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Fig. 2.1 Pendulum
performing oscillations
frequencies which are not necessarily multiples of a basis frequency (almost
periodic oscillations).
5. Chaos : A nonlinear system in steady-state can exhibit a behavior which is not
characterized as equilibrium, periodic oscillation, or almost periodic oscillation.
This behavior is characterized as chaos. As time advances the behavior of
the system changes in a random-like manner, and this depends on the initial
conditions. Although the dynamic system is deterministic, it exhibits randomness
in the way it evolves in time.
6. Multiple modes of behavior : It is possible the same dynamical system to exhibit
simultaneously more than one of the aforementioned characteristics (1)-(5).
Thus, a system without external excitation may exhibit simultaneously more
than one limit cycles. A system receiving a periodic external input may exhibit
harmonic or subharmonic oscillations, or an even more complex behavior in
steady state which depends on the amplitude and frequency of the excitation.
Example 1. Oscillations of a pendulum (Fig. 2.1 ).
The equation of the rotational motion of the pendulum under friction is given by
R
ml 2
D mg sin./l klv )
R
P
ml 2
D mg sin./l kl 2
(2.1)
)
ml R
D mg sin./ kl P
P one has the state-space
By defining the state variables x 1
D and x 2
D
description
x 1 D x 2
(2.2)
g
l sin.x 1 /
k
x 2 D
m x 2
 
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