World Library  
Flag as Inappropriate
Email this Article

Tinkerbell map

Article Id: WHEBN0002673485
Reproduction Date:

Title: Tinkerbell map  
Author: World Heritage Encyclopedia
Language: English
Subject: Chaos theory, List of chaotic maps, Chaotic maps, Exponential map (discrete dynamical systems), Duffing map
Collection: Chaotic Maps
Publisher: World Heritage Encyclopedia

Tinkerbell map

Tinkerbell attractor with a=0.9, b=-0.6013, c=2, d=0.5. Used starting values of x_0 = -0.72 and y_0 = -0.64.

The Tinkerbell map is a discrete-time dynamical system given by:


Some commonly used values of a, b, c, and d are

  • a=0.9, b=-0.6013, c=2.0, d=0.50
  • a=0.3, b=0.6000, c=2.0, d=0.27

Like all chaotic maps, the Tinkerbell Map has also been shown to have periods; after a certain number of mapping iterations any given point shown in the map to the right will find itself once again at its starting location.

The origin of the name is uncertain; however, the graphical picture of the system (as shown to the right) shows a similarity to the movement of Tinker Bell over Cinderella Castle, as shown at the beginning of all films produced by Disney.

Source code

The Java source code that was used to generate the Tinkerbell Map displayed above:


public class TinkerBellMap {
  public static void main(String[] args) throws Exception {
    FileWriter fstream = new FileWriter("TinkerBellMapOutput.txt");
    BufferedWriter out = new BufferedWriter(fstream);
    int time = 0, iterations = 50000;
    double x = -0.72, y = -0.64;
    double a = 0.9, b = -0.6013, c = 2.0, d = 0.5;
    while (time < iterations) {
      double oldX = x;
      x = Math.pow(x,2)-Math.pow(y,2)+a*x+b*y;
      y = 2*oldX*y+c*oldX+d*y;
      out.write(x+" "+y+"\n"); //writing data to a txt file to be read by Mathematica

See also


  • Markov Chain Monte Carlo Estimation of Nonlinear Dynamics from Time SeriesC.L. Bremer & D.T. Kaplan,
  • K.T. Alligood, T.D. Sauer & J.A. Yorke, Chaos: An Introduction to Dynamical Systems, Berlin: Springer-Verlag, 1996.
  • Asymptotic angular stability in non-linear systems: rotation numbers and winding numbersP.E. McSharry & P.R.C. Ruffino,
  • Towards complete detection of unstable periodic orbits in chaotic systemsR.L. Davidchack, Y.-C. Lai, A. Klebanoff & E.M. Bollt,
  • B. R. Hunt, Judy A. Kennedy, Tien-Yien Li, Helena E. Nusse, "SLYRB measures: natural invariant measures for chaotic systems"
  • A. Goldsztejn, W. Hayes, P. Collins "Tinkerbell is Chaotic" SIAM J. Applied Dynamical Systems 10, n.4 1480-1501, 2011

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 World eBook Library 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.