Auxiliary calculations for the analysis of discrete iterated nonlinear systems Numerical input from the progam "Chaos for Java" is used.

The Feigenbaum bifurcation cascade for the logistic map: r:=[3, 3.44948936, 3.54409013, 3.56440714, 3.56875935, 3.56969160, 3.56989125, 3.56993401]; nmax:= nops(r); for n from 2 to nmax-1 do delta := (r[n]-r[n-1])/(r[n+1]-r[n]); rinf := (r[n+1]*r[n-1]-r[n]*r[n])/(r[n+1]-2*r[n]+r[n-1]); end do; The Feigenbaum bifurcation cascade for the sine map: q:=[0.71996083, 0.83326622, 0.85860898, 0.86408412, 0.86525892, 0.86551065, 0.86556458, 0.86557613]; nmax:= nops(q); for n from 2 to nmax-1 do delta := (q[n]-q[n-1])/(q[n+1]-q[n]); qinf := (q[n+1]*q[n-1]-q[n]*q[n])/(q[n+1]-2*q[n]+q[n-1]); end do; JSFH

Estimate of the capacity dimension of the attractor of the logistic map for various parameter values r. Orbits of length N=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Jy1GLzYkUSI2RidGMi8lJ2l0YWxpY0dRJXRydWVGJy8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjwvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yywere used, discarding the first LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Jy1GLzYkUSI0RidGMi8lJ2l0YWxpY0dRJXRydWVGJy8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjwvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy points. 1. r =3.57 restart: with(plots): N:=[28, 44, 68, 112, 197, 366, 688, 1346, 2666, 5296, 10561, 21060, 41684, 79979, 144971]: nmax:= nops(N); for n from 1 to nmax-1 do evalf(ln(N[n+1]/N[n])/ln(2),4); od; logplot([seq(n,n=1..nmax)],[seq(N[n],n=1..nmax)],style=point,gridlines=true,labels=["k","Nk"]); In view of the limited accuracy of the calculation the result is compatible with the result 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 2. r =3.6 restart: with(plots): N:=[390, 778, 1554, 3105, 6208, 12415, 24828, 49625, 98171, 185137, 317263, 478502]: nmax:= nops(N); for n from 1 to nmax-1 do evalf(ln(N[n+1]/N[n])/ln(2),4); od; logplot([seq(n,n=1..nmax)],[seq(N[n],n=1..nmax)],style=point, gridlines=true,labels=["k","Nk"]); 3. r =3.569945672 This is the accumulation point of the period doubling cascade (onset of chaos). Note that it is important to keep al the digits in this number. Even a slight truncation will affect the result. restart: with(plots): N:=[24, 43, 61, 88, 123, 185, 278, 389, 572, 809, 1181, 1750, 2540, 3799, 4343]: nmax:= nops(N); for n from 1 to nmax-1 do evalf(ln(N[n+1]/N[n])/ln(2),4); od; logplot([seq(n,n=1..nmax)],[seq(N[n],n=1..nmax)],style=point, gridlines=true,labels=["k","Nk"]); Result taken from the 'straight part' of N(k) between k=2 and k=13: evalf(ln(N[13]/N[2])/(11*ln(2)),4); This result agrees well with the value found in the literature: 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