= -
2
sin 
- d
,
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Another way to make equation (1) more realistic can be done as follows. Again we add friction
|
= -
2
sin
- d
, |
but instead of a constant force we drive the system
periodically. Imagine that the pivot moves up and down periodically with
amplitude a and frequency
. Since the pendulum is now
being lifted up and down, we can think of this as a periodic
modification of the gravitational constant
g,
|
g(t) = g + a
2 cos
t. |
As before, we take
=
(g/l), where l
= 1. The equation of motion is then
= -d
-
2 (1 +
 cos
t) sin
. |
The phase space of this system is three-dimensional, since we have to
include t next to
and
. Chaotic motion is possible
in a three-dimensional phase space, and indeed, the parametrically
forced damped pendulum can oscillate in a chaotic manner. For certain
parameter values even a strange attractor exists.
| 15 |  |
||
|
|||
| -15 | |||
- |
|
+ |
|
A strange attractor in the stroboscopic phase portrait for
a = 0.45,
= 6.26 and
d = 0.05.
Ever seen a pendulum oscillate according to a strange attractor? Add the following files to DsTool for this animation
Read the instructions if you do not know how to do this. Note that to see the strange attractor in phase space, it is best to "strobe" at the forcing frequency.
Written by Hinke Osinga
Last modified: Fri Feb 26 21:13:51 1999