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# Bifurcation diagram of the logistic map

Rajouté le Friday 18 February 2022

We presented the logistic map here and mentioned the bifurcation diagram that displays the different asymptotic behaviours of the orbits as a function of the parameter. Here you can "draw" it: Hold the mouse button down and scratch’’ to make the diagram appearing. A double-click in the diagram produces a zoom in, which allows you to refine part of the diagram, and see its amazing structure.
The attractor is shown on the vertical line as $r$ varies on the horizontal axis.
Pick up values $r_1$, $r_2$,..., $r_N$, with $N=1000$ such that $r_{j+1}-r_j=0,005$. Then compute the orbit $(x_n)$ of, say, $x_0=0.5$ for each $r_j$. After $50$ iterations, we consider that transient phases are eliminated, that is, $(x_n)$ reached $0$, or became periodic, etc. We thus have a value $x_{50}$ for each $r_j$, denoted by $x_{50}(r_j)$. Finally, we plot $(r_j,x_{50}(r_j))$ for $j=1,2,\ldots,N$.
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