The system
Can be expressed as
x'' + cx '- ax + bx^3 = Acos(wt)
or
x' = y
y' = ax - bx^3 -cy + Acos(wt)
Constants : a=1, b=1, c=1.5, and w=1.
Initial conditions y(0) = -0.5, x(0)=-1.
In our case, A is the variable parameter, and we study x(t)'s behaviour.
At first glance, it's a really interesting system, which exhibits different kinds of behaviours.
Furthermore, however big the value of A, the system is always a periodic / pseudoperiodic one :
( 6 Quicktime movies, MPEG audio 64kbs, 56kB each )
A value |
0.3 |
1 |
10 |
waveform exple |
 |
 |
 |
sample exple |
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A value |
100 |
1000 (1e3) |
1e6 |
waveform exple |
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 |
 |
sample exple |
When listening to the samples, three things are obvious :
- the fundamental remains the same, which for instance was not the case for the Lorenz system
- below a certain point, the system is chaotic with patches of order
- above this point, the system exhibits a "harmonic content + upper noise" behaviour
This first impression is confirmed when looking at a complete spectrum study :
Here A increases logarithmically : 0.1, 0.2, 0.4, 0.8, 1.6..... up to 52428.8 and 104857.6.
Audio samples : complete map
This great variety of behaviours is confirmed by a closer look.
The following movie shows how the system output is alternatively harmonic, inharmonic, and noisy - with different kinds of noisy behaviours.
( 2D false attractor based on x(t) value, with A growing geometrically - A = 0.1*2^(i/4), from 0.1 to 104857.6, along with the audio samples corresponding to x(t)
Quicktime movie, MPEG4 audio 64kbs, 840kB. )
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Click the image to open the movie - Right click to save. |
The same parameter values, this time along with the spectrum.
Note : the whole spectrum is 4096 values wide. For lisibility reasons, the video only shows the 2048 first values, but the 2048 upper values can easily be imagined.
( spectrum corresponding to x(t), with A growing geometrically - A = 0.1*2^(i/4) ,from 0.1 to 104857.6, along with the audio samples corresponding to x(t)
Quicktime movie, MPEG4 audio 64kbs, 600kB. )
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Click the image to open the movie - Right click to save. |
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to be continued |