World Library  
Flag as Inappropriate
Email this Article

Complex squaring map

Article Id: WHEBN0023868295
Reproduction Date:

Title: Complex squaring map  
Author: World Heritage Encyclopedia
Language: English
Subject: List of chaotic maps, Chaos theory, Exponential map (discrete dynamical systems), Duffing map, Synchronization of chaos
Publisher: World Heritage Encyclopedia

Complex squaring map

In mathematics, the complex squaring map, a polynomial mapping of degree two, is a simple and accessible demonstration of chaos in dynamical systems. It can be constructed by performing the following steps:

  1. Choose any complex number on the unit circle whose argument (complex angle) is not a rational fraction of π,
  2. Repeatedly square that number.

This repetition (iteration) produces a sequence of complex numbers that can be described by their complex angle alone. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. In fact, it can be shown that the sequence will be chaotic, i.e. it is sensitive to the detailed choice of starting angle.

Chaos and the complex squaring map

The informal reason why the iteration is chaotic is that the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but angles which differ by multiples of 2π (radians) are identical. Thus, when the angle exceeds 2π, it must wrap to the remainder on division by 2π. Therefore the angle is transformed according to the dyadic transformation (also known as the 2x mod 1 map). As the initial value z0 has been chosen so that its argument is not a rational multiple of π, the forward orbit of zn cannot repeat itself and become periodic.

More formally, the iteration can be written as:

\qquad z_{n+1} = z_n^2

where z_n is the resulting sequence of complex numbers obtained by iterating the steps above, and z_0 represents the initial starting number. We can solve this iteration exactly:

\qquad z_n = z_0^{2^n}

Starting with angle θ, we can write the initial term as z_0 = \exp(i\theta) so that z_n = \exp(i2^n\theta). This makes the successive doubling of the angle clear. (This is equivalent to the relation z_n = \cos(2^n\theta)+i \sin(2^n\theta).)


This map is a special case of the complex quadratic map, which has exact solutions for many special cases.[1] The complex map obtained by raising the previous number to any natural number power z_{n+1} = z_n^p is also exactly solvable as z_n = z_0^{p^n}. In the case p = 2, the dynamics can be mapped to the dyadic transformation, as described above, but for p > 2, we obtain a shift map in the number base p. For example, p = 10 is a decimal shift map.

See also


  1. ^ M. Little, D. Heesch (2004), Chaotic root-finding for a small class of polynomials, Journal of Difference Equations and Applications, 10(11):949–953.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.