Die quadratische Abbildung

Materialien zur Vorlesung Elementare Analysis, Wintersemester 2003 / 4

Für reelle x sei f(x, a)  :=  a x (1 - x) =  
          a x  - a (x^2) .

$DefaultFont = {"Times", 12};

quadrat[x_][a_] := a x (1 - x)

quadratIteration[ start_, anzahl_, a_ ] := 
    Module[ { punkte }, 
    punkte = NestList[ quadrat[#][a]&, start, anzahl ];
    punkte = Flatten[ Transpose[ {punkte, punkte} ]];
    punkte = Partition[ punkte, 2, 1];
    Graphics[{AbsoluteThickness[1.5], Line[ punkte ] }]]

quadratPlot[start_, anzahl_, a_] := 
      Module[{gr}, 
        gr = Plot[quadrat[x][a], {x, 0, 1}, 
            DisplayFunction -> 
              Identity]; 
Show[{Graphics[{RGBColor[1, 0, 
                  0], AbsoluteThickness[1.5], Line[{{0, 0}, {1, 1}}]}], gr, 
            quadratIteration[start, anzahl, a]}, Axes -> True, 
          PlotRange -> {{0, 1}, {0, 1}}, 
          AspectRatio -> 
            1,   
            DisplayFunction -> $DisplayFunction]]

qp[start_, anzahl_, a_] := 
      Module[{gr}, 
        gr = Plot[quadrat[x][a], {x, 0, 1}, 
            DisplayFunction -> 
              Identity]; 
Show[{Graphics[{RGBColor[1, 0, 
                  0], AbsoluteThickness[1.5], Line[{{0, 0}, {1, 1}}]}], gr, 
            quadratIteration[start, anzahl, a]}, Axes -> True, 
          PlotRange -> {{0, 1}, {0, 1}}, AspectRatio -> 1, 
          PlotLabel -> {start, anzahl, a}, 
          DisplayFunction -> $DisplayFunction]]

Für reelle x sei f(x, a)  :=  a x (1 - x) =  
          a x  - a (x^2) .

Es sei nun a = 2 und (Subscript[x, 0]) = 0.1. 
Mit f(x, 
          2)  :=  2 x (1 - x) =  2 x  - 
              2 (x^2) = :h(x) 
gilt dann 

(Subscript[x, 1]) := h((Subscript[x, 0])) , 
(Subscript[x, 2]) := h((Subscript[x, 1])) = h(h((Subscript[x, 0]))) , 
(Subscript[x, 3]) := h((Subscript[x, 2])) = h(h((Subscript[x, 1]))) = 
            h(h(h((Subscript[x, 0])))) 

und allgemein für  n ∈ ℕ 
				(Subscript[x, n + 1]) := h((Subscript[x, n])) .

h[x_] := 2  x - 2  (x^2);
NestList[h, 0.1, 10]

quadratPlot[0.1, 5, 2. ];

gr2 = qp[0.1, 10, 2.75];

gr5 = qp[0.1, 100, 3.3955];

gr4 = qp[0.1, 50, 3.839];

gr5 = qp[0.1, 30, 3.3955];

gr6 = qp[0.55, 50, 4. ];

qpt[start_, anzahl_, a_] := 
      Module[{gr}, 
        gr = Plot[quadrat[x][a], {x, 0, 1}, Ticks -> None, 
            DisplayFunction -> 
              Identity]; 
Show[{Graphics[{RGBColor[1, 0, 
                  0], AbsoluteThickness[1.5], Line[{{0, 0}, {1, 1}}]}], gr, 
            quadratIteration[start, anzahl, a]}, Axes -> True, 
          PlotRange -> {{0, 1}, {0, 1}}, AspectRatio -> 1, 
          PlotLabel -> {anzahl}, 
Ticks -> None, 
          DisplayFunction -> $DisplayFunction]]

Table[qpt[0.1, kk, 3.3955], {kk, 5, 30}]

Table[qpt[0.55, kk, 4. ], {kk, 5, 30}]