Experimentarium Digitale

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:

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 :

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.

For the pleasure of the eyes

Chazottes Jean-René , Monticelli Marc — 2015-09-18T13:35:24Z

All the differential equations we present on this website are physical or biological models. Here you can go through a gallery of phase portraits of two-dimensional systems we find beautiful, without getting preoccupied with what they could potentially model.

Most of the above differential equations are borrowed from the book Systèmes différentiels : étude graphique, by M. Artigue and V. Gautheron (Paris : CEDIC : Fernand Nathan, 1983), which is out-of-print.

Fonction de Liapounov pour le pendule simple

Chazottes Jean-René , Monticelli Marc — 2015-09-18T13:31:21Z

- Systèmes dynamiques / Systèmes dynamiques , javascript , WebGL

Chaos hamiltonien : l'application standard

Chazottes Jean-René , Monticelli Marc — 2015-07-10T16:00:16Z

Il s'agit d'un modèle remarquable qui a permis de mieux comprendre le chaos hamiltonien. Il est défini par une application qui envoie le tore $\mathbb{T}^2$ dans lui-même. On peut représenter ce tore comme un carré de côté $2\pi$ dont on identifie les côtés opposés. Étant donné un point initial $(\theta_0,I_0)$, on calcule son orbite, c-à-d les points $(\theta_1,I_1), (\theta_2,I_2)$,... par récurrence en utilisant l'application

$$ \begin{cases} \theta_{n+1} =\theta_n + I_{n+1}~;(\text{mod}\, 2\pi)\\ I_{n+1} =I_n + K \sin\theta_n~;(\text{mod}\, 2\pi) \end{cases} $$

où $K\geq 0$ est un paramètre.

Dans l'expérience numérique interactive ci-dessous, on peut observer le portrait de phase selon les valeurs de $K$, et y faire des zooms en cliquant directement dans le portrait de phase. (On visualise les orbites d'un milliers de points initiaux.)

On remarque qu'on peut calculer exactement $\theta_n$ et $I_n$ pour tout $n$ dans le cas où $K=0$ (on dit que le système est intégrable). En effet, on obtient facilement que $I_n=I_0$ et $\theta_n=\theta_0+ n I_0$. Donc, l'orbite d'un point $(\theta_0,I_0)$ est confinée dans un cercle qui correspond à un segment horizontal dans notre représentation. Deux cas sont possibles :

soit $I_0=p/q$ où $p,q$ sont des entiers ($q\neq 0$), ce qui donne une orbite périodique de période $q$ : après $q$ itérations, on revient au point de départ. Il y a en particulier des points fixes ;
soit $I_0$ n'est pas un nombre rationnel, auquel cas l'orbite est quasi-périodique : elle finit par passer aussi près que l'on veut de tout point du cercle où elle est confinée (on dit qu'elle est dense dans le cercle en question).
Le portrait de phase ressemble à un « mille-feuille » qui se déforme dès qu'on commence à faire varier $K$ depuis la valeur $0$. Il apparaît très vite des ellipses concentriques autour d'un point fixe ($K=0,01$). Au fur et à mesure qu'on augmente $K$, le portrait de phase devient de plus en plus complexe avec des régions où apparaissent d'autres groupes d'ellipses concentriques, et d'autres qui ressemblent à une « poussière » uniforme de points. Mais un zoom dans l'une de ces régions en apparence uniforme révèle en fait un mélange de groupes d'ellipses et de poussière. Et ainsi de suite !

L'application standard est aussi appelée application de Chirikov ou application de Taylor-Chirikov. On la qualifie de « standard » car elle décrit le comportement générique d'une application à deux dimension préservant les aires et dont le portrait de phase est un mélange de dynamique elliptique (les courbes fermées concentriques sur chacune desquelles on a une rotation déformée) et chaotique (zones « poussiéreuses »). On obtient cette application en approximant ce qu'il se passe dans une section de Poincaré de nombreux systèmes hamiltoniens.

On peut consulter cette page pour plus de détails.

Bifurcations à un paramètre dans des systèmes 2D

Chazottes Jean-René , Monticelli Marc — 2015-06-23T14:52:01Z

Nous illustrons les bifurcations de base pour des systèmes dynamiques de la forme

$$ \begin{cases} \dot{x}=f_\mu(x,y)\\ \dot{y}=g_\mu(x,y) \end{cases} $$

où $\mu$ est un paramètre réel. Dans chacune des expériences numériques interactives qui suit, on visualise les trajectoires de 400 conditions initiales et on peut observer comment le portrait de phase est modifié qualitativement lorsque $\mu$ varie.

Bifurcation nœud-col. Le prototype de cette bifurcation est donné par le système

$$ \begin{cases} \dot{x}= \mu-x^2\\ \dot{y}=-y \end{cases} $$

Lorsque $\mu>0$, on a deux points fixes : un nœud attractif situé au point $(\sqrt{\mu},0)$ et un col au point $(-\sqrt{\mu},0)$. En faisant décroître $\mu$, on constate que ces deux points de rapprochent, se confondent pour $\mu=0$, et disparaissent dès que $\mu<0$ : les points fixes se sont « annihilés ».

Bifurcation transcritique.

$$ \begin{cases} \dot{x}= \mu x-x^2\\ \dot{y}=-y \end{cases} $$

Bifurcation fourche sur-critique.

$$ \begin{cases} \dot{x}= \mu x-x^3\\ \dot{y}=-y \end{cases} $$

Bifurcation fourche sous-critique.

$$ \begin{cases} \dot{x}= \mu x+x^3\\ \dot{y}=-y \end{cases} $$

Bifurcation de Hopf sur-critique.

$$ \begin{cases} \dot{x}= \mu x-y -x(x^2+y^2)\\ \dot{y}=x+\mu y - y(x^2+y^2) \end{cases} $$

Bifurcation de Hopf sous-critique.

$$ \begin{cases} \dot{x}= \mu x-y + x(x^2+y^2)\\ \dot{y}=x+\mu y +y(x^2+y^2) \end{cases} $$

Modèle de compétition cyclique entre trois populations

Chazottes Jean-René , Monticelli Marc — 2015-06-19T13:13:05Z

May et Leonard ont étudié un modèle mettant en jeu trois populations qui sont en compétition de telle sorte que, cycliquement, l'une des trois populations semble dominer pendant une longue période de temps les deux autres avant que cela soit soudainement au tour de l'une des deux autres, et ainsi de suite. Qui plus est, la période durant laquelle l'une des populations semble dominer devient de plus en plus à chaque cycle. Voici les équations gouvernant l'évolution de la densité des trois populations :

$$ \begin{cases} \dot{x}_1 = x_1(1-x_1-\alpha x_2 -\beta x_3)\\ \dot{x}_2 = x_2(1-\beta x_1- x_2 -\alpha x_3)\\ \dot{x}_3 = x_3(1-\alpha x_1-\beta x_2 - x_3) \end{cases} $$

où $0<\beta <1$ et $\alpha+\beta\geq 2$. Lorsque $\alpha+\beta=2$, on peut observer un cycle limite qui va être détruit dès qu'on dépasse la valeur $2$.
Le phénomène remarquable est que les solutions convergent rapidement vers une surface triangulaire légèrement incurvée et qu'on appelle le « simplexe de charge ». Les sommets de ce « triangle » sont en fait les trois points fixes non triviaux, à savoir $\bar{x}_1=1,\bar{x}_2=0,\bar{x}_3=0$, $\bar{x}_1=0,\bar{x}_2=1,\bar{x}_3=0$ et $\bar{x}_1=0,\bar{x}_2=0,\bar{x}_3=1$. Le premier correspond à la domination complète de la population no. 1, le second à celle de la population no. 2, et le troisième à celle de la population no. 3.

Le « triangle » qu'on observe est un exemple de « cycle hétérocline ». C'est un phénomène générique pour des équations différentielles qui possèdent certaines symétries. Ici, nous avons une symétrie par permutation circulaire.