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| | Hi there. Let me begin by introducing the author, her title is Myrtle Cleary. Managing people is his occupation. South Dakota is where me and my husband reside. To perform baseball is the hobby he will by no means stop performing.<br><br>My website; at home std test; [http://netwk.hannam.ac.kr/xe/data_2/38191 official source], |
| [[Image:Michael Wu - TinkerBellMapGraph.gif|thumb|right|Tinkerbell attractor with a=0.9, b=-0.6013, c=2, d=0.5. Used starting values of <math>x_0 = -0.72</math> and <math>y_0 = -0.64</math>.]]
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| The '''Tinkerbell map''' is a discrete-time [[dynamical system]] given by:
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| :<math>x_{n+1}=x_n^2-y_n^2+ax_n+by_n\,</math>
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| :<math>y_{n+1}=2x_ny_n+cx_n+dy_n\,</math>
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| Some commonly used values of a, b, c, and d are
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| *<math>a=0.9, b=-0.6013, c=2.0, d=0.50</math>
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| *<math>a=0.3, b=0.6000, c=2.0, d=0.27</math>
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| 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.
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| 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.
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| ==Source code==
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| The Java source code that was used to generate the Tinkerbell Map displayed above:
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| <source lang="java">
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| import java.io.*;
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| public class TinkerBellMap {
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| public static void main(String[] args) throws Exception {
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| FileWriter fstream = new FileWriter("TinkerBellMapOutput.txt");
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| BufferedWriter out = new BufferedWriter(fstream);
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| int time = 0, iterations = 50000;
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| double x = -0.72, y = -0.64;
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| double a = 0.9, b = -0.6013, c = 2.0, d = 0.5;
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| while (time < iterations) {
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| double oldX = x;
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| x = Math.pow(x,2)-Math.pow(y,2)+a*x+b*y;
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| y = 2*oldX*y+c*oldX+d*y;
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| out.write(x+" "+y+"\n"); //writing data to a txt file to be read by Mathematica
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| time++;
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| }
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| }
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| }
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| </source>
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| ==See also==
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| * [[List of chaotic maps]]
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| == References ==
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| * [http://www.macalester.edu/~kaplan/preprints/bremer.pdf C.L. Bremer & D.T. Kaplan, ''Markov Chain Monte Carlo Estimation of Nonlinear Dynamics from Time Series'']
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| * K.T. Alligood, T.D. Sauer & J.A. Yorke, ''Chaos: An Introduction to Dynamical Systems'', Berlin: Springer-Verlag, 1996.
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| * [http://www.maths.ox.ac.uk/~mcsharry/papers/dynsys18n3p191y2003mcsharry.pdf P.E. McSharry & P.R.C. Ruffino, ''Asymptotic angular stability in non-linear systems: rotation numbers and winding numbers'']
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| * [http://chaos1.la.asu.edu/~yclai/papers/PLA_01_DLKB.pdf R.L. Davidchack, Y.-C. Lai, A. Klebanoff & E.M. Bollt, ''Towards complete detection of unstable periodic orbits in chaotic systems'']
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| * B. R. Hunt, Judy A. Kennedy, Tien-Yien Li, Helena E. Nusse, "SLYRB measures: natural invariant measures for chaotic systems"
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| * [http://www.goldsztejn.com/publications/SIADS2011.pdf A. Goldsztejn, W. Hayes, P. Collins "Tinkerbell is Chaotic" SIAM J. Applied Dynamical Systems 10, n.4 1480-1501, 2011]
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| {{Chaos theory}}
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| [[Category:Chaotic maps]]
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| {{Mathapplied-stub}}
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Hi there. Let me begin by introducing the author, her title is Myrtle Cleary. Managing people is his occupation. South Dakota is where me and my husband reside. To perform baseball is the hobby he will by no means stop performing.
My website; at home std test; official source,