Here is pseudocode for a Mandelbrot variation and the assocuated Mandelbrot point figure as shown on the left-hand side:

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 * y * y + xs;
            		ynew = -x * 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);
	}
}

Two hundred iterations of the Mandelbrot-Lorenz Map (501 pixels x 501 pixels) were generated with the map plotted each ten interals. Here is map pseudocode:

dt=0.02, r=28.0, s=10.0, b=2.667
for (int m = 1; m ≤ 200; 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);
			}
		}
	}
}

Resource for Lorenz calculations:
https://en.wikipedia.org/wiki/Lorenz_system

for (int i = 0; i ≤ 800; i++)
{
	for (int j = 0; j ≤ 800; j++)
	{
		x = 0.0;
		y = 0.0;
		xs = -4.0 + (i / 100.0);
		ys = -4.0 + (j / 100.0);
		k = 0;
		do
		{
			k = k + 1;
			xnew = x * x - y * y + xs;
			ynew = 2.0 * x * y + ys;
			x = xnew;
			y = ynew;
		} while ((k ≤ kmax) && x*x+y*y ≤ 16.0);
		color = gray;
		if (x*x+y*y ≤ 16.0) color = blue;
		Plot respective pixels and points.
	}
}