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Bifurcation diagram of the logistic map

Rajouté le Friday 18 February 2022
Chazottes Jean-René , Monticelli Marc

All the versions of this article: [English] [français]

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.

Let us briefly explain how to compute this diagram.
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 $j=1,2,\ldots,N$.

HTML code to integrate this simulation into your pages:

<iframe style="overflow: hidden;"
src="http://experiences.math.cnrs.fr/simulations/systemesdynamiques/DiagrammeDeBifurcation/" height="600" width="100%" frameborder="0" scrolling="no"></iframe>