Lyapunov fractal

Standard Lyapunov logistic fractal with iteration sequence AB, in the region [2, 4] × [2, 4].

Generalized Lyapunov logistic fractal with iteration sequence AABAB, in the region [2, 4] × [2, 4].

Generalized Lyapunov logistic fractal with iteration sequence BBBBBBAAAAAA, in the growth parameter region (A,B) in [3.4, 4.0] × [2.5, 3.4], known as Zircon Zity.

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 $\lambda$ ) in the a−b plane for given periodic sequences of a and b. In the images, yellow corresponds to $\lambda < 0$ (stability), and blue corresponds to $\lambda >0$ (chaos).

Properties

Lyapunov fractals are generally drawn for values of A and B in the interval $[0,4]$ . For larger values, the interval [0,1] is no longer stable, and the sequence is likely to be attracted by infinity, although convergent cycles of finite values continue to exist for some parameters. For all iteration sequences, the diagonal a = b is always the same as for the standard one parameter logistic function.

The sequence is usually started at the value 0.5, which is a critical point of the iterative function. The other (even complex valued) critical points of the iterative function during one entire round are those that pass through the value 0.5 in the first round. A convergent cycle must attract at least one critical point; therefore all convergent cycles can be obtained by just shifting the iteration sequence, and keeping the starting value 0.5. In practice, shifting this sequence leads to changes in the fractal, as some branches get covered by others; notice for instance how the Lyapunov fractal for the iteration sequence AB is not perfectly symmetric with respect to a and b.

Algorithm for generating Lyapunov fractals

An algorithm, for computing the fractal is summarized as follows.

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

External links

EFG's Fractals and Chaos – Lyapunov Exponents
Elert, Glenn. "Lyapunov Space". The Chaos Hypertextbook.

Fractals

Characteristics	Fractal dimensions Assouad Box-counting Correlation Hausdorff Packing Topological Recursion Self-similarity

Iterated function system	Barnsley fern Cantor set Dragon curve Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Space-filling curve T-square

Strange attractor	Multifractal system

L-system	Space-filling curve

Escape-time fractals	Burning Ship fractal Julia set Filled Lyapunov fractal Mandelbrot set Nova fractal

Random fractals	Brownian motion Brownian tree Diffusion-limited aggregation Fractal landscape Lévy flight Percolation theory Self-avoiding walk

People	Georg Cantor Felix Hausdorff Gaston Julia Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Lewis Fry Richardson Wacław Sierpiński

Other	"How Long Is the Coast of Britain?" Coastline paradox List of fractals by Hausdorff dimension The Beauty of Fractals (1986 book)

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