Tridimensional visualizations of the Verhulst dynamics (Visualisations tridimensionnelles de la dynamique de Verhulst) {a chapter of 'Mathematics - A Virtual Instrument For Exploring Space Time And Beyond'}

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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Copyright © Jean-François Colonna, 2019-2026. Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2019-2026.

Copyright © Jean-François Colonna, 2019-2026.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2019-2026.