Tridimensional visualization of the Verhulst dynamics [Visualisation tridimensionnelle de la dynamique de Verhulst ]

Tridimensional visualization of the Verhulst dynamics [Visualisation tridimensionnelle de la dynamique de Verhulst].




The Verhulst dynamics is defined using the following iteration: 
                    X  = 0.5
                     0

                    X  = RX   (1 - X   )
                     n     n-1      n-1
Here, in this computation, the growing rate 'R' is no longer constant but changes its value periodically using the following arbitrary cycle: 
R3 ==> R3 ==> R3 ==> R2 ==> R2 ==> R2 ==> R1 ==> R1 ==> R1 ==> R2 ==> R3 ==> R2 ==> R1 ==> R1 ==> R1 ==> R3 ==> R3 ==> R1 ==> R1 ==> R1 ==> R2 ==> R3 ==> R2 ==> R1 ==> R1 ==> R1 ==> R2 ==> R2 ==> R2 ==> R3 ==> R3 ==> R3
(based on the sine function on [0,8.pi]) where {R1,R2,R3} are respectively the three coordinates of the current point inside the following domain [2.936,3.413]x[3.500,3.850]x[3.000,4.000]. Only the points corresponding to a dynamical system with a negative Lyapunov exponent are displayed.


See a close-up:

Tridimensional visualization of the Verhulst dynamics


See a set of 4x3 stereograms:

A set of 4x3 stereograms of a tridimensional visualization of the Verhulst dynamics


See a bidimensionnal dynamics:

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


See a related picture:

An interpolation between a pseudo-quaternionic Mandelbrot set (a 'Mandelbulb')and a tridimensional Verhulst dynamics -tridimensional cross-section- LYAP.81.2.Fra


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