The Tinkerbell map is a discrete-time dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

 given by:



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

• a = 0.3, b = 0.6000, c =2, d = 0.27.
• a = 0.9, b = -0.6013, c =2, d = 0.50.

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

Cinderella Castle

Cinderella Castle is the fairy tale castle at the center of two Disney theme parks: the Magic Kingdom at the Walt Disney World Resort, and Tokyo Disneyland at the Tokyo Disney Resort. Both serve as worldwide recognized icons and the flagship attraction for their respective theme parks.-Inspiration...

, 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:

import java.io.*;

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
time++;
}
}
}