The
transition to Chaos and the Feigenbaum constant
an
applet made by António Miguel de
Campos (Portugal)
The figure in the screen shows the «logistic map» - which illustrates the approximation to chaos by period doubling of the logistic equation, defined by: |
x_{n+1}=r x_{n} (1- x_{n}) |
where r is a positive
constant (the «biotic potential») |
The logistic map shows the values of the logistic equation for values of r from 2.9 to 4. The oscillating behavior of the logistic equation can be visualized by pressing the button which reads «Evolution for a value of r » and then pressing the button « + » to increase the value of r. In the textfield to the right of this button, the value for r is shown(multiplied by 1000). |
Notice that, above 3.0, there is an oscillation between two values (corresponding to the first bifurcation in the logistic map) . After 3.45 there is an oscillation between 4 values and then 8 values after 3.54. After that, a caotic behavior soon starts to show up. This behavior is briefly interrupted for certain values (as 3.63, 3.74 and 3.83 where there is an oscillation between 6, 4 and 2 values, respectively) but that is due to «mode locking» . |
The logistic map shows (in the vertical) the values obtained for the various values of r from 2.9 to 4 (in the horizontal). |
The following table shows the type of cycles that can be observed and the value of r for which that cycle appears: |
n |
2^{n} |
r_{n} |
1 | 2 | 3 |
2 | 4 | 3,449490 |
3 | 8 | 3,544090 |
4 | 16 | 3,564407 |
5 | 32 | 3.568750 |
6 | 64 | 3,56969 |
7 | 128 | 3,56989 |
8 | 256 | 3,569934 |
9 | 512 | 3,569943 |
10 | 1024 | 3,5699451 |
11 | 2048 | 3,569945557 |
infinity |
accum. point |
3,569945672 |
Notice that the period doubling happens faster and faster ( 8, 16, 32, ...) and then suddenly ends after the so-called accumulation point and chaos appears. |
The
Feigenbaum constant If we call r_{k} the value of r for which the 2^{k} period becomes unstable and if we determine the limit, as k goes to infinity, of ( r_{k+1 }- r_{k} ) / ( r_{k+2 }- r_{k+1} ) we obtain a value of 4,66920160... |
Surprizingly, that limit value is the same for all functions of this type ( 1D maps with only one quadratic maximum) which approach chaos by period doubling. In fact, that value is one of the universal constants - the Feigenbaum constant- approximately equal to 4.669. |
It was discovered by Feigenbaum about 20 years ago (in 1975) and it characterizes the transition to chaos. |
Fractal figures generated using L-systems
A fractal leaf and the Sierpinski triangle