(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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[ 40288, 1045]*) (*NotebookOutlinePosition[ 80102, 2351]*) (* CellTagsIndexPosition[ 80058, 2347]*) (*WindowFrame->Normal*) Notebook[{ Cell["Dimensie theorie", "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Deze ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks illustreren aspecten van fractale verzamelingen en fractale \ dimensies.\nAan bod komen eerst twee fractale verzamelingen, de Koch \ sneeuwkromme en de Sierpinski zeef.\nDaarna wordt numeriek de box counting \ dimensie van de Henon attractor bepaald. Tenslotte worden nog twee\n\ voorbeelden van vreemde attractoren gegeven bij differentiaalvergelijkingen, \ de Rossler attractor\nen de Lorenz attractor." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Introductie", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Dit notebook is verdeeld in een aantal onderdelen die afzonderlijk \ zijn langs te lopen. Door met de linkermuistoets op een balkje rechts in het scherm te klikken, \ wordt een stuk notebook geselecteerd. Met shift-return (gelijktijdig de shift en de return toets) \ worden de commandos in het geselecteerde gebied uitgevoerd. Zogenaamde initialisatiecellen, waarin \ functies worden gedefinieerd, zijn grijs van kleur. Cellen waarin vervolgens iets uitgerekend of geplot \ wordt, zijn oranje gekleurd.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["De Koch sneeuwvlok", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Drie fractale Koch krommen kunnen aaneengesmeed worden tot \ sneeuwvlokken. Eerst worden de nodige functies gedefinieerd.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Clear[n, Snowflake, start, finish, doline]\), "\[IndentingNewLine]", \(Snowflake[n_Integer?NonNegative] := Show[Graphics[ Nest[\((#1 /. Line[{start_, finish_}] \[RuleDelayed] doline[start, finish])\) &, {Line[{{0, 0}, {1/2, Sqrt[3]/2}}], Line[{{1/2, Sqrt[3]/2}, {1, 0}}], Line[{{1, 0}, {0, 0}}]}, n]], AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]\), "\[IndentingNewLine]", \(\(doline[start_, finish_] := Module[{vec, normaal}, vec = finish - start; \[IndentingNewLine]normaal = Reverse[vec]\ {\(-1\), 1}\ Sqrt[3]/ 6; \[IndentingNewLine]{Line[{start, start + vec/3}], Line[{start + vec/3, start + vec/2 + normaal}], Line[{start + vec/2 + normaal, start + 2\ vec/3}], Line[{start + 2\ vec/3, finish}]}];\)\)}], "Input", PageBreakAbove->False, InitializationCell->True, AspectRatioFixed->True, Background->GrayLevel[0.900008]], Cell["\<\ Het commando Snowflake[i] laat het resultaat van de i-de \ constructiestap zien. Door achtereenvolgens i=0,1,2,3,4,5 te kiezen, wordt de constructie duidelijk. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Do[Snowflake[i], {i, 0, 5}]\)], "Input", PageBreakAbove->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["De Sierpinski zeef", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Een bekende fractaal is de Sierpinski zeef. Opnieuw worden eerst de \ nodige functies gedefinieerd.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\(Clear[Gasket, disp, newF, G, F, sierp, i, j, k, w, trans, group1, group2];\)\), "\[IndentingNewLine]", \(Gasket[level_Integer?NonNegative] := Module[{newF, G, F, sierp, i, j, k, trans, group1, group2}, (*trans = \({{a1, b1, c1, d1, e1, f1}, {a2, b2, c2, d2, e2, f2}, {a3, b3, c3, d3, e3, f3}}\ represents\ the\ coefficients\ of\ the\ affine\ \ transformations\ of\ the\ iterated\ function\ system\ given\ by\ Ti \((x)\) = \ \(\(\(|\)\(ai\ bi\)\(|\)\(\(x\)\(+\)\)\(\ \)\(|\)\(ei\)\(|\)\(i\)\) = 1\)\), 2, 3 | ci\ di | \ \ \ \ | fi | which\ produces\ the\ Sierpinski\ Gasket*) trans = {{1/2, 0, 0, 1/2, 0, 0}, {1/2, 0, 0, 1/2, 1/2, 0}, {1/2, 0, 0, 1/2, 1/4, N[Sqrt[3]]/4}}; \[IndentingNewLine]group1[ list_] := {{list[\([1]\)], list[\([2]\)]}, {list[\([3]\)], list[\([4]\)]}}; \[IndentingNewLine]group2[ list_] := {{list[\([5]\)]}, {list[\([6]\)]}}; \ \[IndentingNewLine]disp[0] = Show[Graphics[Polygon[{{0, 0}, {1/2, Sqrt[3]/2}, {1, 0}}]], AspectRatio \[Rule] Automatic]; \[IndentingNewLine]F = {{0, 0}, {1/2, Sqrt[3. ]/2}, {1, 0}, {0, 0}}; \[IndentingNewLine]G = F; \[IndentingNewLine]For[k = 1, k < level + 1, \(k++\), F = Partition[Flatten[G], 2]; \[IndentingNewLine]G = {}; \[IndentingNewLine]For[i = 1, i < \((Length[trans] + 1)\), \(i++\), newF = F; \[IndentingNewLine]For[j = 1, j < \((Length[F] + 1)\), \(j++\), \(newF = ReplacePart[newF, group1[trans[\([i]\)]] . F[\([j]\)] + group2[trans[\([i]\)]], j];\)]; \[IndentingNewLine]G = Append[G, newF];]; \[IndentingNewLine]G = Partition[Partition[Flatten[G], 2], 4]; \[IndentingNewLine]sierp = Table[Graphics[Polygon[G[\([n]\)]]], {n, 1, Length[G]}]; \[IndentingNewLine]disp[k] = Show[sierp, AspectRatio \[Rule] Automatic];];;]\)}], "Input", PageBreakAbove->False, InitializationCell->True, AspectRatioFixed->True, Background->GrayLevel[0.900008]], Cell["\<\ Het commando Gasket[i] laat de eerste i constructie stappen van de \ Sierpinski zeef zien.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Gasket[6]\)], "Input", PageBreakAbove->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Box counting dimensie van de Henon attractor", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["De Henon afbeelding moet gedefinieerd worden:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(a = 1.4; \ b = .3;\), "\n", \(h[{x_, y_}] := {a - x^2 + b*y, x}\)}], "Input", PageBreakAbove->False, InitializationCell->True, AspectRatioFixed->True, Background->GrayLevel[0.900008]], Cell["\<\ Een plot van de Henon afbeelding wordt verkregen door een beginpunt \ te nemen en dat, middels het commando NestList, te itereren. Het commando NestList[h,{0,1},5000] betekent \ dat begonnen wordt in het punt (0,1) en dat dan 5000 keer geitereerd wordt. Met ListPlot worden de \ punten geplot.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(ListPlot[NestList[h, {0, 1}, 5000], \ AspectRatio \[Rule] 1, PlotRange \[Rule] \ {{\(-2\), 2}, {\(-2\), 2}}, \ \ \[IndentingNewLine]Axes \[Rule] \ False, \ Frame \[Rule] True];\)\)], "Input", PageBreakAbove->False, AspectRatioFixed->True], Cell["\<\ Met de volgende commandos kan numeriek de box counting dimensie van \ de Henon attractor benaderd worden. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\(\(BoxCount[set_, epsilon_] := \n\t Length[Union[Map[Round[#/epsilon] &, set]]]\)\(\n\) \)\), "\[IndentingNewLine]", \(BoxDimensie[N_, epsilon_] := \n\t Log[BoxCount[NestList[h, {0.0, 0.0}, N], epsilon]]/ Log[1/epsilon]\)}], "Input", InitializationCell->True, AspectRatioFixed->True, Background->GrayLevel[0.900008]], Cell["\<\ Door getallen in te vullen in het boven gedefinieerde commando \ boxDimension, wordt een benadering van de box counting dimensie van de Henon attractor verkregen. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(BoxDimensie[300000, .001]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Bij te kleine of te grote roosters is de verkregen schatting van de \ box counting dimensie niet betrouwbaar. Het is daarom nuttig antwoorden voor verschillende roosters \ met elkaar te vergelijken. In de grafiek beneden wordt de logaritme van het aantal getelde vierkanten \ die de attractor overdekken (Log N_d) uitgezet tegen de logaritme van de zijde (-Log d) van de gebruikte \ vierkanten. Voor de plot is de logaritme met basis 10 gebruikt. 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