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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 $t<300$):

enter image description here

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

enter image description here

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

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