Tridimensional visualizations of the Verhulst dynamics [Visualisations tridimensionnelles 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 these 16 computations, the growing rate 'R' is no longer constant but changes its value periodically:



Period={R3,R3,R3,R3,R2,R2,R1,R1 
,R1,R1,R2,R2,R3,R2,R2,R1 
,R1,R1,R2,R2,R3,R2,R2,R1 
,R1,R1,R1,R1,R1}
Period={R3,R3,R3,R3,R2,R2,R2,R1 
,R1,R1,R1,R2,R3,R3,R2,R1 
,R1,R1,R1,R2,R3,R3,R2,R2 
,R1,R1,R1,R1,R1,R1}
Period={R3,R3,R3,R3,R2,R2,R2,R1 
,R1,R1,R1,R2,R2,R3,R2,R2 
,R1,R1,R1,R2,R2,R3,R3,R2 
,R2,R1,R1,R1,R1,R1,R1}
Period={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}

Period={R3,R3,R3,R2,R2,R2,R1,R1 
,R1,R2,R3,R3,R2,R1,R1,R1 
,R2,R3,R2,R2,R1,R1,R1,R1 
,R1}
Period={R3,R3,R3,R3,R2,R2,R1,R1 
,R1,R2,R2,R3,R2,R1,R1,R1 
,R2,R2,R3,R2,R2,R1,R1,R1 
,R1,R1}
Period={R3,R3,R3,R3,R2,R2,R1,R1 
,R1,R1,R2,R3,R2,R2,R1,R1 
,R1,R2,R3,R3,R2,R2,R1,R1 
,R1,R1,R1}
Period={R3,R3,R3,R3,R2,R2,R1,R1 
,R1,R1,R2,R3,R3,R2,R1,R1 
,R1,R2,R2,R3,R3,R2,R2,R1 
,R1,R1,R1,R1}

Period={R3,R3,R3,R2,R2,R1,R1,R1 
,R2,R3,R2,R1,R1,R2,R3,R2 
,R2,R1,R1,R1,R1}
Period={R3,R3,R3,R2,R2,R1,R1,R1 
,R2,R3,R2,R1,R1,R1,R2,R3 
,R2,R2,R1,R1,R1,R1}
Period={R3,R3,R3,R2,R2,R1,R1,R1 
,R2,R3,R3,R2,R1,R1,R2,R3 
,R3,R2,R1,R1,R1,R1,R1}
Period={R3,R3,R3,R2,R2,R1,R1,R1 
,R1,R2,R3,R2,R1,R1,R1,R2 
,R3,R3,R2,R1,R1,R1,R1,R1}

Period={R3,R3,R2,R2,R1,R1,R2,R3 
,R2,R1,R1,R3,R2,R2,R1,R1 
,R1}
Period={R3,R3,R3,R2,R1,R1,R1,R3 
,R2,R1,R1,R2,R3,R2,R1,R1 
,R1,R1}
Period={R3,R3,R3,R2,R1,R1,R1,R2 
,R3,R2,R1,R1,R2,R3,R2,R1 
,R1,R1,R1}
Period={R3,R3,R3,R2,R1,R1,R1,R2 
,R3,R2,R1,R1,R2,R3,R3,R2 
,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.


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