Quadratic Map

Author

Eric W. Weisstein
August 11, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/QuadraticMap.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/QuadraticMap.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

General Equation

RSolve[x[n+1]==a2 x[n]^2+a1 x[n]+a0,x[n],n]

Mandelbrot

RSolve[x[n+1]==x[n]^2+c,x[n],n]

x[n]==x[n-1]^2

RSolve[{x[n]==x[n-1]^2,x[0]==2},x[n],n]

Table[(2^(2^n)),{n,0,5}]

y[n]==y[n-1]^2+1

RSolve[{x[n]==x[n-1]^2+1,x[0]==1},x[n],n]

y[0]:=1
y[n_]:=y[n]=y[n-1]^2+1

Table[y[n],{n,8}]

N[Exp[Sum[(2^(-j-1))Log[1+y[j]^(-2)],{j,0,#}]],110]&/@{5,10,15,20}

c=Exp[Sum[(2^(-j-1))Log[1+y[j]^(-2)],{j,0,22}]];

Block[{$MaxExtraPrecision=100},
Table[Floor[c^(2^n)],{n,8}]
]

z[n]==z[n-1]^2-z[n-1]+1

RSolve[{z[n]==z[n-1]^2-z[n-1]+1,z[1]==2},z[n],n]

z[1]:=2
z[n_]:=z[n]=z[n-1]^2-z[n-1]+1

Table[z[n],{n,8}]

N[Exp[Sum[(2^(-j-1))Log[1+(2z[j]-1)^(-2)],{j,1,#}]],110]&/@{5,10,15,20}

d=Sqrt[6]/2 Exp[Sum[(2^(-j-1))Log[1+(2z[j]-1)^(-2)],{j,1,10}]];

Block[{$MaxExtraPrecision=50},
Table[Floor[d^(2^n)+1/2],{n,8}]
]

a0=(((a1-4) (a1+2))/(4 a2))

With[{a0=(a1-4)(a1+2)/(4a2)},RSolve[x[n+1]==a2 x[n]^2+a1 x[n]+a0,x[n],n]
]

With[{a0=(a1-4)(a1+2)/(4a2)},RSolve[{x[n+1]==a2 x[n]^2+a1 x[n]+a0,x[0]==x0},x[n],n]
]

soln=FullSimplify[With[{a0=(a1-4)(a1+2)/(4a2),r=a1/4+x0 a2/2+Sqrt[(a1+4+2x0 a2)(a1-4+2x0 a2)]/4},a2 x[n]^2+a1 x[n]+a0/.x:>Function[n,(r^(2^n)+r^(-2^n)-a1/2)/a2]
],n∈Integers&&n>0]

Expand[FullSimplify[Table[soln,{n,0,1}]]]//FullSimplify

y[0]:=y0
y[n_]:=With[{a0=(a1-4)(a1+2)/(4a2)},
a2 y[n-1]^2+a1 y[n-1]+a0
]

Expand[Table[y[n],{n,0,2}]]//FullSimplify

check=With[{a0=(a1-4)(a1+2)/(4a2)},x[n+1]-(a2 x[n]^2+a1 x[n]+a0)/.x->Function[n,Evaluate[soln]]
]//Simplify

a0=((a1 (a1-2))/(4 a2))

With[{a0=a1(a1-2)/(4a2)},RSolve[x[n+1]==a2 x[n]^2+a1 x[n]+a0,x[n],n]]

With[{a0=a1(a1-2)/(4a2)},RSolve[{x[n+1]==a2 x[n]^2+a1 x[n]+a0,x[0]==x0},x[n],n]]

Together[Simplify[With[{a0=a1(a1-2)/(4a2),r=a1/4+x0 a2/2},a2 x[n]^2+a1 x[n]+a0/.x:>Function[n,((2r)^(2^n)-a1/2)/a2]
],n∈Integers&&n>0]]

With[{a0=a1(a1-2)/(4a2),r=a1/4+x0 a2/2},
x[n+1]-(a2 x[n]^2+a1 x[n]+a0)/.x:>Function[n,-(((2^(-1-(2^n))) ((2^(2^n)) a1-2 (((a1+2 a2 x0))^(2^n))))/a2)]
]//Simplify

Period-2 Fixed Points

Nest[#/.x[n_]:>((x[n-1])^2)+c&,x[n+2],2]//Expand

Factor[%-x[n]]