Complexity Explorer Santa Fe Institute

This course is no longer in session.

Introduction to Dynamical Systems and Chaos (Fall 2018)

Lead instructor:

6.8 Further Reading (Optional) » References and Additional Reading

Early Papers on Universality and Renormalization:

  • Coullet, P., Tresser, C. Itération déndomorphismes et groupe de renormalisation, J. Phys. Colloque C5, C5-25 - C5-28 (1978).  pdf
  • Feigenbaum, M.J. Quantitative universality for a class of non-linear transformations, J. Stat. Phys. 19 25-52 (1978).  pdf
  • Cvitanovic, Predrag. Universality in chaos. (2nd edition)  Taylor & Francis, 1984.  This is a collection of early articles on universality in chaos, the renormalization group, and experimental implications of universality.


  • Gleick, James. Chaos: Making a new science. Random House, 1997.  The chapter titled "Universality" is an engaging account of Feigenbaum's discovery of universality.  Doesn't say much about what universality and the renormalization group actually are, however.
  • Stewart, Ian. Does God play dice?: The new mathematics of chaos. Penguin UK, 1997.  Chapter 10 has a very nice, not-very-technical discussion of universality and renormalization.



  • Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. Chaos and fractals: new frontiers of science. Springer, 2004.  Chapter 11 contains a very clear account of universality and renormalization.  Highly recommended.
  • Strogatz, Steven. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering.  Westview Press. (2001).  Chapter 10 has a nice treatment of the logistic map, including a clear discussion of renormalization.

  • Tresser, Charles, Pierre Coullet, and Edson de Faria. Period Doubling. Scholarpedia.  September 2013.