(* 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[ 71354, 1812] NotebookOptionsPosition[ 57369, 1586] NotebookOutlinePosition[ 61552, 1670] CellTagsIndexPosition[ 61473, 1665] 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}, { 3.875039850550713*^9, 3.875039850633194*^9}, {3.9080062505517435`*^9, 3.908006250594734*^9}},ExpressionUUID->"92508663-47f1-4deb-9615-\ d2c271e8e453"], Cell["Aula 10: mais sobre fun\[CCedilla]\[OTilde]es; o mapa log\[IAcute]stico \ (parte 1)", "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}, {3.875039852656836*^9, 3.875039852836646*^9}, { 3.908006254069874*^9, 3.9080062542134027`*^9}},ExpressionUUID->"14f8f51b-58e9-4033-aea1-\ fd611ac95bb9"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ee3ea2a1-973b-4e9a-a704-0d8a3c30d1be"], Cell["Iterando fun\[CCedilla]\[OTilde]es", "Section", CellChangeTimes->{ 3.699269664116198*^9, 3.70454748091461*^9, {3.704547516075143*^9, 3.7045475190812902`*^9}, {3.706294276733098*^9, 3.706294304716371*^9}, { 3.839323324298633*^9, 3.839323331059084*^9}, 3.840031288455207*^9, { 3.840714636688951*^9, 3.840714639748958*^9}, {3.841057130911109*^9, 3.84105713513932*^9}, {3.841480515514193*^9, 3.84148051572683*^9}, { 3.84226221732572*^9, 3.842262223457004*^9}, {3.843294513408573*^9, 3.843294517052754*^9}, {3.8438990711047573`*^9, 3.843899073171992*^9}, { 3.845103689939662*^9, 3.845103693111958*^9}},ExpressionUUID->"7da6da63-6378-4f26-b58d-\ d42a7250afa4"], Cell[TextData[{ "Em certas aplica\[CCedilla]\[OTilde]es (por exemplo, na \[AAcute]rea de \ sistemas din\[AHat]micos), queremos aplicar iterativamente uma fun\[CCedilla]\ \[ATilde]o, ou seja, queremos calcular, ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", RowBox[{"f", "(", "x", ")"}], ")"}], TraditionalForm]],ExpressionUUID-> "7d71b2fb-8677-4411-bf8e-41baa817ec01"], " ou ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", RowBox[{"f", "(", RowBox[{"f", "(", "x", ")"}], ")"}], ")"}], TraditionalForm]], ExpressionUUID->"4d7abace-6f1f-4b74-880c-598d0a9db929"], " etc. Algumas fun\[CCedilla]\[OTilde]es intr\[IAcute]nsecas do Mathematica \ s\[ATilde]o \[UAcute]teis nesse caso. Vamos exemplificar com ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{"1", "/", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}]}], TraditionalForm]],ExpressionUUID-> "9e1747bb-81d8-4c95-ba49-b4bcc8de6683"], ":" }], "Text", CellChangeTimes->{{3.6992696952616243`*^9, 3.6992696957779827`*^9}, { 3.704547416597644*^9, 3.704547418388081*^9}, {3.8393233348613234`*^9, 3.839323366214238*^9}, {3.839323576559081*^9, 3.839323576563138*^9}, 3.840031301280365*^9, {3.840714647849332*^9, 3.840714700814382*^9}, { 3.841480523939232*^9, 3.841480544976879*^9}, {3.841480739096553*^9, 3.841480739636643*^9}, {3.8422622643967133`*^9, 3.8422623532037497`*^9}, { 3.842262439308573*^9, 3.8422624573684273`*^9}, {3.8422649573788548`*^9, 3.8422649619920673`*^9}, {3.842358288105185*^9, 3.8423582908266573`*^9}, { 3.8432945364257298`*^9, 3.8432945825057983`*^9}, {3.843294685610161*^9, 3.843294812070178*^9}, {3.8432948733770227`*^9, 3.843294900533929*^9}, { 3.843294935611968*^9, 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fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Nest", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Nest.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Nest.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " cumpre o mesmo papel, e \[EAcute] mais compacta se quisermos iterar um n\ \[UAcute]mero maior de vezes:" }], "Text", CellChangeTimes->{{3.845104008171089*^9, 3.84510405138155*^9}, { 3.845104675355438*^9, 3.845104675360668*^9}},ExpressionUUID->"f9b40690-9acf-43ad-807f-\ f402e4f6bf5d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Nest", "[", RowBox[{"f", ",", "x", ",", "2"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8451040551720324`*^9, 3.8451040851193657`*^9}, 3.875039917460791*^9},ExpressionUUID->"7a3f2fe8-3a48-4419-87d2-\ 2b370a866d6a"], Cell[BoxData[ RowBox[{ RowBox[{"Nest", "[", RowBox[{"f", ",", "x", ",", "4"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.845104093769361*^9, 3.845104154309443*^9}, { 3.8451042090402822`*^9, 3.845104212091942*^9}, 3.87503991815656*^9},ExpressionUUID->"631170c1-2077-45e0-adcd-\ 5077e3b8162b"] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"0d6bbe14-c226-415e-b1fc-fcf294444606"], Cell[TextData[{ "A fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["NestList", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/NestList.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/NestList.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " cria uma lista a partir da aplica\[CCedilla]\[ATilde]o de uma fun\ \[CCedilla]\[ATilde]o a si mesma:" }], "Text", CellChangeTimes->{{3.843295915233144*^9, 3.8432959210819607`*^9}, { 3.843295955496604*^9, 3.843295958139822*^9}, {3.843296011870647*^9, 3.843296086919487*^9}, {3.843297814136956*^9, 3.843297881700449*^9}, { 3.843584988405768*^9, 3.84358498873169*^9}, 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FontWeight->"Regular"], ", o que pode ser interpretado como o resultado da \ aplica\[CCedilla]\[ATilde]o da f\.08un\[CCedilla]\[ATilde]o ao argumento \ \[OpenCurlyDoubleQuote]zero vez\[CloseCurlyDoubleQuote]." }], "ItemParagraph", CellChangeTimes->{{3.845104845077033*^9, 3.845104868948545*^9}, { 3.9080063364612207`*^9, 3.908006388055548*^9}},ExpressionUUID->"5fbbc8e1-b47e-4124-a136-\ c2049f4518b6"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"17774e00-e7a4-4e88-b623-1d0211c771fc"], Cell[TextData[{ "Vamos produzir listas de aplica\[CCedilla]\[OTilde]es iterativas da fun\ \[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "827487d9-bb98-4174-9a4c-1002f4e52c9e"], " come\[CCedilla]ando de diferentes valores de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "a61a3431-9701-4cf9-8c57-f94a59a0dfc8"], ":" }], "Text", CellChangeTimes->{{3.84329829620815*^9, 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"[", "%", "]"}]}], "Input", CellChangeTimes->{ 3.875042122067703*^9},ExpressionUUID->"73691bd7-f845-4ecb-84de-\ 1793075ba124"], Cell[BoxData[{ RowBox[{ RowBox[{"NestList", "[", RowBox[{"f", ",", "2", ",", "15"}], "]"}], " "}], "\[IndentingNewLine]", RowBox[{"N", "[", "%", "]"}]}], "Input", CellChangeTimes->{ 3.875042124892517*^9},ExpressionUUID->"1dcbfeb4-18f8-4c10-9bc5-\ f40cdd573e59"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"916aafb6-db56-46e9-abc4-ac7b00e07c2a"], Cell[TextData[{ "Note que, para todos os valores iniciais de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "82a07037-c80a-4ddc-8c51-6c35ea08da7b"], ", os elementos das listas produzidas no slide anterior parecem convergir \ para o mesmo valor assint\[OAcute]tico. Podemos isolar esse valor \ diretamente, usando a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["FixedPoint", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/FixedPoint.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/FixedPoint.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.843301813765079*^9, 3.843301933487143*^9}, { 3.843306013438015*^9, 3.843306013442376*^9}, {3.843913245975964*^9, 3.843913279330587*^9}, {3.843913358034128*^9, 3.843913392270213*^9}, { 3.843913424300049*^9, 3.843913454248246*^9}, {3.8439148193655033`*^9, 3.843914888020431*^9}, {3.843915028627599*^9, 3.8439150826496363`*^9}, { 3.843915483106853*^9, 3.843915561900948*^9}, {3.845107706853362*^9, 3.845107774792008*^9}, {3.84510784630085*^9, 3.84510785266914*^9}},ExpressionUUID->"644a62b1-ac31-494c-8dfd-\ 7ed730ae7044"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"FixedPoint", "[", RowBox[{"f", ",", "0.5"}], "]"}], " "}], "\[IndentingNewLine]", RowBox[{"FixedPoint", "[", RowBox[{"f", ",", "1."}], "]"}], "\[IndentingNewLine]", RowBox[{"FixedPoint", "[", RowBox[{"f", ",", "2."}], "]"}]}], "Input", CellChangeTimes->{{3.845107780007164*^9, 3.845107826633581*^9}, { 3.8451078593317423`*^9, 3.845107891759548*^9}, 3.845213037001123*^9, 3.875042177535595*^9},ExpressionUUID->"1c6593b6-a4ff-4904-a614-\ 980ccc04af8b"], Cell[TextData[{ "Note que o segundo argumento de ", StyleBox["FixedPoint", FontFamily->"Source Code Pro", FontWeight->"Regular"], " \[EAcute] o valor inicial de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "13e2a46f-a140-4bd2-9dee-70ebd82fb3d8"], ", que escrevemos como um n\[UAcute]mero de ponto flutuante. Fazemos isso \ porque ", StyleBox["FixedPoint", FontFamily->"Source Code Pro", FontWeight->"Regular"], " funciona comparando os valores consecutivos de uma lista iterativa \ produzida internamente, at\[EAcute] que esses dois valores tenha representa\ \[CCedilla]\[OTilde]es ", StyleBox["exatamente", FontVariations->{"Underline"->True}], " iguais. Isso pode n\[ATilde]o acontecer se o valor inicial de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "31861e02-0235-4ef2-8506-1ba7b65a706a"], " for um n\[UAcute]mero exato, como mostram as listas de fra\[CCedilla]\ \[OTilde]es produzidas no slide anterior. Veremos logo mais como modificar o \ teste padr\[ATilde]o de ", StyleBox["FixedPoint", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", mas para isso precisamos primeiro falar de fun\[CCedilla]\[OTilde]es \ puras." }], "Text", CellChangeTimes->{{3.845108195151259*^9, 3.84510819805413*^9}, { 3.84510828538687*^9, 3.845108521665996*^9}, {3.8451140808905888`*^9, 3.845114127239593*^9}, {3.908006472926062*^9, 3.908006477761489*^9}},ExpressionUUID->"2aea6848-64fa-4279-a348-\ 158596fdcb2b"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"3d840658-e7d8-403c-867c-a584b277195f"], Cell[TextData[{ "Note que o \[OpenCurlyDoubleQuote]ponto fixo\[CloseCurlyDoubleQuote] \ localizado por ", StyleBox["FixedPoint", FontFamily->"Source Code Pro", FontWeight->"Regular"], " \[EAcute] a solu\[CCedilla]\[ATilde]o est\[AAcute]vel da equa\[CCedilla]\ \[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"f", "(", "x", ")"}]}], TraditionalForm]],ExpressionUUID-> "977bb292-0018-4142-a98f-18c53121c2e5"], ": " }], "Text", CellChangeTimes->{{3.84331583893407*^9, 3.843315839839013*^9}, { 3.843315893265614*^9, 3.843315919509251*^9}, {3.843470235013055*^9, 3.8434702350181427`*^9}, {3.84347029756699*^9, 3.843470320381606*^9}, { 3.843917372118309*^9, 3.8439174370747337`*^9}, {3.845109251214815*^9, 3.8451092949585342`*^9}},ExpressionUUID->"ea9daf24-7658-4d4d-9dc9-\ 9882aa98b880"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"sol", "=", RowBox[{"Solve", "[", RowBox[{"x", "\[Equal]", RowBox[{"f", "[", "x", "]"}]}], "]"}]}], " "}], "\[IndentingNewLine]", RowBox[{"N", "[", "%", "]"}]}], "Input", CellChangeTimes->{{3.843315922249659*^9, 3.843315936509557*^9}, 3.8433159918784723`*^9, {3.843470272613654*^9, 3.843470277614743*^9}, { 3.843470336038171*^9, 3.843470420519916*^9}, {3.843555696246269*^9, 3.843555696524178*^9}, 3.843584617719832*^9, {3.8439174411252537`*^9, 3.8439174500242653`*^9}, {3.843917552089672*^9, 3.843917554054543*^9}, 3.8439235711713657`*^9, {3.845109298012319*^9, 3.845109326507537*^9}, { 3.8451094765336933`*^9, 3.8451094769859533`*^9}, 3.8750421862191*^9},ExpressionUUID->"ba8b8c62-f258-46b0-9893-3b8d1d653013"], Cell[TextData[{ "Por \[OpenCurlyDoubleQuote]est\[AAcute]vel\[CloseCurlyDoubleQuote] queremos \ dizer a solu\[CCedilla]\[ATilde]o para a qual tende a itera\[CCedilla]\ \[ATilde]o de ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "d8c29659-dd24-4b76-b698-d243f6cbaec2"], " quando ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "8b994dcb-da7e-4fc5-be52-b68669ea0e63"], " \[EAcute] levemente perturbado, como ilustrado a seguir. " }], "Text", CellChangeTimes->{{3.845109347767686*^9, 3.845109372577834*^9}, { 3.845109412843116*^9, 3.845109455261035*^9}, {3.845109485850288*^9, 3.845109532361389*^9}, 3.84510963341779*^9},ExpressionUUID->"4fff3047-e5a4-4496-ab17-\ 40e750c48f53"], Cell[TextData[{ "Primeiro vamos verificar que se trata efetivamente de pontos fixos, \ aplicando ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "1b5fae33-86d1-46d9-8b55-814a62a4f26f"], " a ambas as solu\[CCedilla]\[OTilde]es:" }], "Text", CellChangeTimes->{{3.845109647649426*^9, 3.845109653037158*^9}, { 3.845109710266732*^9, 3.8451097241376247`*^9}},ExpressionUUID->"4fb690eb-0bf0-4f30-8aa3-\ 5eb866a486d5"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x", "/.", RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}], "//", "Simplify", " ", RowBox[{"(*", " ", RowBox[{ "Aplicando", " ", "\[AGrave]", " ", "primeira", " ", "solu\[CCedilla]\[ATilde]o"}], " ", "*)"}], " "}]], "Input", CellChangeTimes->{{3.845109725856978*^9, 3.845109758272209*^9}, { 3.845109789652779*^9, 3.845109791829975*^9}, 3.87504219183082*^9},ExpressionUUID->"593a9df4-7f86-4d72-a13f-\ cf33ddff599f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x", "/.", RowBox[{"sol", "[", RowBox[{"[", "2", "]"}], "]"}]}], "]"}], "//", "Simplify", RowBox[{"(*", " ", RowBox[{ "Aplicando", " ", "\[AGrave]", " ", "segunda", " ", "solu\[CCedilla]\[ATilde]o"}], " ", "*)"}], " "}]], "Input", CellChangeTimes->{{3.845109776347185*^9, 3.845109800965239*^9}, 3.8750421939741783`*^9},ExpressionUUID->"d1172e66-2e2a-4f35-9d9a-\ e73a2b970d5e"], Cell[TextData[{ "Vamos agora invocar ", StyleBox["NestList", FontFamily->"Source Code Pro", FontWeight->"Regular"], " com ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "550322c7-03ca-4ed9-ad77-d391ea508859"], " come\[CCedilla]ando de um valor ligeiramente diferente de cada uma das \ duas solu\[CCedilla]\[OTilde]es acima:" }], "Text", CellChangeTimes->{{3.845109638931375*^9, 3.845109642046425*^9}, { 3.845109762061431*^9, 3.845109765571394*^9}, {3.8451126645409317`*^9, 3.84511272894768*^9}},ExpressionUUID->"ef2850ea-dbc8-48ad-91be-\ 7475fa5a3c05"], Cell[BoxData[ RowBox[{ RowBox[{"NestList", "[", RowBox[{"f", ",", RowBox[{ RowBox[{"1.01", "*", "x"}], "/.", RowBox[{"sol", "[", RowBox[{"[", "2", "]"}], "]"}]}], ",", "15"}], "]"}], "//", "N", " ", RowBox[{"(*", " ", RowBox[{ "Come\[CCedilla]ando", " ", "da", " ", "segunda", " ", "solu\[CCedilla]\[ATilde]o"}], " ", "*)"}], " "}]], "Input", CellChangeTimes->{{3.8451095440755444`*^9, 3.845109592985124*^9}, { 3.845109657372684*^9, 3.845109664108486*^9}, 3.8750421996631927`*^9},ExpressionUUID->"23a8c309-1313-4924-a2f3-\ 46316edf7aaa"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"NestList", "[", RowBox[{"f", ",", RowBox[{ RowBox[{"1.01", "*", "x"}], "/.", RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}]}], ",", "15"}], "]"}], "//", "N", " ", RowBox[{"(*", " ", RowBox[{ "Come\[CCedilla]ando", " ", "da", " ", "primeira", " ", "solu\[CCedilla]\[ATilde]o"}], " ", "*)"}], " "}]], "Input", CellChangeTimes->{{3.8451096014825697`*^9, 3.845109607425693*^9}, { 3.845109690210479*^9, 3.845109694015209*^9}, 3.8750422018455563`*^9},ExpressionUUID->"2b818d79-c2cd-4eb9-8568-\ 94552bb6ee3a"], Cell["\<\ Note que a segunda solu\[CCedilla]\[ATilde]o retorna para si pr\[OAcute]pria, \ enquanto os elementos da lista produzida partindo da primeira solu\[CCedilla]\ \[ATilde]o tendem \[AGrave] segunda solu\[CCedilla]\[ATilde]o. \ \>", "ItemParagraph", CellChangeTimes->{{3.84510984266881*^9, 3.845109887229961*^9}, { 3.845110108430653*^9, 3.845110167521089*^9}, {3.8451103432183*^9, 3.845110359186602*^9}, 3.8451296939106073`*^9}, CellTags-> "SlideShowHeader",ExpressionUUID->"2184bb19-e2a9-4543-bf3c-eceaa64ba3b6"] }, Closed]] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"4b2b51e2-421f-46eb-b547-af24f6515a50"], Cell[TextData[{ "Para obter um crit\[EAcute]rio mais claro a respeito da estabilidade dos \ pontos fixos de uma fun\[CCedilla]\[ATilde]o de uma vari\[AAcute]vel, vamos \ considerar a expans\[ATilde]o em s\[EAcute]rie da fun\[CCedilla]\[ATilde]o no \ entorno de um ponto fixo ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", SuperscriptBox["x", StyleBox["*", "Code"]]}], TraditionalForm]], InitializationCell->True,ExpressionUUID-> "27f9d3c4-fff2-445f-a561-e817fd3f7304"], ":\[LineSeparator]", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{ RowBox[{ RowBox[{"f", "(", SuperscriptBox["x", StyleBox["*", "Code"]], ")"}], "+", RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", SuperscriptBox["x", StyleBox["*", "Code"]], ")"}], RowBox[{"(", RowBox[{"x", "-", SuperscriptBox["x", StyleBox["*", "Code"]]}], ")"}]}], "+", "\[CenterEllipsis]"}], "=", RowBox[{ SuperscriptBox["x", StyleBox["*", "Code"]], "+", RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", SuperscriptBox["x", StyleBox["*", "Code"]], ")"}], RowBox[{"(", RowBox[{"x", "-", SuperscriptBox["x", StyleBox["*", "Code"]]}], ")"}]}], "+", "\[CenterEllipsis]"}]}]}], ","}], TraditionalForm]],ExpressionUUID-> "3b1b49d9-c9b0-478e-8de6-78d1be478fe1"], "\nem que utilizamos o fato de que ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", StyleBox["*", "Code"]], "=", RowBox[{"f", "(", SuperscriptBox["x", StyleBox["*", "Code"]], ")"}]}], TraditionalForm]],ExpressionUUID-> "8b6ae42c-5b62-4fef-bb49-1e9b31c71b07"], " pela defini\[CCedilla]\[ATilde]o de ponto fixo. " }], "Text", CellChangeTimes->{{3.845129696148446*^9, 3.845129861684162*^9}, { 3.845129969727869*^9, 3.845130019903082*^9}, {3.845130052528892*^9, 3.845130132260807*^9}, {3.8451302072986603`*^9, 3.8451305393413067`*^9}, { 3.8454429303822117`*^9, 3.84544294054116*^9}, {3.9080066324978104`*^9, 3.908006632498811*^9}},ExpressionUUID->"72748277-978a-4a2c-9862-\ 2b039b55204a"], Cell[TextData[{ "Se agora partirmos de uma condi\[CCedilla]\[ATilde]o inicial ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "0"], "=", RowBox[{ SuperscriptBox["x", StyleBox["*", "Code"]], "+", "\[Epsilon]"}]}], TraditionalForm]], ExpressionUUID->"3cebf07d-dc23-4562-9080-ef5626b7fed1"], ", com ", Cell[BoxData[ FormBox[ RowBox[{"\[LeftBracketingBar]", "\[Epsilon]", "\[RightBracketingBar]"}], TraditionalForm]],ExpressionUUID->"8661f1c4-4c08-41f1-b1d5-7b7b8cf04081"], " bem pequeno, e calcularmos ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], "=", RowBox[{"f", "(", SubscriptBox["x", "0"], ")"}]}], TraditionalForm]],ExpressionUUID-> "70ca5874-85eb-4d48-8e0e-bc20cb1a4dfc"], ", temos, at\[EAcute] primeira ordem em ", Cell[BoxData[ FormBox["\[Epsilon]", TraditionalForm]],ExpressionUUID-> "890fa1ac-f4b3-4cbf-b01a-5bf3880ad361"], ",\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["x", "1"], "=", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "*"], "+", RowBox[{"\[Epsilon]", " ", RowBox[{"f", "'"}], RowBox[{"(", SuperscriptBox["x", "*"], ")"}]}]}], " ", "\[DoubleRightArrow]", " ", RowBox[{ SubscriptBox["x", "1"], "-", SuperscriptBox["x", "*"]}]}], "=", RowBox[{"\[Epsilon]", " ", RowBox[{"f", "'"}], RowBox[{"(", SuperscriptBox["x", "*"], ")"}]}]}]}], ","}], TraditionalForm]], ExpressionUUID->"2d912a4c-72a1-4015-9299-a1cdd9042c0e"], "\nde modo que o novo valor de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "fbd5d374-e0d9-49d8-b964-a01238a261b1"], " estar\[AAcute] mais pr\[OAcute]ximo do ponto fixo se ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[LeftBracketingBar]", RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", SuperscriptBox["x", "*"], ")"}]}], "\[RightBracketingBar]"}], "<", "1"}], TraditionalForm]],ExpressionUUID-> "d46c67d9-2b29-4641-a4f2-6094c57743c2"], " e mais distante do ponto fixo se ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[LeftBracketingBar]", RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", SuperscriptBox["x", "*"], ")"}]}], "\[RightBracketingBar]"}], ">", "1"}], TraditionalForm]],ExpressionUUID-> "6c370b61-fa76-4f1c-9dc3-8d61510f346e"], ". No primeiro caso, temos um ponto fixo ", StyleBox["est\[AAcute]vel", FontVariations->{"Underline"->True}], "; no segundo, um ponto fixo ", StyleBox["inst\[AAcute]vel", FontVariations->{"Underline"->True}], "." }], "Text", CellChangeTimes->{{3.845130545523119*^9, 3.8451306509643106`*^9}},ExpressionUUID->"96c96dd5-2725-4563-a617-\ 4910b8ff7f51"], Cell["Aplicando essas ideias \[AGrave] fun\[CCedilla]\[ATilde]o com que temos \ trabalhado:", "Text", CellChangeTimes->{{3.845130693580195*^9, 3.845130705278953*^9}},ExpressionUUID->"aa0398cd-0685-4dcf-89b0-\ f87c8c4d8fdc"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"f", "'"}], "[", "x", "]"}], "/.", " ", RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}]}], "//", "N", " "}]], "Input", CellChangeTimes->{{3.845130744235292*^9, 3.845130787710436*^9}, 3.8750422385454116`*^9},ExpressionUUID->"c1d5c68a-71f5-4dbb-844e-\ 8340e6e2721b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"f", "'"}], "[", "x", "]"}], "/.", " ", RowBox[{"sol", "[", RowBox[{"[", "2", "]"}], "]"}]}], "//", "N", " "}]], "Input", CellChangeTimes->{{3.845130799253715*^9, 3.8451307993060913`*^9}, 3.8750422406992617`*^9},ExpressionUUID->"86257bbe-5af2-409c-894b-\ 20c87769e816"], Cell["\<\ Portanto, confirmamos que o primeiro ponto fixo \[EAcute] inst\[AAcute]vel e \ o segundo, est\[AAcute]vel.\ \>", "ItemParagraph", CellChangeTimes->{{3.8451308093996*^9, 3.8451308309649754`*^9}},ExpressionUUID->"826d113a-bf6d-4b83-a110-\ 194708f91cfb"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"4993fa55-f76d-47bf-8cb6-e843c0756681"], Cell["Fun\[CCedilla]\[OTilde]es puras", "Section", CellChangeTimes->{{3.843928639145109*^9, 3.8439286557500896`*^9}, { 3.843928699894526*^9, 3.843928737327956*^9}, {3.84392925592019*^9, 3.843929282312172*^9}, {3.845109979717162*^9, 3.8451099876942244`*^9}},ExpressionUUID->"e214db19-1496-4d51-a1f7-\ 7969f7227bfd"], Cell[TextData[{ "At\[EAcute] aqui, temos atribu\[IAcute]do nomes \[AGrave]s fun\[CCedilla]\ \[OTilde]es que definimos no Mathematica. No entanto, \[EAcute] \ poss\[IAcute]vel trabalhar com \ \[OpenCurlyDoubleQuote]fun\[CCedilla]\[OTilde]es \ puras\[CloseCurlyDoubleQuote], que n\[ATilde]o precisam receber um nome. Isso \ \[EAcute] feito utilizando ", StyleBox[ButtonBox["Function", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Function.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Function.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", mas h\[AAcute] uma nota\[CCedilla]\[ATilde]o compacta que costuma ser \ utilizada, embora seja menos clara. " }], "Text", CellChangeTimes->{{3.845112210033805*^9, 3.845112320054625*^9}},ExpressionUUID->"af163f70-7fe6-4da5-9b3a-\ 9e266a0dc045"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{"3", "*", RowBox[{"x", "^", "2"}]}]}], "]"}], "[", "1", "]"}], " "}]], "Input", CellChangeTimes->{{3.84511232913906*^9, 3.845112337830056*^9}, 3.8750422501895695`*^9},ExpressionUUID->"813d548f-0a8d-495a-aa75-\ 7eed6b3812bd"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{"3", "*", RowBox[{"x", "^", "2"}]}]}], "]"}], "[", "2", "]"}], " "}]], "Input", CellChangeTimes->{{3.845112347089156*^9, 3.845112347145523*^9}, 3.875042252557522*^9},ExpressionUUID->"34b99095-b00d-459d-bf1b-\ cf3890a8635b"], Cell[TextData[{ "O primeiro argumento de ", StyleBox["Function", FontFamily->"Source Code Pro", FontWeight->"Regular"], " \[EAcute] a vari\[AAcute]vel, e o segundo argumento \[EAcute] a express\ \[ATilde]o que define a fun\[CCedilla]\[ATilde]o. Os \[UAcute]ltimos \ colchetes envolvem o valor da vari\[AAcute]vel em que queremos calcular a fun\ \[CCedilla]\[ATilde]o, e n\[ATilde]o s\[ATilde]o obrigat\[OAcute]rios, como \ veremos adiante." }], "ItemParagraph", CellChangeTimes->{{3.845112577422442*^9, 3.845112621914349*^9}},ExpressionUUID->"a0922392-c3bd-4c15-adde-\ 81577196ade3"], Cell["\<\ Expressas com a nota\[CCedilla]\[ATilde]o compacta, as entradas acima \ correspondem ao seguinte:\ \>", "Text", CellChangeTimes->{{3.845112784512451*^9, 3.8451128191863403`*^9}},ExpressionUUID->"ff139f52-5424-425a-afb9-\ 4f7e69e70490"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", "*", RowBox[{"#", "^", "2"}]}], "&"}], "[", "1", "]"}], " "}]], "Input", CellChangeTimes->{{3.8451128270881844`*^9, 3.845112839142391*^9}, 3.8750422589087296`*^9},ExpressionUUID->"5380252d-40a7-4ad0-99f7-\ 99f99332858b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", "*", RowBox[{"#", "^", "2"}]}], "&"}], "[", "2", "]"}], " "}]], "Input", CellChangeTimes->{{3.845112843072373*^9, 3.845112854363484*^9}, 3.8750422609910755`*^9},ExpressionUUID->"1d36b558-a948-469b-9cf8-\ 02156587573c"], Cell[TextData[{ "A vari\[AAcute]vel \[EAcute] representada por ", StyleBox["#", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", enquanto o final da express\[ATilde]o da fun\[CCedilla]\[ATilde]o \ \[EAcute] indicado por ", StyleBox["&", FontFamily->"Source Code Pro", FontWeight->"Regular"], "." }], "ItemParagraph", CellChangeTimes->{{3.845112874411623*^9, 3.84511289682064*^9}},ExpressionUUID->"f129f533-f164-4a36-bc3b-\ e015389163e4"] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"2265f18e-890e-4b10-8aed-719ffb1d5ccb"], Cell[TextData[{ "Fun\[CCedilla]\[OTilde]es puras de mais de uma vari\[AAcute]vel tamb\ \[EAcute]m podem ser definidas. Vamos ilustrar com uma \ fun\[CCedilla]\[ATilde]o que testa se a diferen\[CCedilla]a entre dois n\ \[UAcute]meros tem m\[OAcute]dulo menor que ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "4"}]], TraditionalForm]],ExpressionUUID-> "c77ca293-9615-431b-8843-6e37051b446f"], ". Desta vez, vamos associar a fun\[CCedilla]\[ATilde]o ao nome ", StyleBox["teste", FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.84393082598948*^9, 3.843930827518505*^9}, { 3.843931302776829*^9, 3.843931329214821*^9}, {3.843931448015914*^9, 3.843931455174438*^9}, {3.8439316485591917`*^9, 3.843931757718684*^9}, { 3.844004150101653*^9, 3.84400426326521*^9}, {3.844004298382189*^9, 3.84400429838899*^9}, 3.8451120753356647`*^9, {3.84511294147281*^9, 3.845112956344846*^9}, {3.845113007926779*^9, 3.845113045771599*^9}, { 3.845113100233882*^9, 3.845113111010621*^9}, {3.845113935552493*^9, 3.845113935821535*^9}, {3.845212836746681*^9, 3.845212839560205*^9}},ExpressionUUID->"5ef29a2a-0330-4988-8c52-\ 173723bc616f"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"teste", "=", RowBox[{"Function", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}], ",", RowBox[{ RowBox[{"Abs", "[", RowBox[{"x", "-", "y"}], "]"}], "<", RowBox[{"10", "^", RowBox[{"-", "4"}]}]}]}], "]"}]}], ";"}], " "}]], "Input", CellChangeTimes->{{3.845112078138433*^9, 3.845112108982484*^9}, 3.845112140927153*^9, {3.8451129770792913`*^9, 3.845112996216112*^9}, { 3.845113119985567*^9, 3.84511312414712*^9}, {3.8451139389714813`*^9, 3.8451139392076674`*^9}, 3.8750422799052916`*^9},ExpressionUUID->"5f1769f1-2421-40d7-83df-\ e3544d3cb860"], Cell[BoxData[ RowBox[{ RowBox[{"teste", "[", RowBox[{"2", ",", "3"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.845113138915901*^9, 3.84511314138656*^9}, 3.875042282382063*^9},ExpressionUUID->"fba4139c-7038-4e7e-9c54-\ c7be9c3be9c1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"teste", "[", RowBox[{ RowBox[{"2", "+", RowBox[{"10", "^", RowBox[{"-", "5"}]}]}], ",", "2"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.845113142920627*^9, 3.845113163674726*^9}, { 3.84511394171264*^9, 3.84511394191819*^9}, 3.875042284224992*^9},ExpressionUUID->"179833b8-b310-4c7d-834b-\ a0ddf904d8ec"], Cell["Em nota\[CCedilla]\[ATilde]o compacta, escrever\[IAcute]amos ", "Text", CellChangeTimes->{{3.845113334381442*^9, 3.845113350546816*^9}},ExpressionUUID->"dbe87c05-46ba-4439-ad38-\ 9cb60748ae24"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"teste2", "=", RowBox[{ RowBox[{ RowBox[{"Abs", "[", RowBox[{"#1", "-", "#2"}], "]"}], "<", RowBox[{"10", "^", RowBox[{"-", "4"}]}]}], "&"}]}], ";"}], " "}]], "Input", CellChangeTimes->{{3.845113398265139*^9, 3.845113434596656*^9}, { 3.845113945862681*^9, 3.845113946676579*^9}, 3.8750423399191103`*^9},ExpressionUUID->"21c809f6-202d-4bc0-9cc1-\ 14d836ac558c"], Cell[TextData[{ "Para uma fun\[CCedilla]\[ATilde]o de muitas vari\[AAcute]veis, o ", StyleBox["n", FontFamily->"Source Code Pro", FontWeight->"Regular"], "-\[EAcute]simo argumento \[EAcute] representado por ", StyleBox["#n", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", e novamente o final da express\[ATilde]o \[EAcute] indicado por ", StyleBox["&", FontFamily->"Source Code Pro", FontWeight->"Regular"], "." }], "ItemParagraph", CellChangeTimes->{{3.8451134775495996`*^9, 3.845113508891385*^9}},ExpressionUUID->"9aea7f0d-ced8-4ce3-94fd-\ 0b16e6147195"], Cell[BoxData[ RowBox[{ RowBox[{"teste2", "[", RowBox[{"2", ",", "3"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8451134376050787`*^9, 3.845113439884218*^9}, 3.87504234484902*^9},ExpressionUUID->"581ea401-2f68-44dd-9835-\ 61f386e88a24"], Cell[BoxData[ RowBox[{ RowBox[{"teste2", "[", RowBox[{ RowBox[{"2", "+", RowBox[{"10", "^", RowBox[{"-", "5"}]}]}], ",", "2"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.845113442138323*^9, 3.845113452125916*^9}, { 3.845113950228649*^9, 3.8451139504684343`*^9}, 3.8750423469296865`*^9},ExpressionUUID->"20b673bc-01bd-4bd8-914b-\ 7fec01be7a6a"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"0e4ed51b-6e4d-4bf7-b9e4-9aecccabb67e"], Cell[TextData[{ "Com uma fun\[CCedilla]\[ATilde]o pura, podemos definir um teste de converg\ \[EHat]ncia menos estrito para ", StyleBox["FixedPoint", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", utilizando a op\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["SameTest", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/SameTest.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/SameTest.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Vamos considerar dois elementos consecutivos \ \[OpenCurlyDoubleQuote]iguais\[CloseCurlyDoubleQuote] se sua diferen\ \[CCedilla]a tiver m\[OAcute]dulo menor que ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "11"}]], TraditionalForm]],ExpressionUUID-> "d7215d03-3087-4cfe-9f5c-f7eec60b1caf"], ":" }], "Text", CellChangeTimes->{{3.845113621246573*^9, 3.8451136267376432`*^9}, { 3.845113897457365*^9, 3.8451139242096233`*^9}, {3.845113964789104*^9, 3.845113987457428*^9}, {3.845114027906417*^9, 3.845114066692401*^9}, { 3.8451141998591423`*^9, 3.845114230674667*^9}, {3.845114766961192*^9, 3.845114768984519*^9}},ExpressionUUID->"9a47be2a-1acc-42b5-b67b-\ 728862666ac2"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FixedPoint", "[", RowBox[{"f", ",", "1", ",", RowBox[{"SameTest", "\[Rule]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Abs", "[", RowBox[{"#1", "-", "#2"}], "]"}], "<", RowBox[{"10", "^", RowBox[{"-", "11"}]}]}], "&"}], ")"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.845114031992301*^9, 3.845114047463401*^9}, { 3.845114171116941*^9, 3.845114174806497*^9}, {3.845446693490127*^9, 3.8454467125163813`*^9}, {3.8454486647837143`*^9, 3.845448664861683*^9}},ExpressionUUID->"a6bc0120-1c5d-4072-b31f-\ 4d59bcea605e"], Cell[BoxData[ RowBox[{ RowBox[{"N", "[", "%", "]"}], " "}]], "Input", CellChangeTimes->{{3.8451141832136183`*^9, 3.84511418626961*^9}, 3.8750423318380833`*^9},ExpressionUUID->"9cb65d73-741c-4552-aec2-\ ba303ab57203"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"40fed88a-8620-47ef-b53c-b50de81b6b11"], Cell["O mapa log\[IAcute]stico", "Section", CellChangeTimes->{{3.844004981476076*^9, 3.844005028444255*^9}, { 3.845114897076255*^9, 3.845114899662863*^9}},ExpressionUUID->"f20b05cf-ded6-4dde-ab38-\ b35759e4ea5b"], Cell[TextData[{ "Com a discuss\[ATilde]o sobre fun\[CCedilla]\[OTilde]es, podemos analisar \ um \[OpenCurlyDoubleQuote]modelo de brinquedo\[CloseCurlyDoubleQuote] que \ pode produzir comportamento ca\[OAcute]tico, um fen\[OHat]meno que \ caracteriza os sistemas complexos (foco do pr\[EHat]mio Nobel de \ f\[IAcute]sica em 2021). Esse modelo \[EAcute] o mapa log\[IAcute]stico, \ definido pela equa\[CCedilla]\[ATilde]o iterativa ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", RowBox[{"n", "+", "1"}]], "=", RowBox[{"f", "(", SubscriptBox["x", "n"], ")"}]}], TraditionalForm]],ExpressionUUID-> "bb0d4dd3-30fc-4487-8a7e-e0c9a42f5e87"], ", em que ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{"4", "\[Lambda]", " ", "x", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}]}], TraditionalForm]],ExpressionUUID-> "efc2eee7-8eac-4faf-9f8f-1841f944f854"], ", sendo ", Cell[BoxData[ FormBox["\[Lambda]", TraditionalForm]],ExpressionUUID-> "6a23a1cb-6ed3-4c77-8109-1f6a725aca52"], " \[EAcute] um par\[AHat]metro que vamos variar no intervalo ", Cell[BoxData[ FormBox[ RowBox[{"0", "<", "\[Lambda]", "<", "1"}], TraditionalForm]], ExpressionUUID->"6e6a759a-c3ab-4055-aaf4-30ddcc19b4df"], ". Nosso interesse \[EAcute] acompanhar o que acontece, dado um valor de \ \[Lambda], com o comportamento de ", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]],ExpressionUUID-> "7a16bbba-a364-490d-8f2f-22717ce33c16"], " para ", Cell[BoxData[ FormBox[ RowBox[{"n", "\[GreaterGreater]", "1"}], TraditionalForm]],ExpressionUUID-> "e7cf079a-60ff-422f-aa9c-a0c8cd3eec71"], ". " }], "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}, {3.845443146440558*^9, 3.8454431469576387`*^9}},ExpressionUUID->"de408365-f028-430c-b5e2-\ b4fa36941087"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", RowBox[{"f", ",", "\[Lambda]"}], "]"}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"f", "[", "\[Lambda]_", "]"}], "[", "x_", "]"}], ":=", RowBox[{"4", "*", "\[Lambda]", "*", "x", "*", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}]}], ";"}]}], "Input", CellChangeTimes->{{3.845124570478941*^9, 3.845124572870122*^9}, { 3.8451276756926003`*^9, 3.8451276915597477`*^9}, {3.845127923386724*^9, 3.84512792386327*^9}, {3.845129061090382*^9, 3.845129063986747*^9}, 3.875042428238556*^9},ExpressionUUID->"43fc7f6f-7580-43ef-b8ca-\ 5b32938b6139"], Cell[TextData[{ "A forma peculiar como definimos ", Cell[BoxData[ FormBox["f", TraditionalForm]],ExpressionUUID-> "b313346d-fbad-4d91-9528-c3c78aae22e0"], ", com colchetes separados para ", Cell[BoxData[ FormBox["\[Lambda]", TraditionalForm]],ExpressionUUID-> "389c93fc-019f-4aef-ad74-e3ea622d2e12"], " e para ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "461cf02b-965e-48cc-bfce-a8f20e546f26"], ", vem do fato de que vamos adiante iter\[AAcute]-la em ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "82fe8dc2-c78e-468c-951f-fbf41d82f30c"], ", mantendo ", Cell[BoxData[ FormBox["\[Lambda]", TraditionalForm]],ExpressionUUID-> "0ae2086e-e1f2-4740-8902-4fc5f54b4fed"], " fixo, e as sintaxes de ", StyleBox["Nest", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", ", StyleBox["NestList", FontFamily->"Source Code Pro", FontWeight->"Regular"], " e ", StyleBox["FixedPoint", FontFamily->"Source Code Pro", FontWeight->"Regular"], " impedem o uso da forma ", StyleBox["f[\[Lambda],x]", FontFamily->"Source Code Pro", FontWeight->"Regular"], ". " }], "ItemParagraph", CellChangeTimes->{{3.84512910462147*^9, 3.845129208109707*^9}, { 3.8750423682979827`*^9, 3.8750424221047983`*^9}, 3.9080070388253083`*^9},ExpressionUUID->"d48b23cf-c126-44e5-9827-\ e319be634a2c"], Cell[TextData[{ "O gr\[AAcute]fico de ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "12e60c81-3131-4884-bebd-036c9806e10e"], " \[EAcute] uma par\[AAcute]bola que se anula em ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "0"}], TraditionalForm]],ExpressionUUID-> "8f169945-ad31-4ec1-ac39-f103a640dbe0"], " e em ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "1"}], TraditionalForm]],ExpressionUUID-> "375d002e-81e9-436a-8647-3add14bd5b70"], ", como exemplificamos abaixo para ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.4"}], TraditionalForm]],ExpressionUUID-> "12302e81-7443-4e84-9d1b-1f7891c44411"], ":" }], "Text", CellChangeTimes->{{3.845127730293683*^9, 3.8451277519266233`*^9}, { 3.845129077226615*^9, 3.8451290905756407`*^9}, 3.845443190506402*^9},ExpressionUUID->"fbd00b5c-034e-4c9e-8bc2-\ 6276778ef8a8"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"optam", "=", RowBox[{"{", RowBox[{ RowBox[{"ImageSize", "\[Rule]", " ", "Large"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "\"\\"", "}"}]}]}], "}"}]}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "0.4", "]"}], "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"x\",FontSlant->\"Italic\"]\)\>\"", ",", "\"\<\!\(\*StyleBox[\"f\",FontSlant->\"Italic\"]\)(\!\(\*StyleBox[\"x\",\ FontSlant->\"Italic\"]\))\>\""}], "}"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.845127753530389*^9, 3.845127831927601*^9}, { 3.845127897528433*^9, 3.8451279112691097`*^9}, {3.845127945954247*^9, 3.845127971657958*^9}, {3.845128130265671*^9, 3.845128154483231*^9}, { 3.8451290721369534`*^9, 3.845129094328497*^9}, {3.8451942156575737`*^9, 3.8451942209421673`*^9}, 3.8750424308015127`*^9},ExpressionUUID->"fa314786-42c9-4014-b396-\ 6144df5f6903"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"8865992e-5aaa-48b7-babe-ceabf43bf2e3"], Cell[TextData[{ "Vamos ver o que acontece com os valores de ", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]],ExpressionUUID-> "ff102c3d-e7ca-44f0-a72b-f35986443a27"], " \[AGrave] medida que iteramos a fun\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "cb916fbc-85a9-480c-a23c-54ec9ea811ef"], " a partir de um valor de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "ede32952-3a9b-4abb-9f01-faaea682bb47"], " diferente de 0 ou 1; por concretude, vamos utilizar ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"1", "/", "2"}]}], TraditionalForm]],ExpressionUUID-> "f9ff6cb9-367b-40ab-bc52-b93738be451c"], ". Primeiro vamos supor um valor bem pequeno de \[Lambda]:" }], "Text", CellChangeTimes->{{3.844070009958798*^9, 3.84407008074306*^9}, { 3.845128009746258*^9, 3.84512807508554*^9}, {3.845128189048266*^9, 3.8451281974185266`*^9}, {3.84512947923767*^9, 3.84512949438068*^9}},ExpressionUUID->"9d494f41-a3ae-4c7f-8318-\ 7ba1a29367dc"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"NestList", "[", RowBox[{ RowBox[{"f", "[", "0.1", "]"}], ",", RowBox[{"1", "/", "2"}], ",", "15"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.845128080253821*^9, 3.845128108296287*^9}, { 3.8451292599660683`*^9, 3.8451292812204113`*^9}, {3.845129499316084*^9, 3.845129499893949*^9}, 3.8750424406786084`*^9},ExpressionUUID->"f06aeddc-7039-47de-a8e7-\ e06214798e4a"], Cell[TextData[{ "O resultado parece indicar que ", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]],ExpressionUUID-> "73e49d23-ca8e-48e6-9ac6-8aa79b12d8b8"], " tende ao ponto fixo ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "0"}], TraditionalForm]],ExpressionUUID-> "a40275c2-78ab-45f8-b2f2-d1a43b6c25f9"], ". Como a converg\[EHat]ncia \[EAcute] lenta, vamos aplicar ", StyleBox["FixedPoint", FontFamily->"Source Code Pro", FontWeight->"Normal"], " com um teste menos estrito:" }], "Text", CellChangeTimes->{{3.8451293110375137`*^9, 3.845129379234288*^9}, { 3.845129423357327*^9, 3.845129445609783*^9}},ExpressionUUID->"4460136f-918c-44a5-9735-\ 40444c0ddb42"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"FixedPoint", "[", RowBox[{ RowBox[{"f", "[", "0.1", "]"}], ",", RowBox[{"1", "/", "2"}], ",", RowBox[{"SameTest", "\[Rule]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Abs", "[", RowBox[{"#1", "-", "#2"}], "]"}], "<", RowBox[{"10", "^", RowBox[{"-", "11"}]}]}], "&"}], ")"}]}]}], "]"}], "//", "Chop", " "}]], "Input", CellChangeTimes->{{3.845129508039981*^9, 3.8451295210758953`*^9}, 3.8750424435891647`*^9},ExpressionUUID->"caf2ff82-96f4-4b20-aa71-\ 5a199eef302d"], Cell[TextData[{ "O teste da derivada (em rela\[CCedilla]\[ATilde]o a ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "bb2a4ca0-8e04-4b38-b820-c78d8c21ae86"], ") confirma que se trata de um ponto fixo est\[AAcute]vel:" }], "Text", CellChangeTimes->{{3.8451897512984133`*^9, 3.845189776936459*^9}},ExpressionUUID->"f4d32564-8a9c-4b8f-bc47-\ 9003011e1252"], Cell[BoxData[ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "0.1", "]"}], "[", "x", "]"}], ",", "x"}], "]"}], "/.", RowBox[{"x", "\[Rule]", " ", "0", " "}]}]], "Input", CellChangeTimes->{{3.8451897806908493`*^9, 3.845189845978251*^9}, 3.8750424481349077`*^9},ExpressionUUID->"93894611-3fc3-440f-98c3-\ f48992c3442a"] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"9ce72fc9-f647-41fe-91d6-414b3b141a87"], Cell[TextData[{ "Vamos analisar a estabilidade do ponto fixo ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "0"}], TraditionalForm]],ExpressionUUID-> "edc241f6-6dcb-45f7-98f6-0e8b85ffb7c9"], " para valores arbitr\[AAcute]rios de \[Lambda]:" }], "Text", CellChangeTimes->{{3.844080785935403*^9, 3.8440808511846857`*^9}, { 3.844081051359322*^9, 3.8440810552837553`*^9}, {3.845189946251032*^9, 3.845189994547099*^9}},ExpressionUUID->"a3988b43-0ec7-4560-9c00-\ ea5a75203ee3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], "[", "x", "]"}], ",", "x"}], "]"}], "/.", RowBox[{"x", "\[Rule]", " ", "0"}]}], ")"}], "//", "Simplify", " "}]], "Input", CellChangeTimes->{{3.845190010141101*^9, 3.845190020731094*^9}, 3.87504245703378*^9},ExpressionUUID->"ab18284f-b8df-4725-bbb3-\ a87301ca09c8"], Cell[TextData[{ "Vemos ent\[ATilde]o que esse ponto fixo se torna inst\[AAcute]vel quando ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", ">", RowBox[{"1", "/", "4"}]}], TraditionalForm]],ExpressionUUID-> "4dbd00a3-98c3-4092-ad6d-89faef606bb2"], ". Para melhor acompanhar o fluxo do mapa, nesse caso, vamos produzir gr\ \[AAcute]ficos da itera\[CCedilla]\[ATilde]o para diferentes valores de \ \[Lambda]:" }], "Text", CellChangeTimes->{{3.845190040087376*^9, 3.8451900677742434`*^9}, { 3.8451901059113293`*^9, 3.8451901454769382`*^9}, {3.845190217725511*^9, 3.8451902859677343`*^9}, {3.845190515298428*^9, 3.8451905153020372`*^9}, 3.845195074961197*^9, 3.845196304194281*^9, {3.845448438421937*^9, 3.8454484391973753`*^9}},ExpressionUUID->"b3243aa3-a69b-4a01-8e17-\ 5daded7b8e0d"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"lambda", "=", RowBox[{"{", RowBox[{ "0.1", ",", "0.4", ",", "0.5", ",", "0.6", ",", "0.7", ",", "0.8", ",", "0.88", ",", "0.91"}], "}"}]}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"opgrf", "=", RowBox[{"{", RowBox[{ RowBox[{"Joined", "\[Rule]", RowBox[{"{", RowBox[{"False", ",", "True"}], "}"}]}], ",", RowBox[{"Axes", "\[Rule]", "False"}], ",", RowBox[{"Frame", "\[Rule]", "True"}], ",", RowBox[{"FrameLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"n\",FontSlant->\"Italic\"]\)\>\"", ",", "\"\<\!\(\*StyleBox[SubscriptBox[\"x\", \"n\"],FontSlant->\"Italic\"]\ \)\>\""}], "}"}]}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{"Print", "[", RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"NestList", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"lambda", "[", RowBox[{"[", "i", "]"}], "]"}], "]"}], ",", RowBox[{"1", "/", "2"}], ",", "35"}], "]"}], ",", RowBox[{"NestList", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"lambda", "[", RowBox[{"[", "i", "]"}], "]"}], "]"}], ",", RowBox[{"1", "/", "2"}], ",", "35"}], "]"}]}], "}"}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"\"\<\[Lambda]=\>\"", "<>", RowBox[{"ToString", "[", RowBox[{"lambda", "[", RowBox[{"[", "i", "]"}], "]"}], "]"}]}]}], ",", RowBox[{"Evaluate", "[", "opgrf", "]"}]}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", RowBox[{"Length", "[", "lambda", "]"}]}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.845190518973729*^9, 3.84519063228014*^9}, { 3.845190663140348*^9, 3.8451906992089357`*^9}, {3.845190966026359*^9, 3.8451910523175573`*^9}, {3.845191100442445*^9, 3.845191140284319*^9}, { 3.845193124865024*^9, 3.845193127001524*^9}, {3.845193354570697*^9, 3.8451934496289988`*^9}, 3.845193488718294*^9, {3.845193520274173*^9, 3.845193559386115*^9}, {3.845194034175909*^9, 3.8451940465290527`*^9}, { 3.845194174745172*^9, 3.845194201272317*^9}, {3.845194231293187*^9, 3.84519423539854*^9}, {3.845194332927103*^9, 3.845194434817391*^9}, { 3.845194470092146*^9, 3.8451944773744383`*^9}, {3.8451950792585287`*^9, 3.845195084505178*^9}, {3.845195120483728*^9, 3.8451951495652237`*^9}, { 3.845195368693186*^9, 3.845195455674836*^9}, {3.8451955869234533`*^9, 3.845195588563216*^9}, {3.845195961909034*^9, 3.845196000910573*^9}, { 3.84544738675119*^9, 3.845447387648117*^9}, {3.8454476395764008`*^9, 3.845447640649865*^9}, {3.845447750437708*^9, 3.845447755727961*^9}, { 3.845447845747855*^9, 3.8454478487436657`*^9}, 3.845448184053293*^9, { 3.84544825549638*^9, 3.84544828026046*^9}, 3.875042470690535*^9},ExpressionUUID->"576a716e-9f14-44a0-a9c6-\ be78402c58a1"], Cell[TextData[{ "Note que, quando ", Cell[BoxData[ FormBox["n", TraditionalForm]],ExpressionUUID-> "e01f6b58-b764-435d-a6f0-1c6e949fd111"], " toma valores maiores, para ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.4"}], TraditionalForm]],ExpressionUUID-> "9c6eeadd-737e-4a00-8607-dd1abd187df0"], ", ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.5"}], TraditionalForm]],ExpressionUUID-> "1818387f-741f-43d6-a468-285a42eb28fe"], ", ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.6"}], TraditionalForm]],ExpressionUUID-> "380e6d3e-2fe8-4a52-a192-48220e0dbb9c"], " e ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.7"}], TraditionalForm]],ExpressionUUID-> "88496a83-03eb-4f9a-8051-54883b300f37"], " o mapa gera um ponto fixo, para ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.8"}], TraditionalForm]],ExpressionUUID-> "41752092-9f93-40b1-a591-00ccca71be22"], " h\[AAcute] a altern\[AHat]ncia entre 2 valores de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "76e2c128-f118-416f-b66f-e8852108d783"], ", para ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.88"}], TraditionalForm]],ExpressionUUID-> "a84feed6-4610-4a60-8208-f1a4dc2d5956"], " h\[AAcute] a altern\[AHat]ncia entre 4 valores e para ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.91"}], TraditionalForm]],ExpressionUUID-> "df93487b-5b9f-41d9-8459-6e9b20d4ecd8"], " n\[ATilde]o parece haver um padr\[ATilde]o. 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