincare Maps of Autonomous and Nonautonomous Systems
The usefulness of the Poincare map derives from the fact that there is one-to-one
correspondence between the different types of steady-state behaviour of the underlying
continuous-time dynamic system and the steady-state behaviour of mapping P. Therefore,
from the steady-state behaviour of mapping P, the steady-state behaviour of the
continuous dynamic system can be deduced.
As is clear from the definition of P, a period one solution of the underlying flow
corresponds to a fixed point of the Poincare map. For nonautonomous systems, a period
solution (contains 
subharmonic) corresponds to K different points on the Poincare
map.

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4
of Chaotic Vibrational Systems
120
The 
map can also be used to detect quasi-periodic solutions. As mentioned
earlier, two-periodic solution (meaning two incommensurable minimal periods and is
different from period two solution) lies on a two dimensional torus 
as shown in
figure 4.8. Using coordinate 
on the torus, a two-periodic trajectory may be
written as
(4-18)
where 
and 
are incommensurable. In the nonautonomous case, one of the
frequencies, say, is the forcing frequency of the system. An orbit of the Poincare map
corresponds to sampling (4-l 8) every 
seconds
mod