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# Logistic map

From periodicity to chaos

Rajouté le Friday 18 February 2022

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

The logistic map is $x\mapsto rx(1-x)$ where $r$ is a positive parameter. One can check that when $r\in\, ]0,4]$, if $x\in[0,1]$, then $rx(1-x)\in[0,1]$. Hence, picking $x_0\in [0,1]$, one can construct a sequence $(x_n)$ living in $[0,1]$ by recurrence:

$$x_0\in[0,1], \; x_{n+1}=r x_n(1-x_n),\; n\geq 0.$$

This is an example of a discrete-time dynamical system, with each time step corresponding to an iteration of the logistic map, and $(x_n)$ is the orbit of $x_0$ under this dynamics.
This dynamical system was introduced in 1976 by Robert May in an article entitledSimple mathematical models with very complicated dynamics’’. May proposed it to model the dynamics of a population, for example of insects, with a time step corresponding to one year. One has to interpret $x_n$ as the insect density in year $n$.
The central point to which he draws attention is that, despite the innocent appearance of this model, its behaviour is extraordinarily rich and complex when the parameter $r$ varies. This is what we propose to verify with the following interactive digital experiment.
By clicking in the view, you select $x_0$ and the corresponding orbit is computed and plotted in the view below where you can see the values of $x_n$ as a function of $n$.
What can be observed is the appearance of "chaos" by a "period-doubling cascade". We provide more informations below.

Here is a very partial and rough description of what you will see.
If $0< r\leq 1$, then $(x_n)$ goes to $0$. If $r$ is larger than $1$ but smaller than $3$, $(x_n)$ goes to $\frac{r-1}{r}$, which is the (non-trivial) fixed point of the logistic map, that is, $x$ such that $x=rx(1-x)$, which thus lies at the intersection of the graph of the map and the first bisector $y=x$. (There are two trivial fixed points, namely $0$ and $1$, which always exist, whatever the value of $r$ is.)
If $r$ is larger than $3$ but below $3,57$, you can observe stable periodic oscillations, that is, $(x_n)$ only takes a finite number of values (after a quick transient phase), and these values are powers of $2$.
When $r$≈$3,57$, you will not longer observe oscillations of finite period. You can also observe (by clicking on the button "Sensitivity to initial conditions") that a slightly different value of $x_0$ results in a very different orbit, which a characteristic of deterministic chaos.
Most values of $r$ above the critical value $3,57$ yield this chaotic behaviour. But there are some "windows" of periodicity, for instance for $r=3,83$.
As you will see, these properties do not depend on the initial condition $x_0$ taken in $\left]0,1\right[$ (hence we exclude the trivial fixed points $0$ and $1$).
One can encode all the behaviours as a function of $r$ in a bifurcation diagram.

HTML code to integrate this simulation into your pages:

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