Logistic Map—r== 4

Author

Eric W. Weisstein
February 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DynamicalSystems/LogisticMapR4.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LogisticMapR4.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Solution

RSolve[{x[n+1]==4x[n](1-x[n]),x[0]==x0},x[n],n]

Analytic Derivation

Simplify@(NestList[4#(1-#)&,(Subscript[x, 0]),5]-Table[1/2(1-Cos[2^n ArcCos[1-2(Subscript[x, 0])]]),{n,0,5}])

makeSeries[4, x, 10]

Series[HypergeometricPFQ[{},{1/2}, x/2],{x,0,10}]

Series[Cosh[Sqrt[2x]],{x,0,4}]

which is EIS A000680, (2n)!/(2^n)

CheckPFQ[4 ,{0,1},z]

Solve[Cosh[Sqrt[2z]]==a,z]

1/2(1-f[4^k g[1-2x0]])/.f[z_]:>Cosh[Sqrt[2z]]/.g[z_]:>ArcCosh[z]^2/2

PowerExpand[%]

PowerExpand[FullSimplify[%]]

Chop[Table[(1/2) (1-Cosh[(2^k) ArcCosh[1-2 x0]]),{k,3}] /. x0->2.1]

Chop[Table[(1/2) (1-Cosh[(Sqrt[(2^(2 k)) ((ArcCosh[1-2 x0])^2)])]),{k,3}] /. x0->2.1]

Chop[Table[(1/2) (1-Cos[2^k ArcCos[1-2x0]]),{k,3}] /. x0->2.1]

Chop[Table[-((Sinh[(2^(-1+k)) ArcCosh[1-2 x0]])^2),{k,3}] /. x0->2.1]

Plot

LogisticPlot[a_,x0_,nn_:50,opts___]:=Module[{l=NestList[a#(1-#)&,x0,nn]},
Show[{
ListPlot[l,opts,Joined->True],
ListPlot[l,opts,PlotStyle->{PointSize[.03],Red}]
},AxesOrigin->{1,Automatic},
AxesLabel->{n,(Subscript[x, n])},PlotLabel->r==a]
]

GraphicsGrid[{{LogisticPlot[4,.42],LogisticPlot[4,.71]}},ImageSize->500,Spacings->-30]