Continuing from previous posts on the topic of the derivative[s] of the Mandelbrot Series. I now attempt to examine the regions of stability for this D.E. y'' + y' + y'*y' + y - exp(y*y') = 0 Where I set y equal to the Mandelbrot series , with initial conditions determined by the value of Pixel. I remain uncertain at this stage as to how to interpret the results. I'm assuming that the regions with the fewest iterations are what we require. Another approach to solving second order differential equations is to use the Euler - Heun method ; this however provides less information as it's usually implemented as a purely numerical approach. Non linear differential equations have been difficult to solve as the elementary functions don't provide the scope required. Anyway here's the code . de2bmand(XAXIS){; Edward Montague ; 2nd order non linear differential equation = z. ; y'' + y' + y'*y' + y - exp(y*y') = 0 ; 2nd derivative of mandelbrot series = w. ; 1st derivative of mandelbrot series = u. ; Mandelbrot series = v v = Pixel u = 1 w = 0 z = 0: w = 2*u*u +2*w*v u = 2*v*u + 1 z = w +u + u*u + v - exp(v*u) v = v*v+Pixel |z| < 1 }