There is a unpopular logistic map, that is produced via iterating $x_{n+1} = r (1-x_n^{x_n})$ With rate of growth $r$ as "x axis", being from 0 to $1/(1-{1/e}^{1/e})$ (approx. 3.248869522), and population convergence on y axis, corresponding map would look like:
I have a question. I want to find that tricky intersection point $r_i$, shown by arrow, and $x_{min}$, $x_i$, $x_{max}$ values, at that $r_i$:
And also I would like to know, does it has special name, and what are its properties? Is a population at that growth rate $r_i$ periodic or chaotic? Thanks!
P.S For completeness, lets assume starting point $x_0=1/e$
You should read The Road to Chaos is Filled with Polynomial Curves. While that is written for the logistic family, the same ideas apply to any family of differentiable, unimodal curves. For this particular family, we should define the $n^{\text{th}}$ critical curve $g_n$ to be $$g_n(r) = f_r^n(1/e).$$ If we plot the first few of these together with your bifurcation diagram, we get the following image
As we can see, your point is intersection of two of those curves - the $4^{\text{th}}$ and $5^{\text{th}}$ as it turns out. Thus, we can find the $x$ coordinate of the point by solving $$g_4(r)=g_5(r).$$
A numerical approximation of the point turns out to be $$(3.12099,0.695916),$$ which is plotted in the figure.
The answer to your second question is contained in the paper's Intersection Dichotomy theorem on pages 644 and 645, where they explain that the point is a Misiurewicz point and the dynamics are chaotic.