(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 98162, 2386] NotebookOptionsPosition[ 83523, 2151] NotebookOutlinePosition[ 87860, 2237] CellTagsIndexPosition[ 87781, 2232] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", "FirstSlide", CellTags-> "SlideShowHeader",ExpressionUUID->"ae861ed8-5fa1-4088-84fd-2651cc7b368a"], Cell["Introdu\[CCedilla]\[ATilde]o \[AGrave] F\[IAcute]sica Computacional I \ (2023.2)", "Title", CellChangeTimes->{ 3.559948400406288*^9, {3.699269571766136*^9, 3.699269573578824*^9}, { 3.7099080105038958`*^9, 3.7099080108133383`*^9}, {3.727189496068931*^9, 3.7271894962748613`*^9}, {3.83932299294512*^9, 3.839323000605739*^9}, { 3.877937206335764*^9, 3.8779372064173126`*^9}, {3.909774153113459*^9, 3.909774153345558*^9}},ExpressionUUID->"92508663-47f1-4deb-9615-\ d2c271e8e453"], Cell["Aula 13: o oscilador de Duffing", "Subtitle", CellChangeTimes->{{3.699269582406044*^9, 3.699269584285553*^9}, { 3.8393232649484053`*^9, 3.839323284264048*^9}, 3.840031268565436*^9, { 3.8407028843722677`*^9, 3.840702924713172*^9}, {3.840714600599773*^9, 3.8407146281949797`*^9}, {3.841157909295815*^9, 3.841157911477744*^9}, { 3.841480259563218*^9, 3.8414802737602463`*^9}, {3.8415945589945517`*^9, 3.841594560546528*^9}, {3.842262188714292*^9, 3.8422622084045753`*^9}, { 3.842288735025601*^9, 3.8422887376233883`*^9}, {3.842288769619337*^9, 3.8422887810036163`*^9}, {3.843294467913928*^9, 3.843294500290168*^9}, { 3.843553405919444*^9, 3.8435534093620787`*^9}, {3.843899024345776*^9, 3.8438990498951845`*^9}, {3.844104755695139*^9, 3.8441047787114773`*^9}, { 3.845103660182572*^9, 3.8451036762035027`*^9}, {3.845196864256226*^9, 3.845196866223578*^9}, {3.84521335376473*^9, 3.845213361459647*^9}, { 3.846057914868442*^9, 3.8460579231350527`*^9}, {3.846235828708374*^9, 3.8462358302259893`*^9}, {3.846236430171523*^9, 3.846236433545587*^9}, { 3.847437919397561*^9, 3.847437929769536*^9}, {3.8779372089913487`*^9, 3.877937209066758*^9}, {3.9097741582933464`*^9, 3.9097741583833475`*^9}},ExpressionUUID->"14f8f51b-58e9-4033-aea1-\ fd611ac95bb9"] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"40fed88a-8620-47ef-b53c-b50de81b6b11"], Cell["O oscilador de Duffing", "Section", CellChangeTimes->{{3.844004981476076*^9, 3.844005028444255*^9}, { 3.845114897076255*^9, 3.845114899662863*^9}, {3.847440859665846*^9, 3.847440864367354*^9}},ExpressionUUID->"f20b05cf-ded6-4dde-ab38-\ b35759e4ea5b"], Cell[TextData[{ "J\[AAcute] vimos que pode surgir comportamento ca\[OAcute]tico em sistemas \ matem\[AAcute]ticos simples, como o mapa log\[IAcute]stico. Nele e em outros \ mapas semelhantes, observamos a rota para o caos atrav\[EAcute]s de duplica\ \[CCedilla]\[OTilde]es de per\[IAcute]odo. Agora vamos explorar como surge o \ caos em sistemas f\[IAcute]sicos, representados por osciladores n\[ATilde]o \ lineares. O exemplo com que trabalharemos \[EAcute] o oscilador de Duffing, \ associado a uma energia potencial ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"U", "(", "x", ")"}], "=", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], SubscriptBox["k", "2"], SuperscriptBox["x", "2"]}], "+", RowBox[{ FractionBox["1", "4"], SubscriptBox["k", "4"], SuperscriptBox["x", "4"]}]}]}], TraditionalForm]],ExpressionUUID-> "6de287bb-9de8-466b-968f-1a51f2f08141"], "." }], "Text", CellChangeTimes->{{3.84511593473853*^9, 3.8451160675895443`*^9}, { 3.8451240046530724`*^9, 3.8451240395491037`*^9}, {3.8451242047456007`*^9, 3.84512436548533*^9}, {3.845124472848686*^9, 3.845124478492044*^9}, { 3.845124515707251*^9, 3.845124569246211*^9}, {3.845213375612213*^9, 3.84521339778572*^9}, {3.846057932394002*^9, 3.846057936175067*^9}, { 3.84744086818137*^9, 3.847441069377479*^9}, {3.9097742022722683`*^9, 3.9097742022722683`*^9}},ExpressionUUID->"de408365-f028-430c-b5e2-\ b4fa36941087"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"U", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{ RowBox[{"-", "k2"}], "*", RowBox[{ RowBox[{"x", "^", "2"}], "/", "2"}]}], "+", RowBox[{"k4", "*", RowBox[{ RowBox[{"x", "^", "4"}], "/", "4"}]}]}]}], ";"}], " "}]], "Input", CellChangeTimes->{{3.847441703734817*^9, 3.847441728687311*^9}, 3.84781476902033*^9, 3.847915558882721*^9},ExpressionUUID->"245f6225-9feb-43b1-b3e2-\ 54f424b1dd49"], Cell[TextData[{ "Vamos adotar uma parametriza\[CCedilla]\[ATilde]o em que, nas unidades \ apropriadas, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["k", "2"], "=", "1"}], TraditionalForm]],ExpressionUUID-> "9bdb499f-d63d-4e69-9b18-7fa9aa98d2d8"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["k", "4"], "=", "1"}], TraditionalForm]],ExpressionUUID-> "a3cc6263-116e-451c-9a28-d3b17ecdd4b5"], " e a massa da part\[IAcute]cula presa ao oscilador \[EAcute] ", Cell[BoxData[ FormBox[ RowBox[{"m", "=", "1"}], TraditionalForm]],ExpressionUUID-> "9a6ae8a0-29b5-4835-a4d1-3348922d2e75"], ", e mostrar gr\[AAcute]ficos tanto da energia potencial como da for\ \[CCedilla]a associada." }], "Text", CellChangeTimes->{{3.847441912998021*^9, 3.847441914605616*^9}, { 3.84744197944072*^9, 3.847442070261633*^9}, {3.8474433519167023`*^9, 3.847443374835123*^9}},ExpressionUUID->"6a078818-0cb0-4718-861e-\ 7119a3031f85"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"pars", "=", RowBox[{"{", RowBox[{ RowBox[{"k2", "\[Rule]", " ", "1"}], ",", RowBox[{"k4", "\[Rule]", " ", "1"}], ",", RowBox[{"m", "\[Rule]", " ", "1"}]}], "}"}]}], ";"}], " "}]], "Input", CellChangeTimes->{{3.847441780611301*^9, 3.8474417927783957`*^9}, { 3.847443379441064*^9, 3.84744338087495*^9}, 3.84781477046281*^9, 3.8479155601867332`*^9},ExpressionUUID->"f30d35b1-f39d-4aba-a657-\ 7926810cbe9a"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"optam", "=", RowBox[{"{", RowBox[{ RowBox[{"ImageSize", "\[Rule]", " ", "Large"}], ",", RowBox[{"LabelStyle", "\[Rule]", " ", RowBox[{"{", "\"\\"", "}"}]}]}], "}"}]}], ";"}], " "}]], "Input", CellChangeTimes->{{3.8474415256274347`*^9, 3.847441539215436*^9}, 3.8478147718655453`*^9, 3.847915561233491*^9},ExpressionUUID->"9863ed4b-1664-4c2c-b1e9-\ 8a26e5d3ff53"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"U", "[", "x", "]"}], "/.", "pars"}], ",", RowBox[{ RowBox[{"-", RowBox[{ RowBox[{"U", "'"}], "[", "x", "]"}]}], "/.", "pars"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", " ", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"x\",FontSlant->\"Italic\"]\)\>\"", ",", "\"\<\!\(\*StyleBox[\"U\",FontSlant->\"Italic\"]\)(\!\(\*StyleBox[\"x\"\ ,FontSlant->\"Italic\"]\)), \ \!\(\*StyleBox[\"F\",FontSlant->\"Italic\"]\)(\!\(\*StyleBox[\"x\",FontSlant->\ \"Italic\"]\))\>\""}], "}"}]}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "0.4"}], ",", "0.4"}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.847441502836659*^9, 3.8474415973445063`*^9}, { 3.8474416317106647`*^9, 3.8474416550307903`*^9}, {3.847441738876156*^9, 3.847441821715808*^9}, {3.847441867331349*^9, 3.8474418675821342`*^9}, 3.8478147743963327`*^9, 3.847915565094048*^9},ExpressionUUID->"d7432fea-4357-40e7-89da-\ a9cff7fee6fd"], Cell[TextData[{ "Existem dois pontos de equil\[IAcute]brio est\[AAcute]vel, situados em ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"\[PlusMinus]", "1"}]}], TraditionalForm]],ExpressionUUID-> "c557c273-c6dc-49a4-8632-b63dfe7c079f"], ". Se a part\[IAcute]cula for abandonada do m\[IAcute]nimo situado em ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "1"}], TraditionalForm]],ExpressionUUID-> "f76c410c-c0df-41f3-8d4c-56c5cbc7cea9"], " com velocidade suficientemente elevada, ir\[AAcute] superar a barreira de \ potencial e seguir em dire\[CCedilla]\[ATilde]o ao m\[IAcute]nimo em ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"-", "1"}]}], TraditionalForm]],ExpressionUUID-> "551a7ef4-b7ab-4bd7-8a07-8447465c0372"], " antes de retornar. 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N\[ATilde]o h\[AAcute] solu\[CCedilla]\[ATilde]o anal\[IAcute]tica, de \ modo que temos que resolver numericamente:" }], "Text", CellChangeTimes->{{3.845489934385297*^9, 3.845490016183366*^9}, { 3.8474432223350897`*^9, 3.847443329434802*^9}, {3.847443408527677*^9, 3.847443486639038*^9}, {3.847443532076831*^9, 3.847443685405074*^9}, { 3.8474461070170507`*^9, 3.847446160667893*^9}, {3.847446571890582*^9, 3.847446593659452*^9}, {3.847814803297463*^9, 3.847814812801621*^9}, 3.877955557144431*^9, 3.9097742615254536`*^9},ExpressionUUID->"3effeb8f-90f5-41d6-be6a-\ f5f5bc064e41"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"sol", "=", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"m", "*", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}]}], "\[Equal]", RowBox[{"-", RowBox[{ RowBox[{"U", "'"}], "[", RowBox[{"x", "[", "t", "]"}], "]"}]}]}], ")"}], "/.", "pars"}], ",", RowBox[{ RowBox[{"x", "[", "0", "]"}], "\[Equal]", "1"}], ",", RowBox[{ RowBox[{ 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Como esse tempo m\[AAcute]ximo pode ser grande, para minimizar o erro \ acumulado na solu\[CCedilla]\[ATilde]o num\[EAcute]rica aumentamos a precis\ \[ATilde]o da solu\[CCedilla]\[ATilde]o utilizando a op\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["PrecisionGoal", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/PrecisionGoal.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/PrecisionGoal.html"], FontFamily->"Source Code Pro", FontSize->38, FontWeight->"Normal"], ". 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RowBox[{"t", ",", "0", ",", "tmax"}], "}"}], ",", RowBox[{"PrecisionGoal", "->", "25"}]}], "]"}]}], ";"}], " "}]], "Input",\ CellChangeTimes->{{3.847459154731471*^9, 3.847459185879548*^9}, { 3.847459924058238*^9, 3.8474599309501143`*^9}, 3.847461363418212*^9, 3.847814876930127*^9, {3.877958507420117*^9, 3.8779585076853333`*^9}, { 3.8780216572044773`*^9, 3.878021660114376*^9}, {3.9107907977443466`*^9, 3.9107908031604166`*^9}},ExpressionUUID->"c678563f-309d-4ec9-b93f-\ 4f42425a5088"], Cell[TextData[{ "Vamos agora utilizar a fun\[CCedilla]\[ATilde]o, primeiro escolhendo um \ valor bem pequeno ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["F", "0"], "=", "0.1"}], TraditionalForm]],ExpressionUUID-> "ae5876ec-5f7b-4365-87aa-ac03936f4cf7"], ":" }], "Text", CellChangeTimes->{{3.84745850197913*^9, 3.84745852624242*^9}, { 3.87795851305037*^9, 3.877958633648801*^9}, 3.8780216647378273`*^9},ExpressionUUID->"791c3356-9946-47bb-983b-\ 9cab08a0727d"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"sol", "=", RowBox[{"resolve", "[", RowBox[{"0.1", ",", "200"}], "]"}]}], ";"}], " "}]], "Input", CellChangeTimes->{{3.847447624472376*^9, 3.8474477227423697`*^9}, 3.8474478005444307`*^9, {3.847447832934099*^9, 3.847447837803081*^9}, { 3.84745920631574*^9, 3.8474592133054113`*^9}, 3.847814878448059*^9},ExpressionUUID->"1ae5c1c7-eded-4158-85b3-\ 6cd8ef64db3a"], Cell[TextData[{ "Tamb\[EAcute]m vai ser conveniente definir uma fun\[CCedilla]\[ATilde]o \ para tra\[CCedilla]ar o movimento no espa\[CCedilla]o de fase entre os tempos \ ", Cell[BoxData[ FormBox[ SubscriptBox["t", "min"], TraditionalForm]],ExpressionUUID-> "685eb440-75bf-46b9-a7c4-9c66c5e0a40f"], " e ", Cell[BoxData[ FormBox[ SubscriptBox["t", "max"], TraditionalForm]],ExpressionUUID-> "bd242462-70f5-4e61-a8c9-ec088ace4eec"], ":" }], "Text", CellChangeTimes->{{3.8474592341074867`*^9, 3.847459287836371*^9}, { 3.847459385173738*^9, 3.847459387278059*^9}, {3.8474598711747026`*^9, 3.84745991512983*^9}},ExpressionUUID->"5f2ffef7-3b6c-4e3c-8812-\ 3f30e674c2a1"], Cell[BoxData[ RowBox[{ RowBox[{"espfase", "[", RowBox[{"tmin_", ",", "tmax_"}], "]"}], ":=", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}]}], "}"}], "/.", "sol"}], ",", RowBox[{"{", RowBox[{"t", ",", "tmin", ",", "tmax"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"x\",FontSlant->\"Italic\"]\)\>\"", ",", "\"\<\!\(\*StyleBox[\"v\",FontSlant->\"Italic\"]\)\>\""}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.84745926437646*^9, 3.8474592741575747`*^9}, { 3.84745943819438*^9, 3.847459493265484*^9}, 3.847814882430814*^9},ExpressionUUID->"b52172b4-9b02-4f20-81a7-\ e90745715613"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"8b5b3294-f865-4849-81fa-fe87263b7376"], Cell["Primeiro visualizamos at\[EAcute] o tempo m\[AAcute]ximo da solu\ \[CCedilla]\[ATilde]o:", "Text", CellChangeTimes->{{3.8474595091712847`*^9, 3.8474595176412992`*^9}},ExpressionUUID->"8f4de778-4c8d-404b-b99a-\ 972b9ed2e130"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"espfase", "[", RowBox[{"0", ",", "200"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.847459519963665*^9, 3.847459522782978*^9}, 3.847816855960968*^9},ExpressionUUID->"860b1251-9ab4-4e78-a8b7-\ 221654ad22ce"], Cell[TextData[{ "Como a visualiza\[CCedilla]\[ATilde]o acima tamb\[EAcute]m envolve o \ transiente (o tempo durante o qual as condi\[CCedilla]\[OTilde]es iniciais \ ainda afetam o movimento), vamos restringi-la a uma faixa de tempos pr\ \[OAcute]xima do tempo m\[AAcute]ximo ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", "200"}], TraditionalForm]],ExpressionUUID-> "849d786c-8c7c-4447-8957-710cbece987b"], ":" }], "Text", CellChangeTimes->{{3.8474585429406347`*^9, 3.847458673069179*^9}, { 3.847458841943959*^9, 3.847458871036462*^9}},ExpressionUUID->"f9fe79c7-d6fc-408b-a5e4-\ d190c96d37fb"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"espfase", "[", RowBox[{"150", ",", "200"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8474478774540653`*^9, 3.847447878306818*^9}, { 3.847459542835931*^9, 3.8474595455195*^9}, 3.847816868312057*^9},ExpressionUUID->"052ecd01-74e8-4b29-87b1-\ 07cb2a76c8a3"], Cell[TextData[{ "Agora fica claro que se trata, mais uma vez, de um movimento \ peri\[OAcute]dico. Na realidade, o per\[IAcute]odo desse movimento \[EAcute] \ o per\[IAcute]odo ", Cell[BoxData[ FormBox[ RowBox[{"2", RowBox[{"\[Pi]", "/", "\[Omega]"}]}], TraditionalForm]],ExpressionUUID-> "48fa1eff-8bc3-461b-99a0-b3fd764c7407"], " (", Cell[BoxData[ FormBox[ RowBox[{"\[TildeEqual]", "4.5"}], TraditionalForm]],ExpressionUUID-> "8ed0f5f2-2eb7-461e-bad1-5b3f8fe8cf9c"], ") da for\[CCedilla]a externa, como \[EAcute] poss\[IAcute]vel verificar \ restringindo ainda mais o intervalo de tempo:" }], "Text", CellChangeTimes->{{3.847458687110216*^9, 3.847458775489787*^9}, { 3.847458892378132*^9, 3.847458893615378*^9}, {3.847458939820356*^9, 3.847458961612464*^9}, {3.847459602186405*^9, 3.847459613424664*^9}},ExpressionUUID->"5786effc-5354-459f-81ea-\ 5123885ea14e"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"espfase", "[", RowBox[{"150", ",", RowBox[{ RowBox[{"150", "+", RowBox[{"1.99", "*", RowBox[{"\[Pi]", "/", "\[Omega]"}]}]}], "/.", "pars"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.847448096220454*^9, 3.847448119157854*^9}, { 3.847459561949648*^9, 3.847459575524008*^9}, 3.847816871473543*^9},ExpressionUUID->"8e619f69-3ecc-4a43-b757-\ f6fa4cf13068"], Cell["\<\ Note que a curva no espa\[CCedilla]o de fase quase se fecha, j\[AAcute] que \ utilizamos um intervalo de tempo um pouco menor do que o per\[IAcute]odo.\ \>", "ItemParagraph", CellChangeTimes->{{3.8474587875845747`*^9, 3.8474588266549253`*^9}},ExpressionUUID->"df329a32-1f68-4a23-9133-\ d6b5e9e17bff"] }, Closed]] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ed8a4c5e-4160-4344-95e1-9735acf8ca9d"], Cell[TextData[{ "Agora vamos aumentar ", Cell[BoxData[ FormBox[ SubscriptBox["F", "0"], TraditionalForm]],ExpressionUUID-> "f35c5d2d-2ab6-43e6-9c80-6755485fe55d"], " at\[EAcute] o valor ", Cell[BoxData[ FormBox["0.32", TraditionalForm]],ExpressionUUID-> "c568e026-6d13-42d1-b372-53ec05b20c27"], " e o tempo m\[AAcute]ximo at\[EAcute] ", Cell[BoxData[ FormBox["800", TraditionalForm]],ExpressionUUID-> "815d9d36-51ae-4a69-b917-08e979136bf0"], ", mas vamos visualizar inicialmente apenas at\[EAcute] ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", "200"}], TraditionalForm]],ExpressionUUID-> "20046e0a-3b3d-4f1b-9429-d69cc1053cca"], ":" }], "Text", CellChangeTimes->{{3.845544247714367*^9, 3.845544347753755*^9}, { 3.8455449431778107`*^9, 3.8455450810688334`*^9}, {3.845545162935638*^9, 3.845545171349823*^9}, {3.845545307666675*^9, 3.845545308757752*^9}, { 3.8455480077281313`*^9, 3.845548030406809*^9}, {3.8474596492559977`*^9, 3.847459667481193*^9}, {3.847459700196405*^9, 3.8474597076230288`*^9}, { 3.847459988130185*^9, 3.847459997458757*^9}},ExpressionUUID->"91317753-ba71-48e1-81bd-\ 22e1c6c233ba"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"sol", "=", RowBox[{"resolve", "[", RowBox[{"0.32", ",", "800"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"espfase", "[", RowBox[{"0", ",", "200"}], "]"}], " "}]}], "Input", CellChangeTimes->{{3.846058375557028*^9, 3.846058405111308*^9}, { 3.846058465299769*^9, 3.8460584670000153`*^9}, 3.847441534436879*^9, { 3.847459689972159*^9, 3.847459739841786*^9}, {3.847459972209993*^9, 3.847459980638517*^9}, 3.84781685423771*^9},ExpressionUUID->"8f431743-8686-4bc0-b2b2-\ febd29eb69d5"], Cell["\<\ Note que durante o transiente o oscilador passa do m\[IAcute]nimo \[AGrave] \ direita para o m\[IAcute]nimo \[AGrave] esquerda e parece ficar no entorno \ desse m\[IAcute]nimo. Vamos investigar o que ocorre ap\[OAcute]s o transiente:\ \>", "Text", CellChangeTimes->{{3.845545332462351*^9, 3.845545374368268*^9}, { 3.845547366023691*^9, 3.845547431835915*^9}, {3.845548047767544*^9, 3.8455480523566523`*^9}, 3.845548124768005*^9, {3.845617410281608*^9, 3.845617421271447*^9}, {3.845642312368412*^9, 3.845642323786257*^9}, { 3.847461013731827*^9, 3.847461042586779*^9}, {3.84746114340139*^9, 3.8474611476980133`*^9}, {3.847461302939439*^9, 3.8474613287005796`*^9}},ExpressionUUID->"61c5e8e9-c532-4918-b741-\ 0430b74a3b23"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"espfase", "[", RowBox[{"750", ",", "800"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8474613429781857`*^9, 3.847461346307019*^9}, 3.847816883218783*^9},ExpressionUUID->"c60e8faf-9b7d-48b3-99bd-\ 38e466204b46"], Cell[TextData[{ "Nossa experi\[EHat]ncia com o mapa log\[IAcute]stico sugere que ocorreu uma \ duplica\[CCedilla]\[ATilde]o de per\[IAcute]odo, para ", Cell[BoxData[ FormBox[ RowBox[{"4", RowBox[{"\[Pi]", "/", "\[Omega]"}]}], TraditionalForm]],ExpressionUUID-> "f22c1102-b767-4f3f-aede-849d4965cbfe"], ". Para testar essa hip\[OAcute]tese, vamos visualizar um intervalo de tempo \ correspondente ao per\[IAcute]odo anterior:" }], "Text", CellChangeTimes->{{3.847461421019492*^9, 3.847461479597702*^9}},ExpressionUUID->"0e4466e3-8b42-4c9c-a559-\ b7122bff1729"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"espfase", "[", RowBox[{"750", ",", RowBox[{ RowBox[{"750", "+", RowBox[{"2", RowBox[{"\[Pi]", "/", "\[Omega]"}]}]}], "/.", "pars"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8474614814158*^9, 3.847461502818138*^9}, 3.847816887065837*^9},ExpressionUUID->"28f8acbd-f913-4655-b5b2-\ 1aab22aa08e0"], Cell["E agora quase o per\[IAcute]odo duplicado:", "Text", CellChangeTimes->{{3.8474615144170647`*^9, 3.847461543913992*^9}},ExpressionUUID->"7f3e60bb-ea67-4a51-93a3-\ 27e6d9b2c90c"], Cell[BoxData[ RowBox[{ RowBox[{"espfase", "[", RowBox[{"750", ",", RowBox[{ RowBox[{"750", "+", RowBox[{"3.99", "*", RowBox[{"\[Pi]", "/", "\[Omega]"}]}]}], "/.", "pars"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8474615324804153`*^9, 3.8474615584159393`*^9}, 3.8478168904746647`*^9},ExpressionUUID->"aa478c0f-c88a-4786-9469-\ 47bc67b14c84"] }, Closed]] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"2970e55d-6d34-4167-8ce5-020d6a07dfcd"], Cell[TextData[{ "Aumentamos mais um pouco ", Cell[BoxData[ FormBox[ SubscriptBox["F", "0"], TraditionalForm]],ExpressionUUID-> "8976a0f7-4e56-4a0e-957c-6ff8b80c5cfb"], " at\[EAcute] o valor ", Cell[BoxData[ FormBox["0.338", TraditionalForm]],ExpressionUUID-> "a05bdd94-8e04-4c24-b3ab-6fa8f96e512b"], " e o tempo m\[AAcute]ximo at\[EAcute] 1000. Na \ visualiza\[CCedilla]\[ATilde]o, j\[AAcute] vamos buscar descartar o \ transiente:" }], "Text", CellChangeTimes->{{3.845548133943215*^9, 3.845548163957223*^9}, { 3.845548370073575*^9, 3.8455483701615057`*^9}, {3.847461582550178*^9, 3.847461651431097*^9}, {3.847461873812223*^9, 3.847461874250064*^9}, { 3.8779571617249327`*^9, 3.877957161885922*^9}},ExpressionUUID->"e581e4fa-e30b-4226-bce7-\ a3d945403b16"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"sol", "=", RowBox[{"resolve", "[", RowBox[{"0.338", ",", "1000"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"espfase", "[", RowBox[{"900", ",", "1000"}], "]"}], " "}]}], "Input", CellChangeTimes->{{3.84554816685606*^9, 3.845548210926834*^9}, { 3.845548343664723*^9, 3.845548372744128*^9}, 3.845974398773952*^9, { 3.847461617097424*^9, 3.8474616211576242`*^9}, {3.847461655594159*^9, 3.84746165927274*^9}, 3.847816852418696*^9, {3.877957165323909*^9, 3.8779571713645926`*^9}, {3.877958885049515*^9, 3.877958889157074*^9}, { 3.878021646588191*^9, 3.878021649686966*^9}},ExpressionUUID->"fd8d580a-1b26-4a65-a4ad-\ 5720fa9ec503"], Cell["\<\ A \[OAcute]rbita parece circundar 4 vezes o mesmo m\[IAcute]nimo, sugerindo \ uma nova duplica\[CCedilla]\[ATilde]o do per\[IAcute]odo. Vamos testar:\ \>", "Text", CellChangeTimes->{{3.8455482355661793`*^9, 3.84554831117603*^9}, { 3.847461754819544*^9, 3.847461807819461*^9}, {3.9108024989141665`*^9, 3.910802502292165*^9}},ExpressionUUID->"3afbc804-7a50-49bf-b622-\ 500e1921dfe5"], Cell[BoxData[ RowBox[{ RowBox[{"espfase", "[", RowBox[{"910", ",", RowBox[{ RowBox[{"910", "+", RowBox[{"7.95", "*", RowBox[{"\[Pi]", "/", "\[Omega]"}]}]}], "/.", "pars"}]}], "]"}], " "}]], "Input", CellChangeTimes->{ 3.845548341484728*^9, {3.8455483920158*^9, 3.8455484092542887`*^9}, { 3.84554844256645*^9, 3.845548455045315*^9}, {3.845548485630464*^9, 3.8455485026746387`*^9}, {3.8455486241314363`*^9, 3.845548624334229*^9}, { 3.845548660087085*^9, 3.845548660333696*^9}, 3.845974392608551*^9, { 3.847461816876823*^9, 3.847461834492684*^9}, 3.8478169031449327`*^9, { 3.877957178567613*^9, 3.8779571806562557`*^9}, {3.8779572904506664`*^9, 3.877957294470707*^9}, {3.8779588958740816`*^9, 3.8779589007759705`*^9}, { 3.878021717913804*^9, 3.8780217224452744`*^9}},ExpressionUUID->"c686ef4f-e025-4011-a893-\ b6bb1abf457f"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"38f195cd-9bd1-4b92-aa24-4b496d952422"], Cell[TextData[{ "Finalmente, fazemos ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["F", "0"], "=", "0.35"}], TraditionalForm]],ExpressionUUID-> "f84f7a77-2b76-4aad-815e-090099a19a7b"], " e ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["t", "max"], "=", "3000"}], TraditionalForm]],ExpressionUUID-> "faee1f39-52f4-41b2-a603-68f1960b917d"], ":" }], "Text", CellChangeTimes->{{3.845617539037848*^9, 3.845617584430943*^9}, { 3.845617745723625*^9, 3.845617774703958*^9}, {3.845618831949589*^9, 3.845618832725987*^9}, {3.847461893893098*^9, 3.8474619256401777`*^9}},ExpressionUUID->"cb564d4f-3f10-4263-a901-\ b04efe7c6169"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"sol", "=", RowBox[{"resolve", "[", RowBox[{"0.35", ",", "3000"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"espfase", "[", RowBox[{"2900", ",", "3000"}], "]"}], " "}]}], "Input", CellChangeTimes->{{3.84746196297059*^9, 3.8474619714535093`*^9}, 3.8478168512003183`*^9},ExpressionUUID->"d22f2031-3949-4736-9acb-\ b3e270783b8a"], Cell["\<\ Agora n\[ATilde]o parece haver qualquer sinal de movimento peri\[OAcute]dico. \ Aparentemente estamos vendo uma rota para o caos atrav\[EAcute]s de bifurca\ \[CCedilla]\[OTilde]es de per\[IAcute]odo, da mesma forma como no mapa log\ \[IAcute]stico.\ \>", "Text", CellChangeTimes->{{3.845618098064678*^9, 3.845618131431262*^9}, { 3.845618189423283*^9, 3.845618282497304*^9}, {3.845618415641378*^9, 3.84561842505789*^9}, {3.845618475065123*^9, 3.84561855281361*^9}, { 3.845618631745582*^9, 3.845618631751053*^9}, {3.846058224299605*^9, 3.8460582598552523`*^9}, {3.8474619976975307`*^9, 3.847462089204973*^9}},ExpressionUUID->"263eb1e5-a1fa-41b4-bb67-\ 201d848f0c9d"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"abddd09e-d100-4f15-8e69-35fd58b8982c"], Cell["A se\[CCedilla]\[ATilde]o de Poincar\[EAcute]", "Section", CellChangeTimes->{{3.847462205115141*^9, 3.847462211079854*^9}},ExpressionUUID->"adb2b4c6-a5ea-45b6-b241-\ 1e62c1c24b4c"], Cell[TextData[{ "Para acompanharmos melhor as duplica\[CCedilla]\[OTilde]es de \ per\[IAcute]odo, que parecem estar vinculadas a m\[UAcute]ltiplos de ", Cell[BoxData[ FormBox[ RowBox[{"2", RowBox[{"\[Pi]", "/", "\[Omega]"}]}], TraditionalForm]],ExpressionUUID-> "061d121b-c217-4723-a5a3-4da4c983b7d9"], ", vamos restringir a visualiza\[CCedilla]\[ATilde]o do espa\[CCedilla]o de \ fase a esses m\[UAcute]ltiplos (depois de descartado o transiente), em vez de \ acompanhar a trajet\[OAcute]ria completa. Essa restri\[CCedilla]\[ATilde]o \ define o que se chama de se\[CCedilla]\[ATilde]o ou mapa de Poincar\[EAcute]. \ Em analogia com o que vimos no mapa de H\[EAcute]non, uma \[OAcute]rbita de \ per\[IAcute]odo ", Cell[BoxData[ FormBox[ RowBox[{"2", RowBox[{"\[Pi]", "/", "\[Omega]"}]}], TraditionalForm]],ExpressionUUID-> "8ce3b795-53a1-44ee-aece-b91b24f70349"], " deve aparecer como um \[UAcute]nico ponto, uma \[OAcute]rbita de per\ \[IAcute]odo ", Cell[BoxData[ FormBox[ RowBox[{"4", RowBox[{"\[Pi]", "/", "\[Omega]"}]}], TraditionalForm]],ExpressionUUID-> "e92d8109-0a52-49a4-a582-de07ec1f6aee"], " deve aparecer como dois pontos etc." }], "Text", CellChangeTimes->{{3.8456188067840443`*^9, 3.8456188261106997`*^9}, 3.8456190619743223`*^9, {3.846058282417883*^9, 3.846058285208626*^9}, { 3.847462121707608*^9, 3.847462197475457*^9}, {3.847462227648553*^9, 3.847462423230823*^9}, {3.9097746329641685`*^9, 3.909774635460357*^9}},ExpressionUUID->"22a450a0-a896-4649-b9a9-\ e4956d322ece"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"c4aa0b17-8fca-488c-abec-805518d95f3c"], Cell[TextData[{ "Para produzir o mapa de Poincar\[EAcute], vamos resolver iterativamente a \ equa\[CCedilla]\[ATilde]o de movimento para ", Cell[BoxData[ FormBox["n", TraditionalForm]],ExpressionUUID-> "13fecb88-28b6-4d49-8c72-cb69ea2a68b2"], " intervalos de tempo de dura\[CCedilla]\[ATilde]o igual ao per\[IAcute]odo \ b\[AAcute]sico ", Cell[BoxData[ FormBox[ RowBox[{"T", "=", RowBox[{"2", RowBox[{"\[Pi]", "/", "\[Omega]"}]}]}], TraditionalForm]],ExpressionUUID-> "86771d77-e811-476e-bca7-b81c0cd93e1a"], ", partindo em ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", "0"}], TraditionalForm]],ExpressionUUID-> "a86b6a75-0444-4679-88dd-c8eb40b269fa"], " da condi\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"x", ",", "v"}], ")"}], "=", RowBox[{"(", RowBox[{"0", ",", "0"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "86dc864b-a63f-4340-b251-7b43c2088557"], " e redefinindo a cada intervalo a condi\[CCedilla]\[ATilde]o inicial como o \ resultado para ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "v"}], ")"}], TraditionalForm]],ExpressionUUID-> "d31e929b-e17d-4e66-b49b-7ab8f8c46b7c"], " no final do intervalo anterior. Os valores de ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "v"}], ")"}], TraditionalForm]],ExpressionUUID-> "1c2c8a4b-6e46-4f84-8027-d1a4e8677031"], " ao final dos v\[AAcute]rios intervalos v\[ATilde]o ser armazenados em uma \ lista, ap\[OAcute]s descartarmos os primeiros ", Cell[BoxData[ FormBox[ SubscriptBox["n", "desc"], TraditionalForm]],ExpressionUUID-> "d8b38dad-14ce-44e5-b44e-31910c4813f4"], " pontos (o que esperamos elimine o transiente). Vamos aproveitar para \ introduzir um elemento de programa\[CCedilla]\[ATilde]o muito importante: a \ fun\[CCedilla]\[ATilde]o condicional ", ButtonBox["If", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/If.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/If.html"], "." }], "Text", CellChangeTimes->{{3.847462553181836*^9, 3.847462887847295*^9}, { 3.847462987894732*^9, 3.847463029474605*^9}, {3.84746306718519*^9, 3.847463096862213*^9}, {3.8474631597949142`*^9, 3.8474631598011417`*^9}, { 3.8474659832306547`*^9, 3.8474659880076523`*^9}, {3.847466269560541*^9, 3.847466272889859*^9}, {3.9097745271775966`*^9, 3.9097745374211946`*^9}},ExpressionUUID->"9dae03b9-2ab0-4694-96ae-\ 70df01e83e26"], Cell[BoxData[ RowBox[{ RowBox[{"poincare", "[", RowBox[{"F0_", ",", "n_", ",", "ndesc_", ",", "tam_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"T", ",", "x0", ",", "v0", ",", "ponto", ",", "lista"}], "}"}], ",", 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Poder\[IAcute]amos ter acrescentado \[AGrave] \ fun\[CCedilla]\[ATilde]o ", StyleBox["If", FontFamily->"Source Code Pro", FontWeight->"Regular"], " um terceiro argumento, que seria uma instru\[CCedilla]\[ATilde]o executada \ caso a condi\[CCedilla]\[ATilde]o fosse falsa." }], "ItemParagraph", CellChangeTimes->{{3.847465927963985*^9, 3.8474659568195143`*^9}, { 3.84746600062197*^9, 3.847466021282174*^9}, {3.8474663092379704`*^9, 3.847466335057177*^9}, {3.847466394548321*^9, 3.8474663960659847`*^9}},ExpressionUUID->"a36abcc3-c9c9-47a8-9c5f-\ d7b590868d0f"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"26efd451-f2a1-4c65-9916-cc2c5ae942a3"], Cell[TextData[{ "Vejamos o mapa para um valor de ", Cell[BoxData[ FormBox[ SubscriptBox["F", "0"], TraditionalForm]],ExpressionUUID-> "2101bfce-334d-483e-8944-5a4374f50b41"], " com per\[IAcute]odo ", Cell[BoxData[ FormBox["T", TraditionalForm]],ExpressionUUID-> "c03504f0-128f-4fe2-afb3-f095f9940112"], ":" }], "Text", CellChangeTimes->{{3.84747205173487*^9, 3.847472074717716*^9}},ExpressionUUID->"1a1f9c1c-ede0-423a-9ae1-\ cba6e5718a7d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"poincare", "[", RowBox[{"0.1", ",", RowBox[{"1024", "+", "512"}], ",", "512", ",", "0.02"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.847464576121499*^9, 3.8474645943145847`*^9}, { 3.8474647612755013`*^9, 3.847464829673888*^9}, {3.8474650022214327`*^9, 3.8474650083422403`*^9}, {3.847465158555451*^9, 3.847465162488944*^9}, { 3.84746522908788*^9, 3.847465229125578*^9}, {3.847465266076816*^9, 3.8474652758891*^9}, {3.847465329168455*^9, 3.847465329562769*^9}, { 3.8474653678133802`*^9, 3.847465368240567*^9}, {3.847465494139826*^9, 3.8474654945453043`*^9}, {3.847465681295474*^9, 3.8474656859866037`*^9}, { 3.8474658356237707`*^9, 3.8474658631372547`*^9}, 3.8478168454420156`*^9},ExpressionUUID->"5f040bdd-6bdd-4732-ad06-\ cc48b7e30290"], Cell[TextData[{ "Agora o mapa para um valor de ", Cell[BoxData[ FormBox[ SubscriptBox["F", "0"], TraditionalForm]],ExpressionUUID-> "3a5533ab-b324-47dd-81f4-0c020d7aa3aa"], " com per\[IAcute]odo ", Cell[BoxData[ FormBox[ RowBox[{"2", "T"}], TraditionalForm]],ExpressionUUID-> "f400d7ca-1aa4-4e1c-9bfc-9115ca0b7800"], ":" }], "Text", CellChangeTimes->{{3.8474720877261143`*^9, 3.8474720949133797`*^9}},ExpressionUUID->"ff82d03e-d658-415e-bea9-\ c0f9b2a66d50"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"poincare", "[", RowBox[{"0.32", ",", RowBox[{"1024", "+", "512"}], ",", "512", ",", "0.02"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.847466171472533*^9, 3.847466171683889*^9}, { 3.847466203454846*^9, 3.847466203556706*^9}, 3.847816919635035*^9},ExpressionUUID->"a4045bfc-9ec3-40ca-99d7-\ 616d4898dd53"], Cell[CellGroupData[{ Cell[TextData[{ "Agora o mapa para um valor de ", Cell[BoxData[ FormBox[ SubscriptBox["F", "0"], TraditionalForm]],ExpressionUUID-> "508eaa51-9547-4263-bb8c-6c78584c4348"], " com per\[IAcute]odo ", Cell[BoxData[ FormBox[ RowBox[{"4", "T"}], TraditionalForm]],ExpressionUUID-> "c2c28704-35c0-4c50-80a4-935f4f7180f5"], ":" }], "Text", CellChangeTimes->{{3.847472110574939*^9, 3.847472121407071*^9}},ExpressionUUID->"131133fd-6833-41ae-82c0-\ 0035be1c7b24"], Cell[BoxData[ RowBox[{ RowBox[{"poincare", "[", RowBox[{"0.338", ",", RowBox[{"1024", "+", "512"}], ",", "512", ",", "0.02"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.847466130735868*^9, 3.847466132736714*^9}, 3.8478169204446363`*^9},ExpressionUUID->"c356a485-bc33-4774-82cc-\ fd3a5a67cc66"] }, Open ]] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ab4fb9d2-51c4-4d13-8137-72dfeb06a470"], Cell[TextData[{ "Da nossa experi\[EHat]ncia com o mapa de H\[EAcute]non, esperamos que, na \ aus\[EHat]ncia de movimento peri\[OAcute]dico, observemos um atrator \ estranho. Vamos conferir adotando ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["F", "0"], "=", "0.35"}], TraditionalForm]],ExpressionUUID-> "e09f1a75-5a00-4b61-b2a5-1be451588bb8"], ":" }], "Text", CellChangeTimes->{{3.847472133758671*^9, 3.8474721775767527`*^9}},ExpressionUUID->"8c8fef9e-2d4b-4608-a7c2-\ 8d787b3d1101"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"poincare", "[", RowBox[{"0.35", ",", RowBox[{ RowBox[{"2", "^", "14"}], "+", RowBox[{"2", "^", "13"}]}], ",", RowBox[{"2", "^", "13"}], ",", "0.001"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Demora", " ", "cerca", " ", "de", " ", "1", " ", "min"}], " ", "*)"}], " "}]], "Input", CellChangeTimes->{{3.847472184170257*^9, 3.8474721902558804`*^9}, { 3.847472225144916*^9, 3.847472246888817*^9}, {3.8474725053665953`*^9, 3.847472505870007*^9}, {3.847473157905168*^9, 3.847473158966305*^9}, { 3.8474737471784363`*^9, 3.847473756899522*^9}, 3.8478168438548594`*^9},ExpressionUUID->"ef4da799-04e4-4c63-a8ff-\ 5526e809d874"], Cell[TextData[{ "Assim como no atrator estranho do mapa de H\[EAcute]non, o atrator acima \ ocupa uma \[AAcute]rea finita do espa\[CCedilla]o de fase, embora sua \ estrutura seja fractal, como podemos verificar ampliando-o repetidamente. \ Vamos nos concentrar na regi\[ATilde]o pr\[OAcute]xima de ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"x", ",", "v"}], ")"}], "=", RowBox[{"(", RowBox[{"1.0", ",", "0.5"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "807a499e-8fc4-48a2-aa56-7c45d68a8e26"], ". Os c\[AAcute]lculos associados \[AGrave]s figuras requerem um longo \ tempo, de modo que vamos import\[AAcute]-las de arquivos gerados a partir de \ instru\[CCedilla]\[OTilde]es que aparecem comentadas abaixo. (Compilar as fun\ \[CCedilla]\[OTilde]es neste caso espec\[IAcute]fico n\[ATilde]o diminui \ consideravelmente o tempo de execu\[CCedilla]\[ATilde]o. A fun\[CCedilla]\ \[ATilde]o NDSolve j\[AAcute] deve ser executada da forma mais eficiente.)" }], "Text", CellChangeTimes->{{3.847472330443377*^9, 3.847472448759413*^9}, { 3.847524652987196*^9, 3.847524716909968*^9}, {3.8477191580606327`*^9, 3.8477191777482777`*^9}},ExpressionUUID->"7040ec7b-d6be-4ed9-9d42-\ fa9167217577"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Show", "[", RowBox[{ RowBox[{"poincare", "[", RowBox[{"0.35", ",", RowBox[{ RowBox[{"2", "^", "16"}], "+", RowBox[{"2", "^", "11"}]}], ",", RowBox[{"2", "^", "11"}], ",", "0.001"}], "]"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.8", ",", "1.05"}], "}"}], ",", RowBox[{"{", RowBox[{"0.35", ",", "0.63"}], "}"}]}], "}"}]}]}], "]"}], " ", "*)"}], " ", RowBox[{"(*", " ", RowBox[{ "O", " ", "c\[AAcute]lculo", " ", "requer", " ", "v\[AAcute]rios", " ", "minutos"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"SetDirectory", "[", RowBox[{"NotebookDirectory", "[", "]"}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"Import", "[", "\"\\"", "]"}]}]}]], "Input", CellChangeTimes->{{3.8474724891834183`*^9, 3.8474725436881933`*^9}, { 3.847473741180416*^9, 3.847473742613126*^9}, {3.8475246390024023`*^9, 3.8475246503212957`*^9}, {3.8475247768860407`*^9, 3.847524826587229*^9}, { 3.847869269391364*^9, 3.847869277784021*^9}},ExpressionUUID->"9ab95563-558e-481e-9308-\ 2e31d73899f5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Show", "[", RowBox[{ RowBox[{"poincare", "[", RowBox[{"0.35", ",", RowBox[{ RowBox[{"2", "^", "20"}], "+", RowBox[{"2", "^", "11"}]}], ",", RowBox[{"2", "^", "11"}], ",", "0.001"}], "]"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.86", ",", "0.91"}], "}"}], ",", RowBox[{"{", RowBox[{"0.48", ",", "0.58"}], "}"}]}], "}"}]}]}], "]"}], " ", "*)"}], " ", RowBox[{"(*", " ", RowBox[{ "O", " ", "c\[AAcute]lculo", " ", "requer", " ", "v\[AAcute]rias", " ", "horas"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Import", "[", "\"\\"", "]"}]}]], "Input", CellChangeTimes->{{3.84752478244307*^9, 3.847524785603891*^9}, { 3.847524834575094*^9, 3.84752484162845*^9}, {3.847524907128676*^9, 3.847524926487636*^9}, {3.84786929076635*^9, 3.8478692910476503`*^9}},ExpressionUUID->"b8e51c0d-8c84-45a7-b17f-\ 3e7a2baacecf"], Cell["Note a invari\[AHat]ncia de escala, caracter\[IAcute]stica de um \ fractal.", "ItemParagraph", CellChangeTimes->{{3.847524969800729*^9, 3.847524994212336*^9}},ExpressionUUID->"0931d849-c553-44bf-8bdc-\ 23d3a7a239d4"] }, Closed]] }, Closed]] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"be0abe2e-7ea9-4155-b26d-ec86b9b79f52"], Cell["O diagrama de bifurca\[CCedilla]\[OTilde]es", "Section", CellChangeTimes->{{3.847525113915868*^9, 3.8475251193387737`*^9}},ExpressionUUID->"d617e52d-6810-4f65-91c7-\ 97e36cac5c33"], Cell[TextData[{ "Da mesma maneira que nos mapas log\[IAcute]stico e de H\[EAcute]non, \ podemos produzir um diagrama de bifurca\[CCedilla]\[OTilde]es, mostrando \ agora as coordenadas ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "37610454-318c-4c89-a08b-7f14a2840914"], " dos pontos nos atratores correspondentes a diversos valores da amplitude \ ", Cell[BoxData[ FormBox[ SubscriptBox["F", "0"], TraditionalForm]],ExpressionUUID-> "c5b30c4c-c50a-4f47-a094-4a765fb206dd"], " da for\[CCedilla]a externa. Vamos adotar um procedimento an\[AAcute]logo \ ao do mapa de H\[EAcute]non: primeiro, criamos uma fun\[CCedilla]\[ATilde]o \ que produz, dado um valor de ", Cell[BoxData[ FormBox[ SubscriptBox["F", "0"], TraditionalForm]],ExpressionUUID-> "d843a249-3b2d-4c53-b853-a26ad233b9e2"], ", uma lista contendo os \[UAcute]ltimos ", Cell[BoxData[ FormBox[ RowBox[{"n", "-", SubscriptBox["n", "desc"]}], TraditionalForm]],ExpressionUUID-> "555e82a8-39fe-4623-b9ba-a6e8c939729e"], " entre ", Cell[BoxData[ FormBox["n", TraditionalForm]],ExpressionUUID-> "66ec6c0d-2510-4838-b361-45b88f69fb33"], " pontos na se\[CCedilla]\[ATilde]o de Poincar\[EAcute]; em seguida, criamos \ uma segunda fun\[CCedilla]\[ATilde]o que varre os valores de ", Cell[BoxData[ FormBox[ SubscriptBox["F", "0"], TraditionalForm]],ExpressionUUID-> "0b76f229-cdfc-4e03-9e0c-e6b2fdd83ce6"], ", invocando a primeira fun\[CCedilla]\[ATilde]o e acrescentando os \ resultados a uma grande lista; finalmente, utilizamos a grande lista na produ\ \[CCedilla]\[ATilde]o do diagrama. Vamos deixar a posi\[CCedilla]\[ATilde]o e \ a velocidade iniciais como argumentos das fun\[CCedilla]\[OTilde]es, para \ possibilitar a produ\[CCedilla]\[ATilde]o de um diagrama mais \ \[OpenCurlyDoubleQuote]suave\[CloseCurlyDoubleQuote], evitando que os pontos \ saltem de um atrator com ", Cell[BoxData[ FormBox[ RowBox[{"x", ">", "0"}], TraditionalForm]],ExpressionUUID-> "9f8ef809-bcbe-4eae-b58f-25f940329024"], " para outro com ", Cell[BoxData[ FormBox[ RowBox[{"x", "<", "0"}], TraditionalForm]],ExpressionUUID-> "62b9e055-6c6d-4c7f-80cc-f1529b0dd145"], "." }], "Text", CellChangeTimes->{{3.846494317138795*^9, 3.846494328072805*^9}, { 3.84649841724865*^9, 3.846498499091172*^9}, {3.847525239493621*^9, 3.847525369543956*^9}, {3.8475265581364822`*^9, 3.847526622871665*^9}, { 3.847526674395499*^9, 3.847526747317521*^9}, {3.847721585146187*^9, 3.84772158620277*^9}, {3.847721676925077*^9, 3.847721792788715*^9}, { 3.8477218506150923`*^9, 3.847721866420553*^9}, {3.8478693340377693`*^9, 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