Download An Exploration of Chaos: An Introduction for Natural by G. Faust, M. Haase, J. H. Argyris PDF

By G. Faust, M. Haase, J. H. Argyris

This quantity is meant as a close creation to the speculation of chaos and is addressed to physicists and engineers who desire to be accustomed to this new and interesting technological know-how linked to non-linear deterministic structures. arithmetic are a pre-requisite software.

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Extra resources for An Exploration of Chaos: An Introduction for Natural Scientists and Engineers

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3 Calcul des réduites. Le calcul des réduites d'une fraction continue illimitée est analogue en calcul des réduites d'une fraction continue finie. Pn O n a en = , avec qn pour tout entier n 2: 1 On posera comme dans le cas d'une fraction continue finie P-1 =0 Po =1 Exemple: Calculer la dixième réduite de y'I3, avec Vi3 = [3,1,1,1,1,6] i ai Pi qi -1 0 1 0 +-1 +-0 1 3 3 1 3 Ci ï 2 1 4 1 4 ï 3 1 7 2 7 "2 4 1 11 3 11 '3 5 1 18 5 18 5" 6 6 119 33 119 33 7 1 137 38 137 38 8 1 256 71 9 1 393 109 1 649 180 256 393 109 649 180 TI 10 La dixième réduite est PlO 649 ClO=-=.

6. Propriétés d'approximation des réduites. 5 Toute réduite est une meilleure approximation de x que la réduite précédente. 6 Pour tout irrationnel positif x, on peut trouver une infinité de rationnels de la forme ~ avec (P ,....... q) = 1 qui approchent x à moins de ~ près. Exemple 1: La fraction l~~il étant donnée, trouver une fraction de numérateur et dénominateur inféreurs et donnant une valeur approchée de 15625 à 10-3 près 1024 . Tout d'abord transformation de \5~il en fraction continue Ona 15625 1024 = [15,3,1,6,2,1,3,2,1] et calculons les réduites correspondantes.

Il s'agit de résoudre l'équation x 2 _Ny2 = ±1, avec (x,y) E N 2 et N n'étant pas un carré parfait. 5. Equation de Peil x 2_Ny2 = ±l. Recherche d'une solution particulière. 59 dernière qu'il y ait une solution. 1 1 1 •.. == où On sait que VN = Œn+IPn Œn+lqn + Pn-l + qn-l où les Pn-I,Pn, qn-Iqn sont les numérateurs et dénominateurs des réduités Cn-l = Pn-l et Cn = fu qui précèdent immédiatement le terme 2al, ce qui qn-l qn donne VN = (JN + al) Pn + Pn-l (JN + al) qn + qn-l que l'on peut écrire sous la forme VN (VN +al) qn + qn-IVN = (VN + al)Pn +Pn-l qui est équivalente à : Les deux membres sont égaux si et seulement si { Nqn alqn = alPn + Pn-l + qn-l = Pn ce qui donne { Pn-l :: Nqn - alPn qn-l - Pn - alqn Cependant Pnqn-l - qnPn-1 = (-l)n ce qui donne en remplaçant Pn-l et qn-l par les valeurs précédentes en réduisant Chapitre 4.

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