Identification of the dynamic characteristics of nonlinear structures

characteristic multiplier.
Characteristic multipliers are a generalisation of the eigenvalues at an equilibrium point.
Since a periodic solution corresponds to a fixed point of the Poincare map P, the stability
of the periodic solution is determined by the stability of the fixed point of the Poincare
map. By analogy with the equilibrium point, the stability of fixed point 
of P
(corresponding to initial condition 
is determined by linearising P at 
The linear
discrete-time system
SX
= 
(4-22)
governs the local behaviour of map P near 
Again as in the case of equilibrium point,
the orbit of P for an initial condition 
is, to the first order,
(4-23)
r = l
where 
and 
are the eigenvalue and eigenvector matrices of 
and 
are
scalar constants chosen to achieve the correct initial conditions. The eigenvalues 
are
the characteristic multipliers of the fixed point and determine the amount of contraction (if
1) or expansion (if 
1) near in the direction of 
for one iteration of the
map P.

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4 Identification of Chaotic Vibrational Systems
123
The characteristic multipliers determine the stability of the periodic solution. If all the 
lie within the unit circle 
is in general, complex), then the periodic solution is
asymptotically stable. If some of the 
lie outside the unit circle, then the periodic
solution is not stable. If some of the characteristic multipliers lie on the unit circle, then
the stability of the periodic solution cannot be determined by the multipliers alone.
Lyapunov exponents are a generalisation of the eigenvalues at an equilibrium point and of
characteristic multipliers. They are used to determine the stability of any type of 
state behaviour, including quasi-periodic and chaotic solutions. The definition of the
Lyapunov exponent is as follows. Let [‘m.] be the eigenvalues of a 
(a square
matrix which is a function of time), then the Lyapunov exponents are defined by
lim
t
i = 1, 2, . . . . n
(4-24)
the limit exists.
To explain the physical meaning, the Lyapunov exponents of an equilibrium point are
calculated. Let 
be the eigenvalues of 
then for flow 
= 
which
is a linearised vector field, m;(t) = 
and
lim
t
t
(4-25)
Therefore, in this special case, the Lyapunov exponents are equal to the real parts of the
eigenvalues of 
at the equilibrium point and indicate the rate of contraction 
0)
or expansion 
0) near the equilibrium point.
Lypunov exponents
attractor (including
n
are convenient for 
steady-state behaviour. For an
a chaotic attractor), contraction must outweigh expansion and
therefore 
0. Attractors are classified in terms of Lyapunov exponents as
r = l
follows. For a stable equilibrium point, 
0 for all For a stable limit cycle, 
= 0
and 0 for i 2, 3, . . . . n. For a torus, = 
= 0 and 0 for i = 3, 4, . . . . n.
One feature of chaos, as mentioned earlier, is its sensitive dependence on initial
conditions. Sensitive dependence occurs in an expanding flow, as is illustrated below.

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 Identification of Chaotic Vibrational Svstems
124
Consider a nonautonomous system with a contracting flow 
as shown in 
Suppose that the state of the system can be measured to within an accuracy of then it is
clear that it is more accurate to predict the state at time using the measured state at time
than to measure the state at The larger the elapse time 
the greater the accuracy
of the prediction. Thus for a contracting system, the predictive value of the initial
condition increases with time. On the other hand, consider the opposite case of an
expanding flow as shown in 
It is more accurate to measure the state at 
than to predict it using the measured state at and the predictive value of the initial
condition deteriorates with time. This means that expanding systems exhibit sensitive
dependence on initial conditions, but a purely expanding flow also implies unbounded
behaviour. By definition, a chaotic trajectory is bounded, and therefore it follows that a
chaotic system must contract in some directions and expand in others with the contraction
outweighing the expansion (here we only consider the dissipative/damped dynamical
systems). Hence, for a chaotic/strange attractor, at least one of the Lyapunov exponents
must be positive and this existence of positive Lyapunov exponents distinguishes a
strange attractor from other types of attractor and is one of the main criteria for detecting
chaos.
(a) Contracting Flow
(b) Expanding Flow
Illustration of Contracting Flow and Expanding Flows
4.2.6