The system
This simple, discrete system, exhibits the following behaviour :
| R values | behaviour |
|---|---|
1 to 2 |
constant value |
2 to 2.5 |
periodic, period 2 |
around 2.5 |
cascade of subharmonics, period 2exp(n) |
2.55 to 3 |
chaotic |
above 3 |
to +∞ |
Here is an illustration of the subharmonic cascade : the values U(n) take, depending on the R parameter :
From a spectral point of view, it means that :
- between 2 and 2.5, the signal only has one harmonic, arbitrarily 1600Hz for instance
- at some point, a sub harmonic appears, 800Hz
- and then, two new frequencies appear : 400Hz and 1200Hz
- then 200, 600 around 400, and 1000 & 1400 around 1200.
- etc...
During the chaotic part ( roughly for R ranging from 2.5 to 3 ), the system exhibits different behaviours, like :
- periodic, period 3 or 6
- period doubling phenomemon one way or another
- white like noise
- a main harmonic along with patches of noise
- etc.
Audio samples : main subharmonic cascade
The first movie shows the cascade of lower harmonics that occurs for R values around 2.5.
This cascade is a classic transition from order to chaos.
Macromedia Flash 7+ movie, R from 2.5 to 2.586, arbitrary step.
Audio samples : another subharmonic cascade
The second movie shows another transition to chaos, which also consists in a cascade of harmonics, but a less regular one.
It takes place at the end of a "window of order" which can be found in the chaotic area.
Macromedia Flash 7+ movie, R from 2.85 to 2.8575, arbitrary step.
Audio samples : different aspects of chaos
The third movie shows different aspects of the chaotic area.
Macromedia Flash 7+ movie, arbitrary values of R, from 2.661 to 2.952.