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

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

This image depicts complex temporal phenomena called the logistic equation x_n+1 = r x_n (1-x_n). In spite of its simple form, this equation shows a great variety in its dynamic behavior. For r < 3.57, the x_n-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.

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;

Markus, Mario

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

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

Markus, Mario;

1988

Laupenmühlen Druck, Bochum

JPEG

Still Image