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)?



You must log in to answer this question.