Gallery :

Deterministic Chaos

[Galerie : Chaos Déterministe]

• The Verhulst Dynamics [la Dynamique de Verhulst] :
• Bidimensional Verhulst Dynamics [Dynamique de Verhulst bidimensionnelle] :
• Tridimensional Verhulst Dynamics [Dynamique de Verhulst tridimensionnelle] :
• The Lorenz Attractor [l'Attracteur de Lorenz] :
• Pendulum Systems [Systèmes de Pendules] :
• The N-Body Problem [le Problème des N-Corps] :
• Iterated Function Systems (IFS) [Systèmes de Fonctions Itérées (IFS)] :
• Miscellaneous [Divers] :

Deterministic Chaos [Chaos Déterministe] :

The Verhulst Dynamics [la Dynamique de Verhulst] :

Bidimensional Verhulst Dynamics [Dynamique de Verhulst bidimensionnelle] :

Bidimensional visualization of the Verhulst dynamics.

Bidimensional visualization of the Verhulst dynamics -(grey,orange,red) display negative Lyapunov exponents, (yellow,green,blue) display positive Lyapunov exponents-.

Bidimensional zoom in on the Verhulst dynamics.

Tridimensional visualization of the Verhulst dynamics.

Bidimensional display of the rounding-off errors when computing the Verhulst dynamics.

Tridimensional display of the rounding-off errors when computing the Verhulst dynamics.




Tridimensional Verhulst Dynamics [Dynamique de Verhulst tridimensionnelle] :

Tridimensional visualization of the Verhulst dynamics.	Tridimensional visualization of the Verhulst dynamics.	Tridimensional visualization of the Verhulst dynamics.	Tridimensional visualization of the Verhulst dynamics.	Tridimensional visualization of the Verhulst dynamics -'Time Ships', a tribute to Stephen Baxter-.
Tridimensional visualization of the Verhulst dynamics.




The Lorenz Attractor [l'Attracteur de Lorenz] :

The Lorenz attractor.

The Lorenz attractor.

The Lorenz attractor.

Rotation about the X axis of the Lorenz attractor.

Rotation about the X axis of the Lorenz attractor.

Rotation about the Y (vertical) axis of the Lorenz attractor that can also be viewed as a set of 4x3 stereograms.

Rotation about the Y (vertical) axis of the Lorenz attractor that can also be viewed as a set of 4x3 stereograms.

The Lorenz attractor -sensitivity to initial conditions (displayed as the central point of each frame)-.

The Lorenz attractor -sensitivity to integration methods used (Red=Euler, Green=Runge-Kutta/2nd order, Blue=Runge-Kutta/4th order)-.

Computation of the Lorenz attractor on three different computers (the Red one, the Green one and the Blue one : sensitivity to rounding-off errors).

Rotation about the X axis of the Lorenz attractor (1000 iterations), computed simultaneously on two different computers.

Rotation about the X axis of the Lorenz attractor (5000 iterations), computed simultaneously on two different computers.

Bidimensional display of the integration of the Lorenz attractor system for 448500 different initial conditions.

The evolution of the bidimensional field defining the three coordinates of a sphere using the Lorenz attractor.

The evolution of the bidimensional field defining the three coordinates of a Klein bottle using the Lorenz attractor.

The Lorenz attractor in motion.




Pendulum Systems [Systèmes de Pendules] :

[more information about Pendulum Systems -in french-]



Three pendulums -the Red one, the Green one and the Blue one- and three magnets -white-.

The three pendulums and three magnets problem computed on three different computers (the Red one, the Green one and the Blue one : sensitivity to rounding-off errors).




The N-Body Problem [le Probleme des N-Corps] :

[more information about the N-Body Problem -in french-]



N-body problem integration (N=2) displaying a perfect Keplerian orbit.

N-body problem integration (N=3) with one yellow star and two planets (the red one being very heavy and the blue one being light).

16 earth-like planets initially on the same keplerian trajectory.

N-body problem integration (N=4) with one yellow star and two planets (the red one being very heavy and the blue one and its white satellite being light).

N-body problem integration (N=4 : one star, one heavy planet and one light planet with a satellite) -sensitivity to initial conditions-.

N-body problem integration (N=4 : one star, one heavy planet and one light planet with a satellite) computed on three different computers (the Red one, the Green one and the Blue one : sensitivity to rounding-off errors).

N-body problem integration (N=4 : one star, one heavy planet and one light planet with a satellite) computed with 2 slightly different sets of initial conditions (sensitivity to initial conditions).

N-body problem integration (N=4 : one star, one heavy planet and one light planet with a satellite) computed with 2 different optimization options on the same computer (sensitivity to rounding-off errors).




Iterated Function Systems (IFS) [Systèmes de Fonctions Itérées (IFS)] :

A bidimensional fern computed by means of an 'Iterated Function System' -IFS-.

A bidimensional Sierpinski 'carpet' computed by means of an 'Iterated Function System' -IFS-.

A tridimensional Sierpinski 'carpet' computed by means of an 'Iterated Function System' -IFS-.




Miscellaneous [Divers] :

The Bertrand paradox.

The Bertrand paradox.

The extended Bertrand paradox.

Copyright (c) Jean-François Colonna, 2000-2013.
Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2000-2013.