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Logistic map

From periodicity to chaos

Rajouté le Friday 18 February 2022
Monticelli Marc

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 entitled``Simple 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>