Lyapunov fractal

Standard Lyapunov logistic fractal with iteration sequence AB, in the region [2, 4] × [2, 4].
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].
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.[1]

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 given periodic sequences of a and b. In the images, yellow corresponds to (stability), and blue corresponds to (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]

  1. ^ See Markus & Hess 1989, p. 553.
  2. ^ See Markus & Hess 1989 and Markus 1990.
  3. ^ See Dewdney 1991.