Experimentarium Digitale

Modèle de Lozi (version en couleurs)

Chazottes Jean-René , Monticelli Marc , René Lozi — 2023-12-08T09:05:10Z

Le modèle de Lozi est un système dynamique à temps discret du plan dans lui-même : étant donné un point $(x_0,y_0)$ du plan, son évolution est donnée par

$$ \begin{cases} x_{n+1}= y_n+1-a|x_n|\\ y_{n+1}=bx_n \end{cases} $$

pour $n=0,1,2,\ldots$.Pour plus de détails sur ce modèle, voir cet article.

Ici, nous nous concentrons sur le cas $b=-1$ (cas conservatif). Quand on clique dans la vue, on démarre la trajectoire du point sélectionné avec une certaine couleur (tirée aléatoirement). Cela permet de mieux voir comment les structures se constuisent.

Modèle de Lozi

Chazottes Jean-René , Monticelli Marc , René Lozi — 2023-12-08T08:40:53Z

En 1977, René Lozi a introduit un système dynamique à temps discret du plan dans lui-même en remplaçant le terme quadratique du modèle de Hénon par une valeur absolue, ce qui donne une application affine par morceaux : étant donné un point $(x_0,y_0)$ du plan, son évolution est donnée par

$$ \begin{cases} x_{n+1}= y_n+1-a|x_n|\\ y_{n+1}=bx_n \end{cases} $$

pour $n=0,1,2,\ldots$.

Le modèle de Lozi est beaucoup plus simple à étudier mathématiquement que celui de Hénon tout en ayant la même richesse de comportements.

Quand $|b|<1$, la dynamique est dissipative dans le sens que si on prend une région du plan et qu'on l'itère, sa surface devient strictement plus petite. Pour certaines valeurs des paramètres, ce système dynamique a un attracteur étrange. En fait, Michal Misiurewicz a démontré que pour l'ensemble de paramètres suivant

$$ \Big\{ (a,b)\in \mathbb{R}^2 : b>0, a\sqrt{2} < b +2, 2a + b < 4 \Big\}, $$

on a bien un attracteur étrange.

Quand $|b|=1$, la dynamique est conservative : si on prend une région du plan et qu'on l'itère, cette fois-ci sa surface est inchangée (mais elle se déforme). Dans l'expérience numérique ci-dessous, on pourra constater l'extraordinaire structure du portrait de phase.

Une version en couleurs dans le cas conservatif se trouve là.

Attracteur de Plykin sur la sphère

Chazottes Jean-René , Monticelli Marc — 2022-09-16T06:52:49Z

S. Kuznetsov a proposé un système dynamique explicite donnant un attracteur de Plykin (qui est un attracteur hyperbolique). Ce système évolue sur la sphère unité.

On part d'une condition initiale qui est un point $\boldsymbol{x}_0=(x_0,y_0,z_0)$ qui satisfait $x_0^2+y_0^2+z_0^2=1$. Ensuite on définit son orbite par récurrence :

$$ \boldsymbol{x}_{n+1}=\boldsymbol{f}(\boldsymbol{x}_n) :=\boldsymbol{f}_+(\,\,\boldsymbol{f}_-(\boldsymbol{x}_n)), \, n\geq 0 $$

où

$$ \boldsymbol{f}_{\pm}(\boldsymbol{x})= \begin{pmatrix} \pm z \\ \frac{y\, \mathrm{e}^{\frac{\varepsilon}{2}(x^2+y^2)} \cos\big(\frac{\pi}{2}(z\sqrt{2}+1)\big)\,\pm\, x\, \mathrm{e}^{-\frac{\varepsilon}{2}(x^2+y^2)} \sin\big(\frac{\pi}{2}(z\sqrt{2}+1)\big)}{\sqrt{\cosh( \varepsilon(x^2+y^2) +\varepsilon(y^2-x^2)\frac{\sinh(\varepsilon(x^2+y^2))}{\varepsilon(x^2+y^2)}}} \\ \frac{y\, \mathrm{e}^{\frac{\varepsilon}{2}(x^2+y^2)} \sin\big(\frac{\pi}{2}(z\sqrt{2}+1)\big)\,\mp\, x\, \mathrm{e}^{-\frac{\varepsilon}{2}(x^2+y^2)} \cos\big(\frac{\pi}{2}(z\sqrt{2}+1)\big)}{\sqrt{\cosh(\varepsilon(x^2+y^2)) +\varepsilon(y^2-x^2)\frac{\sinh(\varepsilon(x^2+y^2)}{\varepsilon(x^2+y^2)}}} \end{pmatrix}\cdot $$

Lorenz attractor

Chazottes Jean-René , Monticelli Marc — 2022-03-09T22:11:00Z

To study the possibly complicated behavior of three-dimensional systems, there is no better place to begin than with the famous model proposed by Lorenz in 1963. Before this model appeared, the only types of stable attractors known in differential equations were fixed points and closed trajectories. This model illustrates in particular the sensitive dependence on intial conditions, also known by the large public as the 'butterfly effect' (an expression coined by Lorenz himself).

The Lorenz system is given by the equations

$$ \begin{cases} \dot{x} = \sigma (y-x)\\ \dot{y}=\rho x-y -xz\\ \dot{z}=xy-\beta z \end{cases} $$

where $\sigma,\rho$ and $\beta$ are positive parameters. The 'historical values' (those used by Lorenz in his paper) are $\sigma=10$, $\beta=8/3$ et $\rho=28$.

Explaining how Lorenz got his equations would lead us away. We content ourselves with a few words. Lorenz was interested in setting up a simple model that would explain some of the unpredictable behavior of the weather.

Physical sensible models of atmospheric convection involve partial differential equations, and are extremely complicated to analyze. Lorenz sought a much simpler system. He considered a two-dimensional fluid cell that was heated from below and cooled from above. In Fourier modes, the fluid motion can be described by a system of differential equations involving infinitely many variables. Lorenz made a tremendous simplification by keeping only three variables !

Very roughly speaking, $x$ represents the rate of convective 'overturning', whereas $y$ and $z$ can be thought as the horizontal and vertical temperature, respectively. Notice that $x,y,z$ are thus not representing the position of a point in the ambient space, but instead an abstract three-dimensional phase space.

Regarding the three parameters, $\sigma$ is the Prandtl number (related to the fluid viscosity), $r$ is the Rayleigh number (related to the temperature difference between the top and bottom of the cell, and $b$ is a scaling factor (related to the aspect ratio of the rolls).

In the following digital experiment, you can move the orange bullet which represent the initial condition and then press the 'start' button. You can observe that solutions that start out very differently seem to have the same fate, if we forget the 'transient behavior'. They both eventually wind around the symmetric pair of fixed points, alternating at times which point they encircle. This forms a complicated set, the so-called Lorenz attractor, on which solutions stay for ever.

The previous experiment can be misleading because it can leave the impression that if you start with two very close initial conditions, the resulting solutions travel very close to each other, before they get on the attractor but also once they are on it. The following experiment shows that this is false ! This time you can see how to initially close points evolve on the attractor.

You can observe that the two solutions move quite far apart during their journey around the attractor. Moreover, you can see that the trajectories are nearly identical for a certain time period, but then they differ significantly as one solution winds around one of the symmetric fixed points, while the other solution winds around the other one. No matter how close two solutions start, they always move apart in this manner when they are on the attractor. This is sensitive dependence on initial conditions, one of the main features of a chaotic system.

Moreover, we observe that solutions pass from one 'lobe' of the attractor to the other in an apparently unpredictable manner, leading to an irregular oscillation that never repeats : we have an aperiodic motion. This is called deterministic chaos because the equations are deterministic but the solutions can behave in a seemingly random way. Recall that 'deterministic' means, given the present state, the future (and the past) are completely determined. Mathematically, this means that, given an initial condition $(x_0,y_0,z_0)$, there is a unique solution $(x(t),y(t),z(t))$ passing through $(x_0,y_0,z_0)$ at time $t=0$. But, to predict the future evolution, we need to know exactly the initial condition, which is impossible in practice. In a chaotic system, this leads to unpredictability. To illustrate this, consider a tiny blob if initial conditions around $(x_0,y_0,z_0)$. We observe that, rapidly, this blob smears out over the entire attractor ! This means that a tiny error on the initial condition is amplified quickly in such a way that we only know that we are on the attractor, but we don't know precisely where.

Voir en ligne : Attracteur de Lorenz en 3D

Bifurcation diagram of the logistic map

Chazottes Jean-René , Monticelli Marc — 2022-02-18T09:32:39Z

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:

Turing patterns

Chazottes Jean-René , Monticelli Marc — 2022-02-10T16:43:25Z

Alan Turing was the first to propose a model to account for the very large diversity of patterns in nature, such as animal coats. This model is based on a “reaction-difusion equation” of the form(*)

$\begin{cases} \frac{\partial u}{\partial t}=f(u,v)+A \nabla^2 u\\\frac{\partial v}{\partial t}=g(u,v)+B \nabla^2 v\end{cases}$

where $u(x,y,t)$ is the concentration at point $(x,y)$ and at time $t$ of the activator (which color the skin), and $v(x,y,t)$ is that of the inhibitor (which prevents the activator from being expressed). The positive coefficients $A,B$ are the diffusion coefficients.

In the digital experiment below, we have a portion of the plan $(x,y)$ and we have taken $f(u,v)=u(v-1)-12$, $g(u,v)=16-uv$. What is represented is the minimum of $u(x,y,t)$, in black, and its maximum, in red.

Yu Wang Attractor

Monticelli Marc — 2019-02-12T10:38:50Z

Chen Lee Attractor

Monticelli Marc — 2019-02-12T10:38:14Z

Aizawa Attractor

2019-02-12T10:36:42Z

Cycle limite non convexe

Chazottes Jean-René , Monticelli Marc — 2016-03-15T11:53:00Z