Logistic Map—r==-2

Author

Eric W. Weisstein
February 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DynamicalSystems/LogisticMapR-2.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LogisticMapR-2.html.

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

Solution

V4

RSolve[{x[n+1]==-2x[n](1-x[n]),x[0]==x0},x[n],n]

V7

RSolve[{x[n+1]==-2x[n](1-x[n]),x[0]==x0},x[n],n]

Analytic computations

FullSimplify[NestList[-2#(1-#)&,(Subscript[x, 0]),5]-Table[(1/2)-Cos[(1/3) (π-(((-2))^n) (π-3 ArcCos[(1/2)-(Subscript[x, 0])]))],{n,0,5}]]

makeSeries[-2,x,20]

d=Denominator[CoefficientList[%,x]]

1/Divide@@@Partition[d,2,1]

Product[6k,{k,n}]

FunctionExpand[Product[2k+1,{k,0,n}]]

n even

Table[(((2^(1+n)) Gamma[(3/2)+n])/(Sqrt[π]))(6^n) n!,{n,0,4}]

Sum[((x^(2n+1)(-1)^n)/((((2^(1+n)) Gamma[(3/2)+n])/(Sqrt[π]))(6^n) n!)),{n,0,4}]

Sum[((x^(2n+1)(-1)^n)/((((2^(1+n)) Gamma[(3/2)+n])/(Sqrt[π]))(6^n) n!)),{n,0,∞}]

n odd

Table[(((2^n) Gamma[(3/2)+n-1])/(Sqrt[π]))(6^n) n!,{n,0,4}]

Sum[((x^(2n)(-1)^n)/((((2^n) Gamma[(3/2)+n-1])/(Sqrt[π]))(6^n) n!)) ,{n,0,4}]

Sum[((x^(2n)(-1)^n)/((((2^n) Gamma[(3/2)+n-1])/(Sqrt[π]))(6^n) n!)) ,{n,0,∞}]

PowerExpand[Cos[((Sqrt[(x^2)])/(Sqrt[3]))]+(((Sqrt[3]) x Sin[((Sqrt[(x^2)])/(Sqrt[3]))])/(Sqrt[(x^2)]))]

Series

f=FullSimplify[2 Cos[x/Sqrt[3]+ArcTan[-Sqrt[3]]]]

FullSimplify[SeriesCoefficient[2 Cos[(1/3) (π-(Sqrt[3]) x)],{x,0,n}],n∈Integers&&n>0]

series1=Series[f,{x,0,19}]

series2=makeSeries[-2,x,20]

series1-series2

Solve[2 Cos[(1/3) (π-(Sqrt[3]) z)]==x,z]

FullSimplify[1/2(1-f[(-2)^k g[1-2x0]])/.f[x_]:>2 Cos[(1/3) (π-(Sqrt[3]) x)]/.g[x_]:>(1/3) ((Sqrt[3]) π-3 (Sqrt[3]) ArcCos[(x/2)])]

FullSimplify[NestList[-2#(1-#)&,(Subscript[x, 0]),5]-Table[(1/2)-Cos[(1/3) (π-(((-2))^n) (π-3 ArcCos[(1/2)-(Subscript[x, 0])]))],{n,0,5}]]

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[-2,.31],LogisticPlot[-2,.4]}},ImageSize->500,Spacings->-30]