Bifurcation diagram

Question

Consider the logistic map $x_{n+1}=rx_n(1-x_n)$, whose bifurcation diagram is shown below for $2.4 < r < 4.0$: I need to find a particular value of $r$ so that "attracting $2^k$ periodic points (a result after $k$ instances of period doubling) accumulate". This was one part of my lab assignment, but I do not understand what it is saying. I do see some nearly-empty small gaps in the above bifurcation diagram for certain values of $r$.

Answer 1

You want to compute limit point r of period doubling cascade. One can do it:

• numerically
• visually : zoom into placed of period doubling bifurcation , like as in fig 7 from this paper