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The iterative evolution of complex systems

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  • Erber, T.
  • Gavelek, D.

Abstract

The iterative properties of two classes of functions that map a finite set of N distinct elements into itself are essentially equivalent: (1) functions whose iterates form sequences that imitate random behavior; and (2) functions chosen “at random” from the set of all functions that map N into itself. The orbit structures of these functions are highly constrained. For instance, the expected size of the largest orbit is ∼ 0.8N; the expected size of the largest tree is ∼ 0.5N; the average length of a terminal cycle is (πN/8)12; and the average number of elements without pre-images is N/e. These constraints influence the dynamical behavior of “chaotic” systems whose evolution can be modeled by functional iteration. One practical consequence is the macroscopic stabilization of hysteresis loops in magnetic and mechanical systems.

Suggested Citation

  • Erber, T. & Gavelek, D., 1991. "The iterative evolution of complex systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 177(1), pages 394-400.
    • Handle: RePEc:eee:phsmap:v:177:y:1991:i:1:p:394-400
    DOI: 10.1016/0378-4371(91)90178-F
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    Cited by:

    1. Diamond, P. & Kloeden, P.E. & Kozyakin, V.S. & Pokrovskii, A.V., 1997. "A model for roundoff and collapse in computation of chaotic dynamical systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 44(2), pages 163-185.

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