This example plots points given by the equations:

    X' = A0 + A1 * x + A2 * x * x + A3 * x * y + A4 * y + A5 * y * y
    Y' = A6 + A7 * x + A8 * x * x + A9 * x * y + A10 * y + A11 * y * y

For various values of A1, A2, ... A11.

The program uses a Timer to plot points. The following code shows the Tick event handler that draws the curve. It plots 1,000 points at a time to improve performance and reduce flicker. For each iteration, the routine generates the next (X, Y) point and draws it on the Bitmap named m_Bitmap. After it has finished its 1,000 points, it draws the Bitmap onto the picCanvas control so you can see the result.

Private X As Double
Private Y As Double

Private Sub tmrDrawPoint_Tick(ByVal sender As _
     System.Object, ByVal e As
System.EventArgs) Handles tmrDrawPoint.Tick
    For i As Integer = 1 To 1000
        Dim new_x As Double = A(0) + A(1) * X + A(2) * X * _
             X + A(3) * X * Y +
A(4) * Y + A(5) * Y * Y
        Dim new_y As Double = A(6) + A(7) * X + A(8) * X * _
             X + A(9) * X * Y +
A(10) * Y + A(11) * Y * Y
        X = new_x
        Y = new_y

        Dim pix_x As Integer = (X - m_Wxmin) / m_Wwid * _
            m_Wid
        Dim pix_y As Integer = m_Hgt - (Y - m_Wymin) / _
            m_Whgt * m_Hgt - 1
        If (pix_x >= 0) AndAlso (pix_x < m_Wid) AndAlso _
           (pix_y >= 0) AndAlso (pix_y < m_Hgt) _
        Then
            m_Bitmap.SetPixel(pix_x, pix_y, Color.Blue)
        End If
    Next i

    ' Display the result.
    Dim gr As Graphics = picCanvas.CreateGraphics()
    gr.DrawImage(m_Bitmap, 0, 0)
    gr.Dispose()
End Sub

For more information on this kind of strange attractor, see Strange Attractor by Eric W. Weisstein from MathWorld--A Wolfram Web Resource.