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:
 

xn+1=r xn (1- xn)

where r is a positive constant (the «biotic potential»)
and the initial point x0 lies in the interval [0, 1].

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

2n

rn

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 rk the value of r for which the 2k period becomes unstable and if we determine the limit, as k goes to infinity, of ( rk+1 - rk ) / ( rk+2 - rk+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.

 

Lorenz atractor - the butterfly effect in a strange atractor  

Fractal figures generated using L-systems 

A fractal leaf and the Sierpinski triangle