I have long ago give up trying to find a nice formula for the $n$th iteration of functions in the form $$f_a(x)=x^2+a^2$$ However, it would be interesting to consider the asymptotic growth of the iteration of these functions. It is apparent that $f_a^{\circ n}(x)$ should behave like $\text{b}^{2^n}$, but calculating the value of the base $b$ in terms of $a$ and $x$ has been more difficult than I anticipated. We may solve for $b$ using a limit: $$b_a(x)=\lim_{n\to\infty} \big[f_a^{\circ n}(x)\big]^{2^{-n}}$$ but this is as far as I have gotten, and I can’t make this any neater. The best thing I have managed to find is the rather trivial observation $$b_a(x^2+a^2)=b_a^2(x)$$ and I have noticed that the graph of $b_a(x)$ typically looks like (but is not) a hyperbola.
Can anyone calculate any special values of $b_a(x)$ in closed form, or in terms of an infinite series or integral? Can you find any more interesting (non-trivial) functional or differential equations?
From Wikipedia, it states that the $n$th iteration of $$ax^2+bx+\frac{b^2-2b-8}{4a}$$ is $$\frac{2c^{2^n}+2c^{-2^n}-b}{2a}$$ where $$c=\frac{2ax+b\pm\sqrt{(2ax+b)^2-16}}{4}$$
Thus, we can solve the case $f(x)=x^2-2$.
$$b_{\pm 2i}(x)=\frac{x+\sqrt{x^2-4}}2$$