Graphische Methoden mit Beispielen / Graphical methods with examples / Métodos gráficos con ejemplos

(Lizenzen für die folgenden Bilder werden direkt und ausschließlich durch Mario Markus erteilt: mariomarkus@hotmail.com / Licences for the following pictures are granted directly and exclusively by Mario Markus: mariomarkus@hotmail.com / Licencias para las figuras siguientes son concedidas directa- y exclusivamente por Mario Markus: mariomarkus@hotmail.com)


The predictability of a system (low for certain insect plagues, for the wheather, for heart fibrillation; high for some biorhythms and for the phases of the moon) can be quantified by the average exponent introduced by Lyapunov 100 years ago.

I represent Lyapunov's exponent by using grey or colour shadings on a plane that is defined by parameters that control the system. This representation leads to an easy recognition of important properties of the system: i) crossing branches indicate multistability, i.e. more than mode is possible for identical parameters, depending on the previous history of the sytem; ii) crossing branches looking transparent indicates fractal basins of attraction; iii) the "picture backround" indicates chaos; iv) dark curves on the picture foregrounds indicate superstability, i.e. the sytem is maximally robust against perturbations on these curves (this may be a desired biological or technical feature); v) highly interleaved zones of very different grey (or colour) shadings indicate structural instability.

Exhibitions
 
 

Reports on this work:

In the web: 12345678910 

A.K. Dewdney, Scientific American, Sept. 1991, 178-180 / Pour la Science, Dec. 1991,124-127 / Investigacion y Ciencia, Nov. 1991, 94-97

<-- click to enlarge / click para amplificar
or click here for even higher quality / o aquí para amplificar al máximo

J. Briggs, "Chaos: Neue Expeditionen in fraktale Welten" (Hanser, Muenchen, 1993) pp.151-154 / "Fractals. The Patterns of Chaos" (Simon & Schuster, New York, 1992)


<-- click to enlarge / click para amplificar

GEO-Wissen, "Chaos und Kreativitaet" 2,1990 (several contributions in this issue)

National Geographic 175 (1989) 736-737

Natuur & Techniek 9, 720-724 (1992)

HighTech 9, 24-35 (1988)

WAZ, 17.9. 1988, (W-DO1) / 14.2.2001 (HA4)

El Mercurio (Chile), 27.3.1988, E13

Diário de Notícias (Lisboa), 5.2.1992, 41-42

Daily Telegraph, Aug. 12, 1989, Weekend Mag.; pp. 20-24

<-- click to enlarge / click para amplificar
or click here for even higher quality / o aquí para amplificar al máximo

"Farbrausch aus dem Rechner bleibt doch Wissenschaft", WAZ, Germany, p. Do1, 17.9.1988
<-- click to enlarge / click para amplificar
or click here for even higher quality / o aquí para amplificar al máximo


 
 

Example with the logistic equation. Parameter-sequence: AABAB AABAB ... .
(Abscissa: A. Ordinate: B):


 
 

  Example with the equation above, but with a different A-B-sequence:



 
QUANTITATIVE ILLUSTRATIONS

Logistic equation with Parameter-sequence: ABAABBAAABBB ABAABBAAABBB ... :


 
 

Logistic equation. Parameter-sequence: AAAAABBBBB AAAAABBBBB ... :








Publications:

M. Markus, "Charts for prediction and chance", Imperial College Press, London (2007)

M. Markus, "Chaos in maps with continuous and discontinuous maxima", Computers in Physics, 481-493 and cover (1990)

M. Markus, "Lyapunow-Diagramme", Spektrum der Wissenschaft, April, 66-73 (1995) / "Los diagramas de Lyapunov", Investigacion y Ciencia, Sept., 70-77 (1995)

M. Markus and B. Hess, "Lyapunov exponents of the logistic map with periodic forcing", Computer and Graphics 13, 553-558 (1989)

M. Markus, "A scientist's adventures in Postmodernism", Leonardo 33(3),179-186 (2000)

M. Markus and B. Hess, "Fractals through periodic variation of control parameters of iterated maps on the interval", in 'Visualisierung in Mathematik und Naturwissenschaften' (ed. by H. Juergens and D. Saupe) Springer-Verlag, Heidelberg, pp. 87-101 (1989)

M. Markus and J. Tamames, "Fat fractals in Lyapunov space", in: 'Fractals in the Future' (ed. by C.A. Pickover), St. Martin's Press, N.Y. (1996), pp. 333-348

J. Roessler, M. Kiwi, B. Hess and M. Markus, "Modulated non-linear processes and a novel mechanism to induce chaos", Phys. Rev. A 39, 5954-5960 (1989)








Examples (from "Charts for prediction and chance", Imperial College Press, London 2007):

(Lizenzen für die folgenden Bilder werden direkt und ausschließlich durch Mario Markus erteilt: mariomarkus@hotmail.com / Licences for the following pictures are granted directly and exclusively by Mario Markus: mariomarkus@hotmail.com / Licencias para las figuras siguientes son concedidas directa- y exclusivamente por Mario Markus: mariomarkus@hotmail.com)




 
 
 
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