(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 20202015, 350712] NotebookOptionsPosition[ 20178957, 350038] NotebookOutlinePosition[ 20180573, 350087] CellTagsIndexPosition[ 20180200, 350076] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.394552301337021*^9, 3.394552301584775*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.394552303364355*^9, 3.395450858459072*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Physicist's Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " 6" }], "Title", CellChangeTimes->{{3.394579470096292*^9, 3.3945794805215673`*^9}, 3.395926540537396*^9}, FontColor->RGBColor[0, 0, 1]], Cell[TextData[StyleBox["Third Edition 2007", FontSlant->"Italic"]], "Subtitle", CellChangeTimes->{{3.394579519605548*^9, 3.3945795336356897`*^9}, { 3.394579849753665*^9, 3.3945798497607594`*^9}}], Cell["PHYSICS 200", "Subtitle", CellChangeTimes->{{3.394579873314529*^9, 3.3945798826114264`*^9}}], Cell[CellGroupData[{ Cell["Nicholas Wheeler", "Author", CellChangeTimes->{{3.39457972270331*^9, 3.3945797314311037`*^9}}], Cell["REED COLLEGE", "Institution", CellChangeTimes->{{3.3945797341591387`*^9, 3.39457973632885*^9}, { 3.394579793724296*^9, 3.3945797937303047`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "Author", CellChangeTimes->{{3.395421842987708*^9, 3.39542184299395*^9}}], Cell["", "Institution", CellChangeTimes->{{3.395421843429604*^9, 3.39542184343263*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "Author", CellChangeTimes->{{3.3954218541724863`*^9, 3.3954218541752357`*^9}}], Cell["", "Institution"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Laboratory 6", FontColor->RGBColor[1, 0, 0]]], "Subtitle", CellChangeTimes->{{3.394581250479834*^9, 3.394581255896365*^9}, 3.394810647542058*^9, 3.394904595337902*^9, 3.3950042110191517`*^9, 3.395421873418894*^9, 3.395607533950148*^9}], Cell[CellGroupData[{ Cell["Recursive Processes", "Section", CellChangeTimes->{{3.3950050750221233`*^9, 3.3950050875924597`*^9}, { 3.395422438918091*^9, 3.395422447760174*^9}, {3.3956077154382877`*^9, 3.395607719584688*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Background", FontColor->RGBColor[1, 0, 0]]], "Subsection", CellChangeTimes->{{3.395607835043456*^9, 3.3956078403865137`*^9}}], Cell["\<\ A major shift in the way physicists think can be traced to 1972, and for a \ curious reason. In that year Hewlett-Packard introduced the HP-35, the first \ programmable calculator. It rapidly acquired the status of a cult object. \ \"HP-35 Clubs\" appeared in scientific centers around the world. Physicists \ began to look forward to such dreary events as faculty meetings as \ opportunities to twiddle with their HP-35s, started sharing tricks (slowly\ \[LongDash]those were the days before e-mail), and interest was awakened in \ some mathematical phenomena that on first encounter seemed \ strange\[LongDash]the seeds of much that came after. \ \>", "Text"], Cell[TextData[{ "\tFor example: Select a number ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], " between 0 and 1, and repeatedly hit the Cos key. What you are thus \ computing is" }], "Text"], Cell[BoxData[ StyleBox[ RowBox[{"Cos", "[", RowBox[{"Cos", "[", RowBox[{ RowBox[{"\[Ellipsis]", "[", RowBox[{"Cos", "[", SubscriptBox["x", "0"], "]"}], "]"}], "\[Ellipsis]"}], "]"}], "]"}], FontColor->GrayLevel[0.500008]]], "Input"], Cell[TextData[{ "It was observed that the output converged to 0.739085 ", StyleBox["independently of the value you had assigned to the \"seed\"", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], ". This result is really not unexpected in the light of the following \ calculation " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "\[Equal]", "x"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", ".7"}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"x", "\[Rule]", "0.7390851332151608`"}], "}"}]], "Output", CellChangeTimes->{3.395607985563272*^9}] }, Open ]], Cell[TextData[{ "but is unforgettably impressive when done \"in the hand\". It can be \ phrased this way: ", StyleBox["iteration", FontColor->RGBColor[0, 0, 1]], " of " }], "Text"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"map", ":", " ", "x"}], FontColor->GrayLevel[0.500008]], StyleBox["\[Rule]", FontColor->GrayLevel[0.500008]], RowBox[{ StyleBox[ RowBox[{"Cos", "[", "x", "]"}], FontColor->GrayLevel[0.500008]], StyleBox[" ", FontColor->GrayLevel[0.500008]], StyleBox["yields", FontColor->GrayLevel[0.500008]], StyleBox[" ", FontColor->GrayLevel[0.500008]], StyleBox["0.739085", FontColor->GrayLevel[0.500008]], StyleBox[" ", FontColor->GrayLevel[0.500008]], StyleBox["as", FontColor->GrayLevel[0.500008]], StyleBox[" ", FontColor->GrayLevel[0.500008]], StyleBox["a", FontColor->GrayLevel[0.500008]], " ", StyleBox["fixed", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox["point", FontColor->RGBColor[1, 0, 0]]}]}]], "Input"], Cell[TextData[{ "\tPeople tried other maps, and found that some yielded an ", StyleBox["alternating pair", FontColor->RGBColor[0, 0, 1]], " of fixed points:" }], "Text", CellChangeTimes->{3.395608096878018*^9, 3.396105825184524*^9}], Cell[BoxData[ StyleBox[ RowBox[{ "a", ",", "b", ",", "a", ",", "b", ",", "a", ",", "b", ",", "a", ",", "..."}], "Output"]], "Input", CellChangeTimes->{{3.3956080759624567`*^9, 3.395608084701169*^9}}], Cell[TextData[{ "Other maps yielded still more complicated ", StyleBox["patterns after many iterations", FontColor->RGBColor[0, 0, 1]], ". Some parameterized families of maps" }], "Text"], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"parameterizedmap", ":", " ", "x"}], "\[Rule]", RowBox[{ SubscriptBox["f", "\[Lambda]"], "[", "x", "]"}]}], FontColor->GrayLevel[0.500008]]], "Output"], Cell[TextData[{ "were found upon iteration to possess the property that ", StyleBox["slight \[Lambda]-adjustment produces radical alteration of the \ asymptotic pattern", FontColor->RGBColor[0, 0, 1]], ". And sometimes the asymptotic output appears to be patternless/random, \ even though the map itself contains no element of randomness\[Ellipsis]is \ \"deterministic;\" thus did the mathematical\[LongDash]also (as it develops) \ ubiquitous physical\[LongDash]phenomenon of \"deterministic chaos\" come into \ focus. " }], "Text", CellChangeTimes->{{3.395608229015059*^9, 3.3956082374426622`*^9}}], Cell[TextData[{ "\tPhysicists soon learned to recognize instances/analogs of all of those \ phenomena in a wide range of physical problems. My job here will be to \ illustrate how ", StyleBox["Mathematica", FontSlant->"Italic"], " can be used to approach such subject matter\[LongDash]a subject matter \ that for the most part resists the techniques of classical analysis. " }], "Text", CellChangeTimes->{{3.39560827263936*^9, 3.395608275588821*^9}}], Cell["", "Text", CellChangeTimes->{{3.395608292409066*^9, 3.395608292489352*^9}}], Cell["", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Basic Tools", FontColor->RGBColor[1, 0, 0]]], "Subsection", CellChangeTimes->{{3.395608302508604*^9, 3.395608307369673*^9}}], Cell[TextData[{ "No longer is it necessary to \"hit the Cos key many times.\" One ", StyleBox["can", FontSlant->"Italic"], " do so\[LongDash]proceed" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cos", "[", "0.2", "]"}]], "Input"], Cell[BoxData["0.9800665778412416`"], "Output", CellChangeTimes->{3.395608382647284*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cos", "[", "%", "]"}]], "Input"], Cell[BoxData["0.5569672528096424`"], "Output", CellChangeTimes->{3.395608385066307*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cos", "[", "%", "]"}]], "Input"], Cell[BoxData["0.8488621656582708`"], "Output", CellChangeTimes->{3.395608387430781*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cos", "[", "%", "]"}]], "Input"], Cell[BoxData["0.660837551116615`"], "Output", CellChangeTimes->{3.395608389466476*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cos", "[", "%", "]"}]], "Input"], Cell[BoxData["0.7894784377668683`"], "Output", CellChangeTimes->{3.395608391739059*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cos", "[", RowBox[{"Cos", "[", RowBox[{"Cos", "[", RowBox[{"Cos", "[", RowBox[{"Cos", "[", "0.2", "]"}], "]"}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.395608421695305*^9, 3.395608426497241*^9}}], Cell[BoxData["0.7894784377668683`"], "Output", CellChangeTimes->{3.39560843292304*^9}] }, Open ]], Cell[TextData[{ "\[LongDash]but ", StyleBox["Mathematica", FontSlant->"Italic"], " provides tools that do such things automatically: an informative \ introduction to this subject can be found by commanding" }], "Text", CellChangeTimes->{{3.3956086447011967`*^9, 3.3956087001283083`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", StyleBox["Nest", FontColor->RGBColor[1, 0, 0]]}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Nest\\\", \\\"[\\\", RowBox[{StyleBox[\\\ \"f\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"expr\\\", \\\"TI\\\"], \ \\\",\\\", StyleBox[\\\"n\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) gives an \ expression with \\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\) applied \ \\!\\(\\*StyleBox[\\\"n\\\", \\\"TI\\\"]\\) times to \ \\!\\(\\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\\). \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Nest"]}]], "Print", "PrintUsage", CellChangeTimes->{3.395608463644712*^9}, CellTags->"Info3395583262-1714501"] }, Open ]], Cell[TextData[{ "and following the ", StyleBox["\[RightSkeleton]", FontColor->RGBColor[0, 0, 1]], ". Here we reproduce\[LongDash]with a single command\[LongDash]the \ calculation that we just did \"longhand\" " }], "Text", CellChangeTimes->{{3.3956087029559793`*^9, 3.395608719713077*^9}, { 3.395676488891581*^9, 3.39567657110011*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Nest", "[", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", "0.2", ",", "5"}], "]"}]], "Input", CellChangeTimes->{ 3.395608860012773*^9, {3.395676465148295*^9, 3.3956764689164658`*^9}}], Cell[BoxData["0.7894784377668683`"], "Output", CellChangeTimes->{3.395608867227086*^9}] }, Open ]], Cell["\<\ and here we just as effortlessly extend the interation to depth 100:\ \>", "Text", CellChangeTimes->{{3.395676591210546*^9, 3.395676629232473*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Nest", "[", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", "0.2", ",", StyleBox["100", FontColor->RGBColor[1, 0, 0]]}], "]"}]], "Input", CellChangeTimes->{{3.3956088934566383`*^9, 3.395608894022923*^9}, 3.396138525638371*^9}], Cell[BoxData["0.7390851332151607`"], "Output", CellChangeTimes->{3.395608910471961*^9}] }, Open ]], Cell["The following command", "Text", CellChangeTimes->{{3.395676685829722*^9, 3.395676692092486*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NestList", "[", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", "0.2", ",", StyleBox["40", FontColor->RGBColor[1, 0, 0]]}], "]"}]], "Input", CellChangeTimes->{{3.395608945135215*^9, 3.395608949307453*^9}, 3.396138536670314*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ "0.2`", ",", "0.9800665778412416`", ",", "0.5569672528096424`", ",", "0.8488621656582708`", ",", "0.660837551116615`", ",", "0.7894784377668683`", ",", "0.7042157133419932`", ",", "0.7621195617606608`", ",", "0.7233741721055712`", ",", "0.7495765763314931`", ",", "0.7319774252581913`", ",", "0.7438542614501144`", ",", "0.735864198169292`", ",", "0.7412509562683525`", ",", "0.7376244764483199`", ",", "0.7400682604082975`", ",", "0.7384225298410249`", ",", "0.739531308540739`", ",", "0.7387845105929659`", ",", "0.7392876028296402`", ",", "0.7389487320992274`", ",", "0.7391770077719252`", ",", "0.7390232422893507`", ",", "0.7391268222717291`", ",", "0.7390570503229223`", ",", "0.7391040498977448`", ",", "0.739072390577983`", ",", "0.7390937167488424`", ",", "0.7390793512163929`", ",", "0.7390890280267289`", ",", "0.7390825096176311`", ",", "0.7390869004994727`", ",", "0.7390839427500349`", ",", "0.739085935126266`", ",", "0.739084593037956`", ",", "0.7390854970849157`", ",", "0.7390848881080677`", ",", "0.7390852983222247`", ",", "0.7390850219970461`", ",", "0.7390852081330159`", ",", "0.7390850827495901`"}], "}"}]], "Output", CellChangeTimes->{3.3956089536528587`*^9}] }, Open ]], Cell["\<\ displays the results encountered along the way, so enables you to watch the \ convergence. 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But\[LongDash]our curiosity whetted\ \[LongDash]we abandon the ODE to concentrate on the logistic map itself, \ looking especially to its asymptotic properties. We try a really big \[Tau] \ (\[Tau] = 1.8 ) \ \>", "Text", CellChangeTimes->{{3.395683768572308*^9, 3.395683772928938*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Logistic3", "[", "x_", "]"}], ":=", RowBox[{"x", "+", RowBox[{ StyleBox["1.8", FontColor->RGBColor[1, 0, 0]], " ", "x", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}]}]}]], "Input", CellChangeTimes->{{3.3956838461715727`*^9, 3.39568384724621*^9}, { 3.3956846662645082`*^9, 3.395684667898904*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Xvalues3", "=", RowBox[{"NestList", "[", RowBox[{"Logistic3", ",", "0.1", ",", "50"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.3956839218458147`*^9, 3.395683931792272*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ListPlot", "[", RowBox[{"Xvalues3", ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "1.5"}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"iterations", ",", "x"}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", RowBox[{"{", RowBox[{"Automatic", ",", " ", RowBox[{"{", "1", "}"}]}], "}"}]}], ",", RowBox[{ "PlotLabel", "\[Rule]", "\"\\""}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Bold", ",", "Red"}], "]"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.395684017386237*^9, 3.3956840554590073`*^9}, { 3.395684100980661*^9, 3.395684101640498*^9}, {3.395684195166665*^9, 3.3956842685485773`*^9}, {3.3956844283122168`*^9, 3.395684429070003*^9}, { 3.3956844897488422`*^9, 3.395684543820425*^9}}], Cell[BoxData[ GraphicsBox[ {Hue[0.67, 0.6, 0.6], PointBox[{{1., 0.1}, {2., 0.262}, {3., 0.6100407999999999}, {4., 1.038244640203648}, {5., 0.9667715133293299}, {6., 1.024595351148975}, { 7., 0.9792348427441656}, {8., 1.0158359806441204`}, {9., 0.986879814575374}, {10., 1.010186297661663}, {11., 0.9916641926825761}, { 12., 1.0065435716233992`}, {13., 0.9946880697080176}, {14., 1.0041987543474176`}, {15., 0.99660926335354}, {16., 1.0026918945461578`}, {17., 0.997833441029828}, {18., 1.0017247980161494`}, {19., 0.9986148067163267}, {20., 1.001104700858159}, {21., 0.999114042658298}, {22., 1.0007073530166213`}, {23., 0.9994332169597807}, {24., 1.000452848194749}, {25., 0.9996373523155233}, {26., 1.000289881423564}, { 27., 0.9997679436049174}, {28., 1.000185548185759}, {29., 0.9998514994807601}, {30., 1.0001187607210642`}, {31., 0.9999049660357526}, {32., 1.0000760109147802`}, {33., 0.9999391808683894}, {34., 1.0000486486471483`}, {35., 0.9999610768222378}, {36., 1.000031135815185}, {37., 0.9999750896028619}, {38., 1.0000199272007604`}, {39., 0.9999840575246237}, {40., 1.0000127535228085`}, {41., 0.9999897968889789}, {42., 1.0000081623014305`}, {43., 0.9999934700389339}, {44., 1.0000052238921002`}, {45., 0.9999958208371995}, {46., 1.0000033432988027`}, {47., 0.9999973253408381}, {48., 1.0000021397144527`}, {49., 0.9999982882201968}, {50., 1.0000013694185683`}, {51., 0.9999989044617698}}]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, AxesLabel->{ FormBox["iterations", TraditionalForm], FormBox["x", TraditionalForm]}, LabelStyle->Directive[Bold, RGBColor[1, 0, 0]], PlotLabel->FormBox["\"Logistic map with \[Tau] = 1.800\"", TraditionalForm], PlotRange->{0, 1.5}, PlotRangeClipping->True, Ticks->{Automatic, {{1, FormBox["1", TraditionalForm]}}}]], "Output", CellChangeTimes->{ 3.395684272311953*^9, 3.3956844366023903`*^9, {3.395684505882172*^9, 3.395684549217606*^9}, 3.39568467699323*^9}] }, Open ]], Cell["\<\ and find that the iterative process still approaches 1.00000 asymptotically \ (as a fixed point). 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And at \[Tau] = 2.542 (I take this value from page 5 of the 1987 thesis \ of Stacey Irvine) " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"Logistic5", "[", "x_", "]"}], ":=", RowBox[{"x", "+", RowBox[{ StyleBox["2.542", FontColor->RGBColor[1, 0, 0]], " ", "x", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}]}]}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Xvalues5", "=", RowBox[{"NestList", "[", RowBox[{"Logistic5", ",", "0.1", ",", "50"}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"ListPlot", "[", RowBox[{"Xvalues5", ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "1.5"}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"iterations", ",", "x"}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", RowBox[{"{", RowBox[{"Automatic", ",", " ", RowBox[{"{", "1", "}"}]}], "}"}]}], ",", RowBox[{ "PlotLabel", "\[Rule]", "\"\\""}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Bold", ",", "Red"}], "]"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.395684901857607*^9, 3.395684959477054*^9}}], Cell[BoxData[ GraphicsBox[ {Hue[0.67, 0.6, 0.6], PointBox[{{1., 0.1}, {2., 0.32878}, {3., 0.8897579948871999}, {4., 1.139099484068677}, {5., 0.7363242854040065}, {6., 1.2298557006741087`}, { 7., 0.5112593887180008}, {8., 1.1464371296312024`}, {9., 0.7196837227888677}, {10., 1.2325044182444755`}, {11., 0.5040619770198792}, {12., 1.139520034890997}, {13., 0.7353779405724086}, {14., 1.2300440867559972`}, {15., 0.5107486617559917}, {16., 1.145954975015489}, {17., 0.7207855717979517}, {18., 1.2323725567265862`}, {19., 0.5044211305134526}, {20., 1.1398714435773194`}, {21., 0.7345864933121724}, {22., 1.2301981416414485`}, {23., 0.5103308748055082}, {24., 1.1455595748369722`}, {25., 0.7216882822618663}, {26., 1.2322599268626508`}, {27., 0.5047278324198804}, {28., 1.14017101262063}, {29., 0.7339113042545349}, { 30., 1.2303270496825525`}, {31., 0.509981191758582}, {32., 1.1452279470703433`}, {33., 0.7224447855142335}, {34., 1.2321623483387463`}, {35., 0.5049934959448311}, {36., 1.1404301111703798`}, {37., 0.7333269623897816}, {38., 1.2304367421467974`}, {39., 0.5096835674120528}, {40., 1.1449452003154246`}, {41., 0.7230893407114508}, {42., 1.2320769139983354`}, {43., 0.5052260564386283}, {44., 1.1406566301839112`}, {45., 0.732815817139858}, {46., 1.2305312707657674`}, {47., 0.5094270374713508}, {48., 1.1447011323831446`}, {49., 0.7236453960388356}, {50., 1.2320015130623032`}, {51., 0.505431274213279}}]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, AxesLabel->{ FormBox["iterations", TraditionalForm], FormBox["x", TraditionalForm]}, LabelStyle->Directive[Bold, RGBColor[1, 0, 0]], PlotLabel->FormBox["\"Logistic map with \[Tau] = 2.542\"", TraditionalForm], PlotRange->{0, 1.5}, PlotRangeClipping->True, Ticks->{Automatic, {{1, FormBox["1", TraditionalForm]}}}]], "Output", CellChangeTimes->{3.395684963873713*^9}] }, Open ]], Cell[TextData[{ "we are led asymptotically to a ", StyleBox["quartet of points that present themselves in a fixed cyclic order", FontColor->RGBColor[0, 0, 1]], ". " }], "Text"], Cell[TextData[{ "\t", StyleBox["CobwebPlot", "Input"], " can be used to clarify the situation: we define" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"logisticmap", "[", RowBox[{"\[Tau]_", ",", "x_"}], "]"}], ":=", RowBox[{"x", "+", RowBox[{"\[Tau]", " ", "x", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}]}]}]], "Input", CellChangeTimes->{3.3956850749590054`*^9}], Cell["\<\ and, after initially setting \[Tau] = 1.800, proceed\ \>", "Text", CellChangeTimes->{{3.395685244219419*^9, 3.3956852463800087`*^9}, { 3.39568546337892*^9, 3.395685486101098*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CobwebPlot", "[", RowBox[{ RowBox[{"logisticmap", "[", RowBox[{ StyleBox["1.800", FontColor->RGBColor[1, 0, 0]], ",", "x"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1.5"}], "}"}], ",", "0.1", ",", "50"}], "]"}]], "Input", CellChangeTimes->{{3.395685120690195*^9, 3.3956851248543043`*^9}, { 3.395685494184119*^9, 3.395685494343154*^9}}], Cell[BoxData[ GraphicsBox[{{{}, 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We again detect the scent of some \ kind of \"universality\" principle\[Ellipsis]in which connection we note that \ both systems are (not obviously, the way I have presented them) ", StyleBox["nonlinear forced dissipative", FontColor->RGBColor[0, 0, 1]], " systems. " }], "Text", CellChangeTimes->{{3.395704698783392*^9, 3.395704703056085*^9}}], Cell[TextData[{ "I will not pursue this rich topic, because to do so would risk being \ seduced by the physics, and our present assignment is simply to become \ acquainted with the utility of ", StyleBox["Mathematica", FontSlant->"Italic"], ". So I leave the ", StyleBox["Poincare maps", FontColor->RGBColor[0, 0, 1]], ", ", StyleBox["Liapunov exponents", FontColor->RGBColor[0, 0, 1]], ", etc. for you to explore on your own, if someday you are so inclined." }], "Text", CellChangeTimes->{3.395704772427663*^9}], Cell[TextData[{ "You will appreciate that DEs are like iterated maps, in that they tell the \ state point where to go, where to go, where to go\[Ellipsis] And that such \ rich structure as we have seen is exposed only by numerical analysis (which \ ", StyleBox["Mathematica", FontSlant->"Italic"], " takes in easy stride). 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Select ", StyleBox[" ", FontWeight->"Bold"], "Blossoms.jpg file and\[LongDash]with the cursor blinking at the ", Cell[BoxData[ FormBox[ StyleBox["\[Wedge]", FontColor->RGBColor[1, 0, 0]], TraditionalForm]]], " in the ", StyleBox["Import[ ]", "Input"], " command\[LongDash]hit the ", StyleBox["Choose", FontColor->RGBColor[0.5, 0, 0.5]], " button: the desired file ", StyleBox["\"path\"", "Input", FontColor->GrayLevel[0.5]], " will be inserted between the square brackets, as shown below. 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