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Fractals through periodic variation of control parameters of iterated maps on the interval

Chapter

Zusammenfassung

Wir untersuchen eindimensionale Iterationen bei periodischer Änderung eines Kontrollparameters zwischen zwei Werten A and B . Der Lyapunov-Exponent λ wird berechnet, und die Funktion λ(A, B) in der A-B-Ebent graphisch dargestellt. Verschiedenen Werten von \ entsprechen dabei verschiedene Graustufen im Bild. Es ergeben sich vielfältige fraktale Strukturen, und das graphische Verfahren gestattet unmittelbar die Feststellung von Eigenschaften des Systems: Koexistenz von zwei oder mehr Attraktoren, geschlossene und sich kreuzende superstabile Kurven, Ordnung aus alternierenden chaotischen Prozessen und Chaos bei überraschend niedrigen Kontrollparametern durch sogenannte „immerwährende Transienten“.

Abstract

One-dimensional iterations are examined under periodic variation of a control parameter between two values A and B . The Lyapunov exponent λ is calculated, and the function λ(A, B) is represented graphically on the A-B-plane using different grey levels for different λ ’s. An overwhelming richness of fractal structures is obtained. The graphical technique allows the straightforward identification of system properties: coexistence of two or more attractors, closed and crossing superstable lines, order arising counter intuitively from alternating chaotic processes, and chaos resulting from “permanent transients” at surprisingly low control parameter values.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  1. 1.Max-Planck-Institut für ErnährungsphysiologieDortmund 1Deutschland

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