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Etonnament l'itération de nombreuses fonctions simples répétées de nombreuses fois en prenant pour valeur le résultat de l'itération précédente : x ( t+1) = f (x (t)) aboutit à des courbes de bifurcation similaires
avec des constantes de bifurcation identiques ( Constantes de Feigenbaum )
Ces transitions appparaissent aussi dans de nombreux processus physiques, notemment ceux décrit par des équations différentielles non linéires. L' étude mathématique et physique de ces phénomènes se nomme la Théorie du Chaos.
Ci- dessous :
Diagramme de bifurcation obtenue après itérations de la fonction logistique
f ( n ) = a × n × ( 1 - n ).
"We applied the well known formula f ( n ) = a × n × ( 1 - n ).
The formula was run 1500 times for each value of a from about 1.5 to 4.2 with n starting at 0.0001.
For those coming by about the Feigenbaum's number, there are 5 x 60 digits after the decimal point:
Une courbe similaire avec des constantes identiques peut être obtenue avec par exemple la fonction x(t+1) = r sin x(t)
Les exposants de Lyapounov
L' autosimilarité des structures, attracteurs et attracteurs étranges