Basic Mandelbrot-Lorenz Map

On the left-hand side, the Mandelbrot pixel and point figures are presented and pseudocode is shown.

One hundred iterations of the Mandelbrot-Lorenz Map were generated with the map plotted each ten iterations.
On the right-hand side, the Mandelbrot-Lorenz Map at 40 iterations is presented (X-Y, X-Z, Y-Z respectively) and pseudocode is shown where:

• dt=0.02
• r=28.0
• s=10.0
• b=2.667

Mandelbrot Pixel and Point

for (int i = 0; i ≤ 500; i++)
{
	for (int j = 0; j ≤ 500; j++)
	{
		x = 0.0;
		y = 0.0;
		xs = -2.5 + (i / 100.0);
		ys = -2.5 + (j / 100.0);
		k = 0;
		do
		{
            		xnew = x*x - y*y + xs;
            		ynew = 2.0*x*y + ys;
			x = xnew;
			y = ynew;
			k = k + 1;
		} while ((k ≤ 64) && (x*x + y*y ≤ 6.25));
		xarray[i][j] = x;
		yarray[i][j] = y;
		zarray[i][j] = k;
		PlotPixel(i, j, color);
		PlotPoint(x, y, color);
	}
}

Mandelbrot-Lorenz Map

for (int m = 1; m ≤ 100; m++)
{
	for (int i = 0; i ≤ 500; i++)
	{
		for (int j = 0; j ≤ 500; j++)
		{
			x = xarray[i][j];
			y = yarray[i][j];
			z = zarray[i][j];
			dx = s * (y - x);
			dy = r * x - y - x * z;
			dz = x * y - b * z;
			xarray[i][j] = x = x + dx * dt;
			yarray[i][j] = y = y + dy * dt;
			zarray[i][j] = z = z + dz * dt;
			if (m % 10 == 0)
			{
				PlotPoint(x, y, color);
				PlotPoint(x, z, color);
				PlotPoint(y, z, color);
			}
		}
	}
}

Mandelbrot	Build: (f(x,y), g(x,y))	Escape: h(x,y)>value	Plot
Basic Mandelbrot Set	(x² - y², 2.0*x*y)	x² + y² > 6.25	Pixel, Point, Lorenz
Mandelbrot Variant #1	(x*y², -y*x²)	x³ + y² > 6.25	Pixel, Point, Lorenz
Mandelbrot Variant #2	(x*y², 2.0*x*y)	x³ + y² > 6.25	Pixel, Point, Lorenz

Julia Pixel and Point

for (int i = 0; i ≤ 500; i++)
{
	for (int j = 0; j ≤ 500; j++)
	{
		xs = 0.0;
		ys = 0.0;
		x = -2.5 + (i / 100.0);
		y = -2.5 + (j / 100.0);
		k = 0;
		do
		{
            		xnew = x*x - y*y + xs;
            		ynew = 2.0*x*y + ys;
			x = xnew;
			y = ynew;
			k = k + 1;
		} while ((k ≤ 64) && (x*x + y*y ≤ 6.25));
		xarray[i][j] = x;
		yarray[i][j] = y;
		zarray[i][j] = k;
		PlotPixel(i, j, color);
		PlotPoint(x, y, color);
	}
}

Julia-Lorenz Map

for (int m = 1; m ≤ 100; m++)
{
	for (int i = 0; i ≤ 500; i++)
	{
		for (int j = 0; j ≤ 500; j++)
		{
			x = xarray[i][j];
			y = yarray[i][j];
			z = zarray[i][j];
			dx = s * (y - x);
			dy = r * x - y - x * z;
			dz = x * y - b * z;
			xarray[i][j] = x = x + dx * dt;
			yarray[i][j] = y = y + dy * dt;
			zarray[i][j] = z = z + dz * dt;
			if (m % 10 == 0)
			{
				PlotPoint(x, y, color);
				PlotPoint(x, z, color);
				PlotPoint(y, z, color);
			}
		}
	}
}