M6 - complex temporal phenomena is the so-called logistic equation xⁿ⁺¹ = r xⁿ(1-xⁿ).

M6 - complex temporal phenomena is the so-called logistic equation xⁿ⁺¹ = r xⁿ(1-xⁿ).

This image depicts the so-called 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 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 exponent λ 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 M6 is an enlarged detail of picture M5. At smaller scales, structures become visible which are similar to the large-scale picture. The procedure of enlarging and rediscovering structures can be continued indefinitely, thus defining a so-called “selfsimilar object,” or a “fractal.” Within certain ranges of enlargement, many natural objects have a fractal structure: trees, cauliflower, blood-vessel systems, the rings of Saturn ...

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

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