The points to which the equations converge is called an attractor.

Some equations are drawn towards a collection of points that is not easily defined but that somehow has a coherent shape. These points are called a strange attractor.

Clifford Pickover discovered that the following equations geneate points that are drawn to a strange attractor.

    X(n) = Sin(A * Y(n - 1)) - Z(n - 1) * Cos(B * X(n - 1))
    Y(n) = Z(n) * Sin(C * X(n - 1)) - Cos(D * Y(n - 1))
    Z(n) = Sin(X(n - 1))

Here A, B, C, and D are constants.

The following code plots these points.

' Draw the curve.
Private Sub DrawCurve()
    Const XMIN As Double = -2.1
    Const XMAX As Double = 2.1
    Const YMIN As Double = -2.1
    Const YMAX As Double = 2.1
    Const ZMIN As Double = -1.2
    Const ZMAX As Double = 1.2

    Dim xoff As Double
    Dim yoff As Double
    Dim zoff As Double
    Dim xscale As Double
    Dim yscale As Double
    Dim zscale As Double
    Dim x As Double
    Dim y As Double
    Dim z As Double
    Dim x2 As Double
    Dim y2 As Double
    Dim i As Integer

    ' Get the drawing parameters.
    GetParameters()
    picCanvas.ForeColor = PointColor

    Select Case SelectedPlane
        Case PlaneTypes.plane_XY
            xoff = m_Wid / 2
            yoff = m_Hgt / 2
            xscale = m_Wid / (XMAX - XMIN)
            yscale = m_Hgt / (YMAX - YMIN)
        Case PlaneTypes.plane_XZ
            xoff = m_Wid / 2
            zoff = m_Hgt / 2
            xscale = m_Wid / (XMAX - XMIN)
            zscale = m_Hgt / (ZMAX - ZMIN)
        Case PlaneTypes.plane_YZ
            yoff = m_Wid / 2
            zoff = m_Hgt / 2
            yscale = m_Wid / (YMAX - YMIN)
            zscale = m_Hgt / (ZMAX - ZMIN)
    End Select

    ' Compute the values.
    x = X0
    y = Y0
    z = Z0
    i = 0
    Do While Running
        ' Move to the next point.
        x2 = Sin(A * y) - z * Cos(B * x)
        y2 = z * Sin(C * x) - Cos(D * y)
        z = Sin(x)
        x = x2
        y = y2

        ' Plot the point.
        Select Case SelectedPlane
            Case PlaneTypes.plane_XY
                m_Bm.SetPixel(CInt(x * xscale + xoff), _
                    CInt(y * yscale + yoff), PointColor)
            Case PlaneTypes.plane_XZ
                m_Bm.SetPixel(CInt(x * xscale + xoff), _
                    CInt(z * zscale + zoff), PointColor)
            Case PlaneTypes.plane_YZ
                m_Bm.SetPixel(CInt(y * yscale + yoff), _
                    CInt(z * zscale + zoff), PointColor)
        End Select

        ' To make things faster, only DoEvents
        ' every 100 times.
        i = i + 1
        If i > 100 Then
            i = 0
            picCanvas.Refresh()
            Application.DoEvents()
        End If
    Loop
End Sub

For more information on fractals, see my book Visual Basic Graphics Programming.