Lyapunov fractal

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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.[1]

File:Lyapunov-fractal-AB.png
Standard Lyapunov logistic fractal with iteration sequence AB, in the region [2, 4] × [2, 4].
File:Lyapunov fractal segment.png
Detail of the Lyapunov fractal in the form of a swallow. Iteration sequence AB, in the region [3.81, 3.87] x [3.81, 3.87].
File:Lyapunov-fractal-AABAB.png
Generalized Lyapunov logistic fractal with iteration sequence AABAB, in the region [2, 4] × [2, 4].
File:Lyapunov-fractal.png
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.

A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} ) in the a−b plane for given periodic sequences of a and b. In the images, yellow corresponds to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda < 0} (stability), and blue corresponds to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda > 0} (chaos).

Lyapunov fractals were discovered in the late 1980s[2] by the Germano-Chilean physicist Mario Markus from the Max Planck Institute of Molecular Physiology. They were introduced to a large public by a science popularization article on recreational mathematics published in Scientific American in 1991.[3]

Properties

Lyapunov fractals are generally drawn for values of A and B in the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [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.[4] 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.[5] 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. For instance, the Lyapunov fractal for the iteration sequence AB (see top figure on the right) is not perfectly symmetric with respect to a and b.

Algorithm

The algorithm for computing Lyapunov fractals works as follows:[6]

  1. Choose a string of As and Bs of any nontrivial length (e.g., AABAB).
  2. Construct the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} formed by successive terms in the string, repeated as many times as necessary.
  3. Choose a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b) \in [0,4] \times [0,4]} .
  4. Define the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_n = a} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n = A} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_n = b} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n = B} .
  5. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0 = 0.5} , and compute the iterates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{n+1} = r_n x_n (1 - x_n)} .
  6. Compute the Lyapunov exponent:
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} is approximated by choosing a suitably large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} and dropping the first summand as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0 (1 - 2x_0) = r_n \cdot 0 = 0} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0=0.5} .
  7. Color the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)} according to the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} obtained.
  8. Repeat steps (3–7) for each point in the image plane.

More Iterations

More dimensions

File:Lyapunov fractal animation.webm
Animation of a 3D Lyapunov fractal with the sequence ABBBCA

Lyapunov fractals can be calculated in more than two dimensions. The sequence string for a n-dimensional fractal has to be built from an alphabet with n characters, e.g. "ABBBCA" for a 3D fractal, which can be visualized either as 3D object or as an animation showing a "slice" in the C direction for each animation frame, like the example given here.

Notes

  1. ↑ See Markus & Hess 1989, p. 553.
  2. ↑ See Markus & Hess 1989 and Markus 1990.
  3. ↑ See Dewdney 1991.
  4. ↑ See Markus 1990, p. 483.
  5. ↑ See Markus 1990, p. 486.
  6. ↑ See Markus 1990, pp. 481, 483 and Markus & Hess 1998.

References

  • Dewdney, A.K. (1991). "Leaping into Lyapunov Space". Scientific American. 265 (3): 130–132. doi:10.1038/scientificamerican0991-178.
  • Markus, Mario; Hess, Benno (1989). "Lyapunov exponents of the logistic map with periodic forcing". Computers and Graphics. 13 (4): 553–558. doi:10.1016/0097-8493(89)90019-8.
  • Markus, Mario (1990). "Chaos in Maps with Continuous and Discontinuous Maxima". Computers in Physics. 4 (5): 481. doi:10.1063/1.4822940.
  • Markus, Mario; Hess, Benno (1998). "Chapter 12. Lyapunov exponents of the logistic map with periodic forcing". In Clifford A. Pickover (ed.). Chaos and Fractals. A Computer Graphical Journey. Elsevier. pp. 73-78. doi:10.1016/B978-0-444-50002-1.X5000-0. ISBN 978-0-444-50002-1.
  • Markus, Mario, "Die Kunst der Mathematik", Verlag Zweitausendeins, Frankfurt ISBN 978-3-86150-767-3

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