Looking at Chaos

An article in Babbage's Bag is all about chaos and this seems like a good time to have some fun plotting some chaos diagrams and using the scrolling technique described in the recent project Stripchart Scrolling.

The basic idea is very simple. The iteration

x<-ax(1-x)

is performed to see what values of x are generated for various values of the constant a.

The values of a that are of interest are between about 2.95 and 4.00. For this range the iteration stays within the interval 0 to 1 and chaos sets in at about a = 3.5.

When the iteration is started it takes some time to settle down and then it cycles though a finite number of values. The values that the iteration cycles though form the attracting set of the process and it’s these we need to plot.

Start a new C# project and place a button and a picturebox on the form.

[画像:form]

The form you need to show chaos

The button’s click routine does the work of plotting. First we need to make the pictureBox have a persistent drawing surface in the form of a bitmap:

private void button1_Click(
 object sender, EventArgs e)
{
 Bitmap BImage1 = new Bitmap(
 pictureBox1.ClientSize.Width,
 pictureBox1.ClientSize.Height);
 pictureBox1.BackgroundImage = BImage1;

Double a;
Graphics G = pictureBox2.CreateGraphics();
for (int y = 0; y < BImage1.Height; y++)
{
 a = 2.95 + (4 - 2.95) *
 y / BImage1.Height;

We then initialise x to some convenient value, the eventual result doesn’t depend much on what this is, and then iterate 500 times to allow transients to die away:

Double x = 0.2;
for(int i = 1;i<500;i++)
{
 x = a * x * (1 - x);
}

The next 500 iterations are used to find out what values of x occur in the iteration after the transients have gone:

for ( int i= 1; i < 500; i++)
{
 x = a * x * (1 - x);
 BImage1.SetPixel(
 (int)(x * BImage1.Width), 
 y,Color.Green);
 int y1 =(int) (x * pictureBox2.Height);
 G.DrawLine(Pens.Red, 0, y1, 0, y1 + 1);
 pictureBox2.Scroll(1, 0);
 Application.DoEvents();
 }

Finally we Refresh the pictureBox and make sure to dispose of the Graphics object:

 pictureBox1.Refresh();
 }
 G.Dispose();
}

If you run this finished program you will see the bifurcation diagram for the map - see the next page.

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