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NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 51628, 2181]*) (*NotebookOutlinePosition[ 75764, 3057]*) (* CellTagsIndexPosition[ 75720, 3053]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The Logistic Map -- Supplement", "Title"], Cell["R.G. Palmer, 9/14/99", "Subsubtitle"], Cell[CellGroupData[{ Cell["f(y) -- f(f(y)) -- f(f(f(y))) -- ...", "Section"], Cell["\<\ Another neat trick is to make similar plots of the function \ iterated 2 (or more times); if f(y) = a y (1-y), then plot f(f(y)) or \ f(f(f(y))), etc.\ \>", "Text"], Cell["Don't even try to understand it ...", "Text"], Cell[BoxData[ \(logplot[i_, y0_, a_, n_]\ := \ Module[{f, y}, \[IndentingNewLine]f[y_] := Nest[\((a\ # \((1 - #)\))\)&, y, i]; Plot[f[y], \ {y, 0, 1}, \ Epilog -> {Line[{{0, 0}, {1, 1}}], Apply[\(({Line[{{#1, #2}, {#2, #2}}], Line[{{#2, #2}, {#2, #3}}]})\)&, \ Partition[NestList[f, y0, n], 3, 1], {1}]}]]\)], "Input"], Cell[TextData[{ "The ", StyleBox["logplot[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " function takes ", StyleBox["four", FontSlant->"Italic"], " arguments: the nested i value, the initial y value, the a value, and the \ number of steps (n).\n1. ", StyleBox["logplot[1,y,a,n]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " with f(y) (and ", StyleBox["logplot[y,a,n])", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ".\n2. ", StyleBox["logplot[2,y,a,n]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " with f(f(y)).\n3. ", StyleBox["logplot[3,y,a,n]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " with f(f(f(y))).\n4. ..." }], "Text"], Cell[TextData[{ "Here's a 2-cycle, at a=3.2 with f(y) (", StyleBox["without", FontSlant->"Italic"], " the transient spiral):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a\ = \ 3.2; \ny\ = \ 0.04; \nDo[y\ = \ a\ y\ \((1 - y)\), {100}]; \n logplot[1, y, a, 25]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.735756 [ [.21429 .00222 -9 -9 ] [.21429 .00222 9 0 ] [.40476 .00222 -9 -9 ] [.40476 .00222 9 0 ] [.59524 .00222 -9 -9 ] [.59524 .00222 9 0 ] [.78571 .00222 -9 -9 ] [.78571 .00222 9 0 ] [.97619 .00222 -3 -9 ] 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CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, 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If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. 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