Chapter 4. Universal Behavior of Quadratic Maps

  1. Prof. Dr. H. G. Schuster1 and
  2. Lecturer Wolfram Just2

Published Online: 23 AUG 2005

DOI: 10.1002/3527604804.ch4

Deterministic Chaos: An Introduction, Fourth Edition

Deterministic Chaos: An Introduction, Fourth Edition

How to Cite

Schuster, H. G. and Just, W. (2005) Universal Behavior of Quadratic Maps, in Deterministic Chaos: An Introduction, Fourth Edition, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, FRG. doi: 10.1002/3527604804.ch4

Author Information

  1. 1

    Christian Albrecht University Kiel, Department of Theoretical Physics, Germany

  2. 2

    Queen Mary/University of London, School of Mathematical Sciences, United Kingdom

Publication History

  1. Published Online: 23 AUG 2005
  2. Published Print: 26 JAN 2005

ISBN Information

Print ISBN: 9783527404155

Online ISBN: 9783527604807



  • chaos;
  • deterministic chaos;
  • quadratic maps;
  • pitchfork bifurcation;
  • doubling transformation;
  • supercycles;
  • self-similarity;
  • universal power spectrum;
  • external noise;
  • cycle elements;
  • Hausdorff dimension;
  • power spectrum;
  • logistic map;
  • chaotic bands;
  • period doubling;
  • phase transitions;
  • bifurcation route


This chapter contains sections titled:

  • Parameter Dependence of the Iterates

  • Pitchfork Bifurcation and the Doubling Transformation

    • Pitchfork Bifurcations

    • Supercycles

    • Doubling Transformation and α

    • Linearized Doubling Transformation and δ

  • Self-Similarity, Universal Power Spectrum, and the Influence of External Noise

    • Self-Similarity in the Positions of the Cycle Elements

    • Hausdorff Dimension

    • Power Spectrum

    • Influence of External Noise

  • Behavior of the Logistic Map for rr

    • Sensitive Dependence on Parameters

    • Structural Universality

    • Chaotic Bands and Scaling

  • Parallels between Period Doubling and Phase Transitions

  • Experimental Support for the Bifurcation Route