Logistic map

Monticelli Marc — 2022-02-18T08:54:58Z

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:

Hénon's attractor

Chazottes Jean-René , Monticelli Marc — 2022-02-17T09:55:27Z

By playing with the parameters of the Lorenz equations and using a Poincaré section, Pomeau and Ibanez demonstrated the formation mechanism of a Smale horseshoe. Pomeau presented his work at a seminar attended by Michel Hénon who then conceived a very simple model of quadratic transformation of the plane which simulates, when a parameter varies, the mechanism of formation of a horseshoe: it is the famous Hénon model.

The model is defined as follows. Given $(x_0,y_0)$ in the place, one computes its orbit $(x_1,y_1),(x_2,y_2),...$ by successive iterations:

$$ \begin{cases} x_{n+1} =y_n+1-ax_n^2 \\ y_{n+1} =b x_n \end{cases} $$

where $a,b$ are parameters.

Numerical exploration of this model shows, for certain values of the parameters, the existence of a “strange attractor” which has a fractal structure. The values in Hénon's paper are $a=1.4$, $b=0.3$.
The fact that this attractor really exists, and is not just a numerical belief, remained an open problem until the late 1980s. It was in 1991 that Benedicks and Carleson first showed rigorously the existence of these attractors. Their theorem is a mathematical tour-de-force but does not cover the values of the parameters $a=1.4$ et $b=0.3$.

In the following interactive experiment, we run 100 initial points at the same time to generate the attractor faster.
By clicking in the plane you can zoom in to see more details of the strange attractor, when it appears.

HTML code to integrate this simulation into your pages :

Experimentarium Digitale

Logistic map

Hénon's attractor