Identification of the dynamic characteristics of nonlinear structures

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4.2.1 DYNAMICAL SYSTEMS THEORY
In general, dynamic systems can be divided into three different categories: 
autonomous
dynamic systems, nonautonomous dynamic systems 
and 
discrete-time dynamic systems.
Both autonomous and nonautonomous dynamic systems are described by differential
equations (ordinary or partial differential equations) while discrete-time dynamic systems
are expressed in terms of iterative maps. All three types of systems are defined and
discussed and some of the useful facts from the theory of differential equations are
presented.
An 
autonomous dynamical system is defined by the state equation as
(4-l)
d
where x, 
is the state at time instant t and
is called the vector
field. Since the vector field does not depend on time, the initial time may always be taken
as 
The solution of equation (4-l) with initial condition 
at time t=O is called a
trajectory (in n-dimensional space) and is denoted by 
The mapping (which is
continuous as compared with the discrete mapping of discrete-time dynamical systems)
is called the flow of the system since is a continuous trajectory starts at 
and is like fluid flowing in the state-space. The dynamic system described in (4-l) is
linear if f(x) is a linear function of state variable x. Free vibrational mechanical systems
belong to this category.
An 
nonautonomous dynamical system, on the other hand, is defined by the
time-varying state equation as
The vector field depends on time and, unlike the autonomous case, the initial time cannot
be arbitrarily set to 0. The solution of (4-2) passing through the point at 
is denoted
as 
Again, the system is linear if f(x) is linear with respect to x.
If for a nonautonomous system, there exists a constant T 0 such that 
= 
for all x and all t, then the system is said to be 
time periodic 
with period T. The smallest
such T is called the 
minimal period. 
In this Chapter, all nonautonomous systems are
assumed to be time periodic e.g., time invariant systems with periodic input force.

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 Identification of Chaotic Vibrational Systems
110
An 
time periodic nonautonomous system can always be converted to an
(n+l)*-order autonomous system by appending an extra state 
Therefore, the
corresponding autonomous system is given by
= 
x(0) = 
(4-3)
= 
(4-4)
Since is time periodic with period T, the new system described by (4-3) and (4-4) is
periodic in with period 
Therefore, the state-space is transformed from Euclidean
space 
to cylindrical space 
where S [0, 2x) is a circle. The solution in the
new state-space is
 
(4-5)
where the modulo function (x mod y gives the remainder of x divided by y, e.g., 3 mod 2
restricts to be within the semi-closed interval 
Using this transformation,
results for autonomous systems can be applied to the time periodic nonautonomous case.
As for discrete-time dynamic systems, any map 
defines a discrete-time
dynamic system by the state equation
= 
k = 0, 1, 2, 
where 
is called the state, and maps state 
to the next 
State 
Starting with an
initial condition 
repeated applications of the map gives rise to a sequence of points
called an orbit of the discrete-time system. Examples of discrete-time dynamic
systems are given below.
Although the research presented in this Chapter focuses on continuous time vibrational
systems, discrete-time systems will be discussed for two reasons. First, 
mapping technique, which replaces the analysis of flow of continuous-time system with
the analysis of a discrete-time system, is an extremely useful tool for studying dynamical
systems. Second, due to this correspondence between flows (of continuous-time dynamic
systems) and maps (of discrete-time dynamic systems), maps will be used to illustrate
important concepts without getting into details of solving differential equations.

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4
Identification of Chaotic Vibrational Systems
111
The simplest one-dimensional discrete-time system the population
which has been found to be chaotic is described by the logistic equation
(4-7)
growth model 
For some values of after certain iterations until the transient component dies, the 
will settle to one specific value (period one solution). While for other values of
oscillates between 2 values (period 2 solution), 4 values (period 4 solution) and so on.
However, there are some parameter regions in which never repeats its value as
iteration continues, as shown in Fig.4.2 and such phenomenon is the earliest observation
of what we call chaos today.
Fig.4.2 Bifurcation 
of Logistic Map for 2.7
Another discrete-time system which exhibits chaotic behaviour is the quadratic map
studied by 
= 
(4-g)
In the case when 
and 
for initial condition 
the sequence of points
generated by the mapping 
is shown in Fig.4.3. Although the sequence of points
never repeats, they settle to restricted areas on the x-y plane and exhibit a very 
constructed pattern (as will be discussed, the pattern is very finely defined as it is fractal).

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of Chaotic Vibrational Systems
112
Fig. 4.3 The 
Attractor
The characteristics of logistic map and 
map are briefly discussed here because they
will be referred to in later discussions.
4.2.2 
STEADY-STATE BEHAVIOUR AND LIMIT SETS OF
DYNAMIC SYSTEMS
Dynamic systems are classified in terms of their steady-state solutions and limit sets.
Steady state refers to the asymptotic behaviour of the solution of a dynamic system as
The difference between the solution and its steady state is called the transient.
A point y is defined as the 
 point 
of x if, for every neighbourhood U of x, flow 
repeatedly enters U as
e.g., the equilibrium point of a dynamic system.
The set of all limit points is called the