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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]
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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]
. X = [[R+1]*X ] - [R*[X *X ]]
n n-1 n-1 n-1
Y = [[R+1]*Y ] - [[R*Y ]*Y ]
n n-1 n-1 n-1
computed with the same initial conditions:
X = 0.5
0
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 = Xin 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.
:
{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 = Xfor:
Y ~ X
n n
during the first iterations: