# Can a boundary between bounded and unbounded solutions of a differential equation have a shape of a fractal?

Iterative maps (like $$z_{n+1}=z_n^2+c$$, where $$z\in \mathbb{C}$$, $$z_0=c$$) sometimes have a fractal-like boundary between bounded and unbounded sequences (Mandelbrot set). Are there differential equations that have a similar property of the fractal boundary between bounded and unbounded solutions?

I have tried to build the analogue of the Mandelbrot fractal with a differential equation, $$z+z'+\frac{1}{2!} z''+\frac{1}{3!} z'''+ \frac{1}{4!}z''''=z^2+c$$ with initial conditions $$z(0)=c$$, $$z'(0)=z^2+c-z$$, $$z''(0)=z'''(0)=0$$. The result looks the following (color of each pixel corresponds to the time $$t$$ when $$|z(t)|=100$$ for the first time; Calculations were performed for $$t<300$$):

Even though the plot may seem to resemble Mandelbrot fractal, with a closer look the boundary appears fairly smooth:

Do all boundaries between bounded and unbounded solutions for differential equation look this simple (meaning, they are not fractals)?