click to view the MPEG movie (cliquez pour voir le film MPEG)

Bidimensional display of the rounding-off errors when computing the Verhulst dynamics [Visualisation bidimensionnelle des erreurs d'arrondi dans le calcul de la dynamique de Verhulst].

This picture displays the two following Verhulst dynamics (for 1000 iterations):
                    X  = [[R+1]X   ] - [R[X   X   ]]
                     n          n-1        n-1 n-1

Y = [[R+1]Y ] - [[RY ]Y ] n n-1 n-1 n-1
computed with the same initial conditions:
                    X  = 0.5

Y = 0.5 0
They are strictly equivalent mathematically speaking and thus the point set {Xn,Yn} should be on a straight line with equation:
                    Y = X
in the bidimensional space. Unfortunately, when the R parameter increases (from 2.0 -picture 01- to 3.0 -picture 16-) the rounding-off errors become huge and give birth to two different numerical Verhulst dynamics. Then, the {Xn,Yn} points appear "anywhere" inside a square.

Each point {Xn,Yn} is displayed as a small sphere. Its color is chosen according to a color scale of increasing luminance: {Blue,Red,Magenta,Green,Cyan,Yellow,White} when 'n' vary from 1 to 1000. Points with a small 'n' value (most of the blue ones) are always on the straight line with equation:
                    Y = X
                    Y  ~ X
                     n    n
during the first iterations:

This double precision (64 bits) computation was made on a SGI O200 under IRIX 6.5 and C compiler release 7.2.1 [see the same computation on a PC under LINUX 2.2 and C compiler release 2.95.2].

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(CMAP28 WWW site: this page was created on 12/18/2000 and last updated on 04/02/2019 08:41:23 -CEST-)

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