I'm new to dynamical systems, but I've stumbled onto an interesting system and I'm wondering if the orbit is bounded or not for a particular initial condition. The function in question is:
$f(x) = \begin{cases}
\sqrt{x} &\quad\text{if }x \geq 2\\
\frac{x}{2-x} &\quad \text{if } x < 2
\end{cases} $
and initial condition $x_0 = \frac{1 + \sqrt{5}}{2}$. Does anyone know if this has a bounded or dense orbit?
Thank you!