In mathematics Lyapunov fractals (also known as Markus-Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values a and b.

A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent) in the a-b plane for a given periodic sequence of as and bs.

Algorithm for generating Lyapunov fractals

An algorithm for computing the fractal is summarized as follows:

1. Choose a string of A's and B's of any nontrivial length (e.g., AABAB).
2. Construct the sequence $S$ formed by successive terms in the string, repeated as many times as necessary.
3. Choose a point $(a,b)\; \backslash in\backslash times\; [0,4$.
4. Define the function $r\_n\; =\; a$ if $S\_n\; =\; A$, and $r\_n\; =\; b$ if $S\_n\; =\; B$.
5. Let $x\_0\; =\; 0.5$, and compute the iterates $x\_\{n+1\}\; =\; r\_n\; x\_n\; (1\; -\; x\_n)$.
6. Compute the Lyapunov exponent $\backslash lambda\; =\; \backslash lim\_\{N\; \backslash rightarrow\; \backslash infty\}\; \{1\; \backslash over\; N\}\; \backslash sum\_\{n\; =\; 1\}^N\; \backslash log\_2\; \backslash left|\{dx\_\{n+1\}\; \backslash over\; dx\_n\}\backslash right|\; =\; \backslash lim\_\{N\; \backslash rightarrow\; \backslash infty\}\; \{1\; \backslash over\; N\}\; \backslash sum\_\{n\; =\; 1\}^N\; \backslash log\_2\; |r\_n\; (1\; -\; 2x\_n)|$. In practice, $\backslash lambda$ is approximated by choosing a suitably large $N$.
7. Color the point $(a,b)$ according to the value of $\backslash lambda$ obtained.
8. Repeat steps (3-7) for each point in the image plane.

In the image shown for the string BBBBBBAAAAAA, yellow corresponds to , and black to blue corresponds to $\backslash lambda\; >\; 0$.

External links

• ^* EFG's Fractals and Chaos - Lyapunov Exponents
• ^* Lyapunov Space - The Chaos Hypertextbook by Glenn Elert

Fractals

Fractale de Lyapunov | Fractal de Lyapunov | Fraktal Ljapunova