M5 - complex temporal phenomena is the so-called logistic equation xn+1 = r xn (1-xn).

DynamicPattern_P_006.jpg

Title

M5 - complex temporal phenomena is the so-called logistic equation xn+1 = r xn (1-xn).

Description

This image depicts complex temporal phenomena called the logistic equation xn+1 = r xn (1-xn). In spite of its simple form, this equation shows a great variety in its dynamic behavior. For r < 3.57, the xn-series are periodic. For r ≥ 3.57, however, the series are chaotic, except for some intervals of r which are called “periodic windows.” The behavior of the logistic equation for periodically varying r is displayed. The colors in this picture represent the Lyapunov exponentλ as a function of A (abscissa) and B (ordinate). When A changes from negative values to zero, the color changes from blue to white; when λ changes from zero to positive values, the color changes from black to blue.

The r-sequence is given by ABABAB ... The case r = A = B
(“classic” logistic equation) corresponds to the diagonal from the lower left to the upper right corner. Picture M5 shows the neighborhood of a periodic window (period 3). The asymmetry properties of the system with respect to the diagonal indicate
that the sequences ABABAB ... and BABABA ... are not equivalent.

Subject

Pattern formation (Physical sciences); Physics; Visualization; Mathematics; Mathematics in art; Lyapunov exponents; Nonlinear Dynamics; Pattern Formation; Chemical Waves; Reaction Diffusion Systems; Nonlinear Waves; Traveling Waves; Chemical Oscillations; Oscillating chemical reactions;

Creator

Markus, Mario

Source

Dynamic Pattern Formation in Chemistry and Mathematics: Aesthetics in the Sciences: Catalogue of an Exhibition

Contributor

Müller, Stefan C; Plesser, Theo; Max-Planck-Institut für Ernährungsphysiologie Dortmund, West-Germany;

Markus, Mario;

Date

1988

Publisher

Laupenmühlen Druck, Bochum

Rights

Format

JPEG

Type

Still Image

Citation

Markus, Mario, “M5 - complex temporal phenomena is the so-called logistic equation xn+1 = r xn (1-xn).,” History of the Belousov-Zhabotinsky Reaction, accessed December 6, 2022, http://woosterdigital.org/BZ-history/items/show/192.