Tridimensional visualization of the Verhulst dynamics with a tridimensional non linear transformation of the coordinates [Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation non linéaire tridimensionnelle des coordonnées].

The Verhulst dynamics is defined using the following iteration:
                    X  = 0.5
                    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 ==> R3 ==> R2 ==> R2 ==> R2 ==> R1 ==> R1 ==> R1 ==> R1 ==> R2 ==> R2 ==> R3 ==> R3 ==> R2 ==> R1 ==> R1 ==> R1 ==> R1 ==> R2 ==> R2 ==> R3 ==> R3 ==> R2 ==> R2 ==> R1 ==> R1 ==> R1 ==> R1 ==> R1 ==> R1
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.

This process gives birth to the following tridimensionnal structure:

Then a tridimensional non linear transformation of the coordinates is applied. It is defined using the pseudo-projection of an interpolation between the Bonan-Jeener's triple Klein bottle and a "double sphere":

{tX=,tY=,tZ=} ==>

See some related pictures (possibly including this one):



See a bidimensionnal dynamics:

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