Lyapunov fractal

In mathematics, Lyapunov charts (also known as Lyapunov fractals and Markus - Lyapunov fractals, after Alexander Mikhailovich Lyapunov ) fractal derived from an extension of the logistic equation in which the degree of growth of the population, r, periodically between two values ​​a and b varies.

The logistic equation is:

With an initial value that is usually set at.

In this case, or is selected for the n- th iteration of the logistic equation according to the value at the point n in an endless sequence, which is formed from the concatenation sequence of simple patterns ( sequences) of symbols, for example, with the pattern ( aababab ). Then, for values ​​(a, b ) of intervals - to get interesting characters - mostly in the area and, in each case calculated the iterates of the logistic equation and calculated the Lyapunov exponent:

Is the value of, one selects for the point with the coordinates (a, b ), for example, as a yellow color, it is greater than zero (leading to exponential growth, chaos), is chosen, for example, blue in color. Accordingly, the color values ​​you can still grade depending on the size of. The result is the Lyapunov diagram, which are often fractal nature. One example is the diagram zircon Zity formed with and of the sequence and ( bbbbbbaaaaaa ).

Swell

• Mario Markus: Lyapunov charts. Spektrum der Wissenschaft 1995/4, 66-73.