(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 79533, 2202] NotebookOptionsPosition[ 65312, 1951] NotebookOutlinePosition[ 68499, 2015] CellTagsIndexPosition[ 68420, 2010] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", "FirstSlide", CellTags-> "SlideShowHeader",ExpressionUUID->"9c5761b0-3686-48c1-8154-19410c1e62b8"], Cell["Introdu\[CCedilla]\[ATilde]o \[AGrave] F\[IAcute]sica Computacional I \ (2023.2)", "Title", CellChangeTimes->{ 3.9086270022129*^9},ExpressionUUID->"99685dfb-5458-4006-812e-865118742b8b"], Cell["Aula 11: o mapa log\[IAcute]stico (parte 2)", "Subtitle", CellChangeTimes->{ 3.908627033296776*^9},ExpressionUUID->"3fe80e06-7cea-4539-ad21-\ 2279761c61e2"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"fd05f88e-12a3-4c0f-94b5-09f4827276cb"], Cell[CellGroupData[{ Cell["O mapa log\[IAcute]stico", "Section", CellChangeTimes->{{3.9086271077088933`*^9, 3.90862711030834*^9}},ExpressionUUID->"dbcdbb78-32c4-4305-987f-\ 2cc5531c3098"], Cell[TextData[{ "Na aula passada, come\[CCedilla]amos a investigar 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-> "529b36fc-f4fd-4dd5-a342-61e2f610f035"], ", em que ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{"4", "\[Lambda]", " ", "x", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}]}], TraditionalForm]],ExpressionUUID-> "162aba92-a6e9-4bc5-891c-7d9d8b530e5c"], "." }], "Text", CellChangeTimes->{ 3.908627123425235*^9},ExpressionUUID->"3f1902cb-91e5-4264-94a8-\ 7e11e9ec9963"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"f", "[", "\[Lambda]_", "]"}], "[", "x_", "]"}], ":=", RowBox[{"4", "*", "\[Lambda]", "*", "x", "*", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " "}]}]], "Input", InitializationCell->True, CellLabel->"In[2]:=",ExpressionUUID->"22f2944f-9122-403e-b51a-6f1e66f88d99"], Cell[TextData[{ "Conclu\[IAcute]mos a aula mostrando o comportamento do mapa ap\[OAcute]s \ muitas aplica\[CCedilla]\[OTilde]es, para diversos valores do par\[AHat]metro \ \[Lambda]. 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O esquema \ a seguir implementa uma fun\[CCedilla]\[ATilde]o que produz a visualiza\ \[CCedilla]\[ATilde]o do mapa, para um par\[AHat]metro ", Cell[BoxData[ FormBox["\[Lambda]", TraditionalForm]],ExpressionUUID-> "f1c0754f-8f03-48f5-9145-92bdf3002c2c"], ", come\[CCedilla]ando de um certo valor inicial ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", SubscriptBox["x", "0"]}], TraditionalForm]],ExpressionUUID-> "cb1a63d3-90b3-49ae-a45d-60b699480823"], ". A visualiza\[CCedilla]\[ATilde]o tem por pano de fundo gr\[AAcute]ficos \ de ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "37d29147-b2da-460a-a5df-a41294b5dee3"], " e da fun\[CCedilla]\[ATilde]o identidade, e \[EAcute] produzida a partir \ de uma lista que une consecutivamente os pontos de coordenadas ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", "0"}], ")"}], TraditionalForm]], ExpressionUUID->"c0388bc6-6309-4bff-9bdc-e64c41f3014c"], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["x", "1"]}], ")"}], TraditionalForm]],ExpressionUUID-> "9cac2476-cd20-4df6-bd67-60a52be0295b"], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "1"], ",", SubscriptBox["x", "1"]}], ")"}], TraditionalForm]],ExpressionUUID-> "4ecc7c68-bd65-4c39-b467-ed50619a9ae6"], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "1"], ",", SubscriptBox["x", "2"]}], ")"}], TraditionalForm]],ExpressionUUID-> "af430fcb-653e-4a9f-864f-992e611537fc"], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "2"], ",", SubscriptBox["x", "2"]}], ")"}], TraditionalForm]],ExpressionUUID-> "9b520084-83d5-4256-8488-d30da630e63c"], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "2"], ",", SubscriptBox["x", "3"]}], ")"}], TraditionalForm]],ExpressionUUID-> "d2b7ff65-646f-4925-aab0-710b95f03e3c"], " etc., com ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "i"], "=", RowBox[{"f", "(", SubscriptBox["x", RowBox[{"i", "-", "1"}]], ")"}]}], TraditionalForm]],ExpressionUUID-> "f71da0ee-3fa5-42b0-93c4-4b6b5c1feda4"], ". A lista \[EAcute] constru\[IAcute]da com o aux\[IAcute]lio da fun\ \[CCedilla]\[ATilde]o ", StyleBox["Do", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", tendo como primeiro argumento um conjunto de instru\[CCedilla]\[OTilde]es \ separadas por ponto e v\[IAcute]rgula e contidas coletivamente entre par\ \[EHat]nteses. Note o uso da fun\[CCedilla]\[ATilde]o ", StyleBox["AppendTo", FontFamily->"Source Code Pro", FontWeight->"Regular"], " para acrescentar elementos a uma lista, bem como os par\[EHat]nteses (com \ fundo colorido avermelhado) que delimitam o conjunto de instru\[CCedilla]\ \[OTilde]es no primeiro argumento de ", StyleBox["Do", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", instru\[CCedilla]\[OTilde]es que devem ser repetidas para cada valor de ", Cell[BoxData[ FormBox["i", TraditionalForm]],ExpressionUUID-> "518a79cb-9b1a-476e-a905-3c02f52567ab"], " entre ", Cell[BoxData[ FormBox["1", TraditionalForm]],ExpressionUUID-> "6adfd097-b7d6-40a3-a282-934474047b29"], " e ", Cell[BoxData[ FormBox["n", TraditionalForm]],ExpressionUUID-> "95498f11-aafc-4297-9ce3-2c565d127798"], ", como definido no terceiro argumento. A pr\[OAcute]pria fun\[CCedilla]\ \[ATilde]o que definimos faz uso de um conjunto de \ instru\[CCedilla]\[OTilde]es contidas entre par\[EHat]nteses (com fundo \ colorido amarelo) e separadas por ponto e v\[IAcute]rgula. Isso \[EAcute] um \ exemplo da programa\[CCedilla]\[ATilde]o de um ", StyleBox["procedimento", FontVariations->{"Underline"->True}], ", ou seja, de uma sequ\[EHat]ncia de instru\[CCedilla]\[OTilde]es. Esse \ \[EAcute] um tipo de programa\[CCedilla]\[ATilde]o bastante comum e presente \ em linguagens como Python, C, C++ e FORTRAN." }], "Text", CellChangeTimes->{ 3.908627507249659*^9, {3.908814561782674*^9, 3.908814566727716*^9}},ExpressionUUID->"5f24f455-72a2-42db-90c7-\ 68a6e794c88d"], Cell[BoxData[ RowBox[{ RowBox[{"grafmapa", "[", RowBox[{"\[Lambda]_", ",", "x0_", ",", "n_"}], "]"}], ":=", " ", "\[IndentingNewLine]", RowBox[{ StyleBox["(", Background->RGBColor[1, 1, 0]], StyleBox["\[IndentingNewLine]", Background->RGBColor[0.87, 0.94, 1]], RowBox[{ RowBox[{"xx", "=", "x0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Definimos", " ", "uma", " ", "vari\[AAcute]vel", " ", "auxiliar", " ", "para", " ", "aplicar", " ", "o", " ", "mapa"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"lista1", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"xx", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"xx", ",", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], "[", "xx", "]"}]}], "}"}]}], "}"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{"Inicializamos", " ", "a", " ", "lista"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Do", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox["(", Background->RGBColor[1, 0.85, 0.85]], " ", RowBox[{"(*", " ", RowBox[{ "Come\[CCedilla]amos", " ", "as", " ", "instru\[CCedilla]\[OTilde]es", " ", "que", " ", "ser\[ATilde]o", " ", "repetidas", " ", "no", " ", "la\[CCedilla]o"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"xx", "=", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], "[", "xx", "]"}]}], ";", RowBox[{"AppendTo", "[", RowBox[{"lista1", ",", RowBox[{"{", RowBox[{"xx", ",", "xx"}], "}"}]}], "]"}], ";", RowBox[{"AppendTo", "[", RowBox[{"lista1", ",", RowBox[{"{", RowBox[{"xx", ",", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], "[", "xx", "]"}]}], "}"}]}], "]"}]}], "\[IndentingNewLine]", StyleBox[")", Background->RGBColor[1, 0.85, 0.85]]}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "n"}], "}"}]}], " ", RowBox[{"(*", " ", RowBox[{ "O", " ", "la\[CCedilla]o", " ", "\[EAcute]", " ", "repetido", " ", "n", " ", "vezes"}], " ", "*)"}], "\[IndentingNewLine]", "]"}], ";", RowBox[{"(*", " ", RowBox[{ RowBox[{ "Aqui", " ", "se", " ", "encerra", " ", "o", " ", "la\[CCedilla]o"}], ";", " ", RowBox[{ "abaixo", " ", "fazemos", " ", "os", " ", "gr\[AAcute]ficos"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"grafico0", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], "[", "x", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"PlotRangePadding", "->", "None"}], ",", RowBox[{"FrameLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"x\",FontSlant->\"Italic\"]\)\>\"", ",", "\"\<\!\(\*StyleBox[\"f\",FontSlant->\"Italic\"]\)(\!\(\*StyleBox[\"\ x\",FontSlant->\"Italic\"]\))\>\""}], "}"}]}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Show", "[", RowBox[{"{", RowBox[{"grafico0", ",", RowBox[{"ListLinePlot", "[", RowBox[{"lista1", ",", RowBox[{"PlotStyle", "\[Rule]", "Red"}]}], "]"}]}], "}"}], "]"}]}], "\[IndentingNewLine]", StyleBox[")", Background->RGBColor[1, 1, 0]]}]}]], "Input", InitializationCell->True, CellChangeTimes->{{3.9086275620729055`*^9, 3.9086275671562047`*^9}, { 3.908823611384212*^9, 3.9088236140511065`*^9}, {3.908823662519965*^9, 3.9088236675492687`*^9}},ExpressionUUID->"a7c55489-1ce5-41dc-84f0-\ a655ba7d703d"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"72d2fb75-e5fa-4640-b70b-787d4e88c7ad"], Cell["Vamos produzir gr\[AAcute]ficos para diversos valores de \[Lambda]:", \ "Text", CellChangeTimes->{ 3.9086275974556246`*^9},ExpressionUUID->"dadf2276-6499-494e-9d42-\ a5b91213566f"], Cell[BoxData[ RowBox[{"grafmapa", "[", RowBox[{"0.7", ",", "0.5", ",", "10"}], "]"}]], "Input", CellLabel->"In[13]:=",ExpressionUUID->"8e49ac6d-f6eb-4524-ad9b-fd0f77029737"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"grafmapa", "[", RowBox[{"0.75", ",", "0.5", ",", "50"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.9088236491536417`*^9, 3.9088236493963118`*^9}},ExpressionUUID->"d1f0b64c-cfb5-4b86-aae8-\ 294a3485193a"], Cell[TextData[{ "Exatamente para ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", RowBox[{"3", "/", "4"}]}], TraditionalForm]],ExpressionUUID-> "98802344-5a2b-4fd8-b2e3-4a4cbb0d7c41"], ", o mapa converge para o ponto fixo muito lentamente. (Note que a curva \ vermelha corresponde a 100 itera\[CCedilla]\[OTilde]es do mapa.)" }], "Text", CellMargins-> Dynamic[{{ 0.185 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}], 0.03 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}]}, { 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}], 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}]}}], CellChangeTimes->{ 3.9086276600530825`*^9},ExpressionUUID->"f7539c52-74d6-48fc-8016-\ ca7bdcf6ba7d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"grafmapa", "[", RowBox[{"0.8", ",", "0.5", ",", "50"}], "]"}]], "Input", CellChangeTimes->{{3.9088236549765477`*^9, 3.908823655183276*^9}},ExpressionUUID->"ea5b7b6f-c247-4391-a021-\ 96a8c9b6b8f4"], Cell[TextData[{ "Para ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.8"}], TraditionalForm]],ExpressionUUID-> "9a214165-9c94-49c7-bd47-07ac5a02345c"], ", a representa\[CCedilla]\[ATilde]o do mapa n\[ATilde]o mais converge para \ um ponto, mas sim para um quadril\[AAcute]tero que representa a \ altern\[AHat]ncia entre dois valores de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "0671ee27-ae89-4b28-9bdf-5c60bb8650fa"], "." }], "Text", CellMargins-> Dynamic[{{ 0.185 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}], 0.03 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}]}, { 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}], 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}]}}], CellChangeTimes->{ 3.9086276809080453`*^9},ExpressionUUID->"e05f6c98-e7b3-4e5d-bd64-\ 9a3b98c4480c"] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ece5ddef-2fd4-4000-80eb-6b5e79943bc4"], Cell[TextData[{ "Quando, ap\[OAcute]s muitas aplica\[CCedilla]\[OTilde]es do mapa, \ h\[AAcute] a altern\[AHat]ncia entre dois valores de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "e65fe3a9-2f5e-49be-8fe6-60217c07a1c7"], ", dizemos que h\[AAcute] um \[OpenCurlyDoubleQuote]ciclo 2\ \[CloseCurlyDoubleQuote] do mapa, ou seja, a cada duas aplica\[CCedilla]\ \[OTilde]es o valor de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "227899f0-29ae-4e46-a9eb-eb041a70e0d4"], " se repete. Essas solu\[CCedilla]\[OTilde]es correspondem ent\[ATilde]o a \ solu\[CCedilla]\[OTilde]es da equa\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"f", "(", RowBox[{"f", "(", "x", ")"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "e6bfaf3d-5a05-4181-a316-29242a5ac46c"], ": " }], "Text", CellChangeTimes->{ 3.9086277492110515`*^9},ExpressionUUID->"334109b3-32af-4f0d-9f1d-\ b20c17ec9d3b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol2", "=", RowBox[{"Solve", "[", RowBox[{"x", "\[Equal]", RowBox[{"Nest", "[", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], ",", "x", ",", "2"}], "]"}]}], "]"}], " "}]], "Input", CellLabel->"In[16]:=",ExpressionUUID->"c60e0e1e-f6b6-471f-8874-e063dbf435b4"], Cell[TextData[{ "As duas primeiras solu\[CCedilla]\[OTilde]es acima correspondem aos pontos \ fixos de ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "689d6a46-21dd-4b26-8b2d-1e7d57fd8aa9"], ", que j\[AAcute] discutimos, como pode ser visto no gr\[AAcute]fico abaixo, \ para ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.8"}], TraditionalForm]],ExpressionUUID-> "30a9fef2-c43a-4be2-a129-f1c65d7b660e"], ". 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Vamos fazer um gr\[AAcute]fico dessas solu\[CCedilla]\[OTilde]es, \ juntamente com o gr\[AAcute]fico do ponto fixo est\[AAcute]vel para ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"1", "/", "4"}], "<", "\[Lambda]", "<", RowBox[{"3", "/", "4"}]}], TraditionalForm]],ExpressionUUID-> "44a100f5-ed8d-4558-a731-eda2aed7b074"], ":" }], "Text", CellChangeTimes->{ 3.90862791918265*^9},ExpressionUUID->"32f3caab-0ea4-4d9f-bc5b-e079e00a3e56"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "/.", RowBox[{"sol2", "[", RowBox[{"[", "2", "]"}], "]"}]}], ",", RowBox[{"x", "/.", RowBox[{"sol2", "[", RowBox[{"[", "3", "]"}], "]"}]}], ",", RowBox[{"x", "/.", RowBox[{"sol2", "[", RowBox[{"[", "4", "]"}], "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Lambda]", ",", RowBox[{"1", "/", "4"}], ",", "1"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"FrameLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"\[Lambda]\",FontSlant->\"Italic\"]\)\>\"", ",", "\"\<\!\(\*SuperscriptBox[StyleBox[\"x\",FontSlant->\"Italic\"], \"*\"]\ \)\>\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.9088237154183645`*^9, 3.9088237171748238`*^9}},ExpressionUUID->"d1bc0c3a-d394-4ed4-b7ca-\ 7b6c18c7cb3f"], Cell[TextData[{ "Note que o ciclo 2 surge de uma ", StyleBox["bifurca\[CCedilla]\[ATilde]o", FontVariations->{"Underline"->True}], " do ponto fixo quando este se torna inst\[AAcute]vel em ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.75"}], TraditionalForm]],ExpressionUUID-> "3f76eee6-06b7-4ef1-8edf-81499d657734"], "." }], "Text", CellMargins-> Dynamic[{{ 0.185 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}], 0.03 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}]}, { 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}], 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}]}}], CellChangeTimes->{ 3.9086279398244967`*^9},ExpressionUUID->"9d2d9920-565e-4a9e-ba02-\ 0138e278bb79"] }, Closed]] }, Closed]] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"950ef03f-ef35-40db-9ce7-9de5562b8bf7"], Cell[TextData[{ "Para analisarmos a estabilidade do ciclo 2, precisamos calcular a derivada \ de ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", RowBox[{"f", "(", "x", ")"}], ")"}], TraditionalForm]],ExpressionUUID-> "e9a5191c-19be-46c4-8c59-facab5f79e4a"], ":" }], "Text", CellChangeTimes->{ 3.9086279587164526`*^9},ExpressionUUID->"d0a9e523-f4fa-4c55-88c7-\ e61fe59307da"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"Nest", "[", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], ",", "x", ",", "2"}], "]"}], ",", "x"}], "]"}], "/.", RowBox[{"sol2", "[", RowBox[{"[", "3", "]"}], "]"}]}], "//", "FullSimplify"}]], "Input", CellLabel->"In[20]:=",ExpressionUUID->"80722212-f81c-4782-875d-458ea89d1080"], Cell[BoxData[ RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{"Abs", "[", "%", "]"}], "<", "1"}], ",", "\[Lambda]", ",", "PositiveReals"}], "]"}]], "Input", CellChangeTimes->{{3.908643936675864*^9, 3.9086439409535127`*^9}},ExpressionUUID->"dc7a4e62-b33e-4c05-a467-\ 6c0a626a8964"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "%", "]"}]], "Input", CellLabel->"In[22]:=",ExpressionUUID->"496dfbea-b099-4a0e-a9eb-c1ab507da987"], Cell[TextData[{ "Portanto, o ciclo 2 n\[ATilde]o \[EAcute] mais est\[AAcute]vel se ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.88"}], TraditionalForm]],ExpressionUUID-> "54f6e158-4769-4b16-84a5-b1bd5782c797"], ", caso em que vimos a altern\[AHat]ncia entre 4 valores ap\[OAcute]s muitas \ itera\[CCedilla]\[OTilde]es do mapa. \[CapitalEAcute] claro que agora \ precisamos estudar os pontos fixos de ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["f", RowBox[{"(", "4", ")"}]], "(", "x", ")"}], "=", RowBox[{"f", "(", RowBox[{"f", "(", RowBox[{"f", "(", RowBox[{"f", "(", "x", ")"}], ")"}], ")"}], ")"}]}], TraditionalForm]], ExpressionUUID->"060a7f5b-bbb7-437c-ad07-7eaff815fc07"], ". Para esse caso, no entanto, n\[ATilde]o h\[AAcute] mais solu\[CCedilla]\ \[ATilde]o anal\[IAcute]tica fechada, de modo que temos que buscar solu\ \[CCedilla]\[OTilde]es num\[EAcute]ricas, \[OpenCurlyDoubleQuote]descontando\ \[CloseCurlyDoubleQuote] aquelas correspondentes aos pontos fixos de ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", RowBox[{"f", "(", "x", ")"}], ")"}], TraditionalForm]],ExpressionUUID-> "9948f7bc-22e3-43c4-9836-6da46c0eae00"], ". 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cc6953b661f3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"grafmapa", "[", RowBox[{"0.88", ",", "0.373084", ",", "50"}], "]"}]], "Input", CellChangeTimes->{{3.908823750612084*^9, 3.9088237519163127`*^9}},ExpressionUUID->"28e64705-2522-4467-8d89-\ 6a13e8c901ba"], Cell[TextData[{ "Note que a derivada da quarta iterada de ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "4192b3f0-2571-4c2d-9bfc-1aff66452bfb"], " \[EAcute] negativa nos pontos do ciclo 4:" }], "Text", CellMargins-> Dynamic[{{ 0.185 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}], 0.03 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}]}, { 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}], 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}]}}], CellChangeTimes->{ 3.9086281097314167`*^9},ExpressionUUID->"cd9d23cc-a537-4ccd-a20c-\ 3bb8ef3f420f"], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{ RowBox[{"grafmapa", "[", RowBox[{"0.88", ",", "0.373084", ",", "50"}], "]"}], ",", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Nest", "[", RowBox[{ RowBox[{"f", "[", "0.88", "]"}], ",", "x", ",", "4"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"PlotStyle", "->", "Gray"}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{ 3.908628120939082*^9, 3.90862841550583*^9, {3.9088237658521557`*^9, 3.9088237669788513`*^9}},ExpressionUUID->"f4fb5619-9a5e-4fcb-9eef-\ 5cfd9d88c035"] }, Closed]] }, Closed]] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"72ee8388-64ea-4694-8497-05e665d09508"], Cell[TextData[{ "Analisar a estabilidade do ciclo 4 requer uma abordagem puramente num\ \[EAcute]rica, determinando a solu\[CCedilla]\[ATilde]o simult\[AHat]nea das \ equa\[CCedilla]\[OTilde]es ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{ SuperscriptBox["f", RowBox[{"(", "4", ")"}]], "(", "x", ")"}]}], TraditionalForm]], ExpressionUUID->"3483ca4d-95f7-4d60-85b3-e9214767adf1"], " e ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", " ", RowBox[{ RowBox[{ SuperscriptBox["f", RowBox[{"(", "4", ")"}]], "(", "x", ")"}], "/", "d"}], " ", "x"}], "=", RowBox[{"-", "1"}]}], TraditionalForm]],ExpressionUUID-> "0721ba60-2b56-431e-aa5e-aad163a75047"], ":" }], "Text", CellChangeTimes->{ 3.908628145902891*^9, {3.908628278211878*^9, 3.9086282905747695`*^9}},ExpressionUUID->"090ed4c2-a7a2-4267-94c5-\ c855f3129b9d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "\[Equal]", RowBox[{"Nest", "[", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], ",", "x", ",", "4"}], "]"}]}], ",", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"Nest", "[", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], ",", "x", ",", "4"}], "]"}], ",", "x"}], "]"}], "\[Equal]", RowBox[{"-", "1"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "0.5"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Lambda]", ",", "0.9"}], "}"}]}], "}"}]}], "]"}]], "Input", CellLabel->"In[29]:=",ExpressionUUID->"32ceba8d-9039-4354-8658-360b59a84e81"], Cell[TextData[{ "Para valores de \[Lambda] logo acima de ", Cell[BoxData[ FormBox["0.886023", TraditionalForm]],ExpressionUUID-> "f0fc6346-a169-459c-ac20-61c21360bd4d"], ", o ciclo 4 d\[AAcute] lugar a um ciclo 8, cuja estabilidade pode ser \ estudada tamb\[EAcute]m numericamente, pela solu\[CCedilla]\[ATilde]o simult\ \[AHat]nea das equa\[CCedilla]\[OTilde]es ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{ SuperscriptBox["f", RowBox[{"(", "8", ")"}]], "(", "x", ")"}]}], TraditionalForm]], ExpressionUUID->"b1ed09c2-8c4d-4845-bb90-01fe7c235993"], " e ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", " ", RowBox[{ RowBox[{ SuperscriptBox["f", RowBox[{"(", "8", ")"}]], "(", "x", ")"}], "/", "d"}], " ", "x"}], "=", RowBox[{"-", "1"}]}], TraditionalForm]],ExpressionUUID-> "2309b462-703b-4aa0-b3c4-2a1b570ed7e4"], ":" }], "Text", CellChangeTimes->{{3.908628194191818*^9, 3.908628225106185*^9}, { 3.9086282991428876`*^9, 3.9086283077504005`*^9}},ExpressionUUID->"c779ca75-ee8c-4212-9707-\ ebce8647f66d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "\[Equal]", RowBox[{"Nest", "[", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], ",", "x", ",", "8"}], "]"}]}], ",", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"Nest", "[", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], ",", "x", ",", "8"}], "]"}], ",", "x"}], "]"}], "\[Equal]", RowBox[{"-", "1"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "0.55"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Lambda]", ",", "0.9"}], "}"}]}], "}"}]}], "]"}]], "Input", CellLabel->"In[30]:=",ExpressionUUID->"ca0d065a-07b4-43eb-8ae2-d549201556ff"], Cell[TextData[{ "Vemos que esse processo de ", StyleBox["duplica\[CCedilla]\[ATilde]o de per\[IAcute]odo", FontVariations->{"Underline"->True}], " por bifurca\[CCedilla]\[OTilde]es ocorre para valores de ", Cell[BoxData[ FormBox["\[Lambda]", TraditionalForm]],ExpressionUUID-> "5678cfb1-7b76-492d-b3a8-c9d3b6936411"], " cada vez mais pr\[OAcute]ximos entre si: ", Cell[BoxData[ FormBox["0.75", TraditionalForm]],ExpressionUUID-> "ea176356-bbe0-4498-9e88-163fb5d3035d"], ", ", Cell[BoxData[ FormBox["0.862372", TraditionalForm]],ExpressionUUID-> "74333ecf-493b-4cbb-b2f3-85ad58eac01c"], ", ", Cell[BoxData[ FormBox["0.886023", TraditionalForm]],ExpressionUUID-> "b13c26f2-f358-483b-8f47-5e75e6940e46"], ", ", Cell[BoxData[ FormBox["0.891102", TraditionalForm]],ExpressionUUID-> "8fcc5b2b-1508-4ffa-b6b3-265b83c477f7"], ". A an\[AAcute]lise torna-se cada vez mais complexa, de modo que \[EAcute] \ \[UAcute]til fazermos um estudo \[OpenCurlyDoubleQuote]panor\[AHat]mico\ \[CloseCurlyDoubleQuote] com aux\[IAcute]lio de um gr\[AAcute]fico." }], "Text", CellChangeTimes->{ 3.9086283277523766`*^9},ExpressionUUID->"65bb5832-953c-4073-8fb5-\ 54e95b60903e"] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"0a38dcf4-d647-442e-91ab-58034f6db37d"], Cell[TextData[{ "O gr\[AAcute]fico que vamos produzir toma por base a itera\[CCedilla]\ \[ATilde]o do mapa por um grande n\[UAcute]mero de vezes, seguida pelo \ descarte dos primeiros elementos da lista de itera\[CCedilla]\[OTilde]es, que \ comp\[OTilde]em o ", StyleBox["transiente", FontVariations->{"Underline"->True}], " que leva aos ", StyleBox["atratores", FontVariations->{"Underline"->True}], " (pontos fixos, ciclos 2, ciclos 4 etc.). Fazemos isso para diversos \ valores de ", Cell[BoxData[ FormBox["\[Lambda]", TraditionalForm]],ExpressionUUID-> "cc8872b0-8c46-4cb9-9444-62dba8b94b60"], ", e em seguida produzimos um gr\[AAcute]fico dos elementos preservados para \ cada valor." }], "Text", CellChangeTimes->{ 3.908643275993351*^9},ExpressionUUID->"86c5bcec-d142-4650-bb82-\ 7c04ea899395"], Cell[TextData[{ "Primeiro vamos definir uma fun\[CCedilla]\[ATilde]o que produz uma lista a \ partir de ", Cell[BoxData[ FormBox["n", TraditionalForm]],ExpressionUUID-> "845c6c07-f9fa-4515-854c-e6feed7920aa"], " aplica\[CCedilla]\[OTilde]es do mapa, eliminando os primeiros ", Cell[BoxData[ FormBox["m", TraditionalForm]],ExpressionUUID-> "a8b1158e-0dcf-41e4-b654-713351d22d4f"], " elementos com o aux\[IAcute]lio de ", StyleBox["Drop", FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{ 3.9086432892884407`*^9},ExpressionUUID->"6a93f9e9-b110-4ee5-bcc0-\ 60b1b0c72756"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"itera", "[", RowBox[{"\[Lambda]_", ",", "m_", ",", "n_"}], "]"}], ":=", RowBox[{"Drop", "[", RowBox[{ RowBox[{"NestList", "[", RowBox[{ RowBox[{"f", "[", "\[Lambda]", "]"}], ",", "0.5", ",", "n"}], "]"}], ",", RowBox[{"m", "+", "1"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"O", " ", "argumento", " ", "m"}], "+", RowBox[{ "1", " ", "vem", " ", "do", " ", "fato", " ", "de", " ", "que", " ", "NestList", " ", "come\[CCedilla]a", " ", "da", " ", RowBox[{"itera\[CCedilla]\[ATilde]o", " ", "'"}], RowBox[{"0", "'"}]}]}], " ", "*)"}], " "}]], "Input",ExpressionUUID->\ "d900a697-ac98-452d-945a-c403ac69a9cd"], Cell[TextData[{ "Em seguida, vamos definir uma fun\[CCedilla]\[ATilde]o que vai produzir um \ ponto cuja abscissa \[EAcute] definida pelo valor de \[Lambda] e cuja \ ordenada indicar\[AAcute] um elemento da lista de valores de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "90055374-9e7e-401a-9c11-6060ffee27fd"], ":" }], "Text", CellChangeTimes->{ 3.908643379983594*^9},ExpressionUUID->"dfbccfbc-89f0-4254-beb9-\ 616fed648501"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"ponto", "[", "x_", "]"}], ":=", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"\[Lambda]", ",", "x"}], "}"}], "]"}], " "}]], "Input",ExpressionU\ UID->"d4a34a22-c0d3-4f20-8063-06ad5d75a209"], Cell[TextData[{ "Note o uso de ", StyleBox[ButtonBox["Point", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Point.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Point.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que ser\[AAcute] combinado mais adiante com ", StyleBox[ButtonBox["Graphics", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Graphics.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Graphics.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", fun\[CCedilla]\[ATilde]o que gera desenhos geom\[EAcute]tricos em duas \ dimens\[OTilde]es e que j\[AAcute] utilizamos quando discutimos anima\ \[CCedilla]\[OTilde]es. Por exemplo, para desenhar os tr\[EHat]s \ v\[EAcute]rtices de um tri\[AHat]ngulo equil\[AAcute]tero, utilizando tamb\ \[EAcute]m a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["PointSize", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/PointSize.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/PointSize.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " para ajustar o tamanho dos pontos:" }], "Text", CellChangeTimes->{{3.9086434083667417`*^9, 3.9086434520477347`*^9}},ExpressionUUID->"ecff448c-3ec6-412f-9232-\ 59dbd65e853e"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", "0.1", "]"}], ",", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], "]"}], ",", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"0.5", ",", RowBox[{ RowBox[{"Sqrt", "[", "3", "]"}], "/", "2"}]}], "}"}], "]"}], ",", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}], "]"}]}], "}"}], "]"}]], "Input",Expressi\ onUUID->"ac7eec93-7fbb-4921-9db1-378118dbe59d"], Cell[TextData[{ "Para produzir o gr\[AAcute]fico, vamos varrer um n\[UAcute]mero ", Cell[BoxData[ FormBox[ SubscriptBox["n", "\[Lambda]"], TraditionalForm]],ExpressionUUID-> "43f04c57-5683-40f8-8de3-06e1d22f8063"], " de valores de \[Lambda] entre ", Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", "min"], TraditionalForm]],ExpressionUUID-> "7af0282c-d6f1-4d51-9647-c949ba873a57"], " e ", Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", "max"], TraditionalForm]],ExpressionUUID-> "fee330fb-16b5-4718-a2ac-cfbf442f63d7"], ". Para isso, definimos uma outra fun\[CCedilla]\[ATilde]o, que faz uso de ", StyleBox["Map", FontFamily->"Source Code Pro", FontWeight->"Regular"], " para aplicar a fun\[CCedilla]\[ATilde]o ", StyleBox["ponto[", FontFamily->"Source Code Pro", FontWeight->"Regular"], "\[Lambda]] a cada elemento da lista de itera\[CCedilla]\[ATilde]o produzida \ por ", StyleBox["itera", FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{ 3.9086434820971804`*^9},ExpressionUUID->"1ee2b9a2-8dbe-4744-8063-\ 0847aa9a3b15"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"graflambda", "[", RowBox[{ "\[Lambda]min_", ",", "\[Lambda]max_", ",", "n\[Lambda]_", ",", "m_", ",", "n_"}], "]"}], ":=", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", "0.001", "]"}], ",", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"Map", "[", RowBox[{"ponto", ",", RowBox[{"itera", "[", RowBox[{"\[Lambda]", ",", "m", ",", "n"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"\[Lambda]", ",", "\[Lambda]min", ",", "\[Lambda]max", ",", RowBox[{ RowBox[{"(", RowBox[{"\[Lambda]max", "-", "\[Lambda]min"}], ")"}], "/", "n\[Lambda]"}]}], "}"}]}], "]"}]}], "}"}], "]"}]}]], "Input",Express\ ionUUID->"d83dcd25-8dde-456f-82d8-25cfad24d49c"], Cell[TextData[{ "Veja abaixo como funciona o primeiro argumento de ", StyleBox["Table", FontFamily->"Source Code Pro", FontWeight->"Regular"], " na fun\[CCedilla]\[ATilde]o acima, no caso \[Lambda]=0.8:" }], "Text", CellChangeTimes->{ 3.908643502778941*^9},ExpressionUUID->"42af828d-d22f-4d8c-b859-\ 2bdfe84bdb0a"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Lambda]", "=", "0.8"}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{"itera", "[", RowBox[{"\[Lambda]", ",", "98", ",", "100"}], "]"}], "\[IndentingNewLine]", RowBox[{"Map", "[", RowBox[{"ponto", ",", RowBox[{"itera", "[", RowBox[{"\[Lambda]", ",", "98", ",", "100"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Clear", "[", "\[Lambda]", "]"}]}], "Input",ExpressionUUID->\ "52f58575-d6bb-4a4b-9b33-5b514954c60a"], Cell["\<\ E agora um exemplo da tabela, com 6 valores de \[Lambda] entre 0 e 1: \ \>", "Text", CellChangeTimes->{ 3.908643528155534*^9},ExpressionUUID->"117ac904-ccc7-4973-8a1c-\ 77ac8fd18c99"], Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Map", "[", RowBox[{"ponto", ",", RowBox[{"itera", "[", RowBox[{"\[Lambda]", ",", "98", ",", "100"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"\[Lambda]", ",", "0", ",", "1", ",", "0.2"}], "}"}]}], "]"}]], "Input",ExpressionUUID->"8d020c30-9318-4b43-a3c2-4ccf13bb68d0"] }, Closed]] }, Closed]] }, Closed]] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"2ecb9827-6d58-4eb7-ac48-9877d3d7aaac"], Cell[TextData[{ "O gr\[AAcute]fico propriamente dito, a depender do computador, provoca \ consider\[AAcute]vel lentid\[ATilde]o no Mathematica. Por isso, optamos por \ produzir e salvar o gr\[AAcute]fico sem exibi-lo. Para fazer isso em seu \ computador, retire o coment\[AAcute]rio da segunda entrada abaixo e \ execute-a, mas saiba que a execu\[CCedilla]\[ATilde]o, a depender do \ computador, pode demorar bastante. Para contornar uma eventual \ lentid\[ATilde]o, vamos ler uma figura previamente preparada. Para cada um de \ ", Cell[BoxData[ FormBox[ RowBox[{"1000", " "}], TraditionalForm]],ExpressionUUID-> "fc8d4093-ef72-4562-8354-ad8771b2595e"], "valores de \[Lambda] entre ", Cell[BoxData[ FormBox["0.72", TraditionalForm]],ExpressionUUID-> "82302eaa-ea6a-4326-8688-494cebb9cb22"], " e 1, o gr\[AAcute]fico mostra os \[UAcute]ltimos ", Cell[BoxData[ FormBox["1024", TraditionalForm]],ExpressionUUID-> "f68766c0-ed5b-408e-a3eb-4bcb6cd5f2e1"], " valores de ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "cc5cddf4-826a-4848-b6ea-1279b4e183a4"], " obtidos pela itera\[CCedilla]\[ATilde]o do mapa ", Cell[BoxData[ FormBox["2048", TraditionalForm]],ExpressionUUID-> "8dee4259-658f-451b-b930-6b761555485b"], " vezes." }], "Text", CellChangeTimes->{ 3.9086435704762363`*^9},ExpressionUUID->"e3eddba9-8062-4206-a89d-\ e19b5f3aee4f"], Cell[BoxData[ RowBox[{ RowBox[{"SetDirectory", "[", RowBox[{"NotebookDirectory", "[", "]"}], "]"}], ";"}]], "Input",ExpressionU\ UID->"eac0607b-6564-4b97-8711-508e9914d217"], Cell[BoxData[ RowBox[{"(*", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"grafico", "=", " ", RowBox[{ RowBox[{"Show", "[", RowBox[{ RowBox[{"graflambda", "[", RowBox[{"0.72", ",", "1", ",", "1000", ",", "1024", ",", "2048"}], "]"}], ",", " ", RowBox[{"Evaluate", "[", "optam", "]"}], ",", " ", RowBox[{"FrameLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\[Lambda]\>\"", ",", "\"\<\!\(\*StyleBox[\"x\",FontSlant->\"Italic\"]\)\>\""}], "}"}]}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}]}], ",", " ", RowBox[{"AspectRatio", "\[Rule]", "1"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Export", "[", RowBox[{"\"\\"", ",", "grafico"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"Clear", "[", "grafico", "]"}]}], " ", "*)"}]], "Input",ExpressionU\ UID->"0ebf7bc5-8467-4efc-86d8-5c2eb702a259"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Import", "[", "\"\\"", "]"}]], "Input",ExpressionUUID->"1aa17faf-2d58-4af4-b4af-13d0de5ce6a3"], Cell[TextData[{ "Note que o padr\[ATilde]o para \[Lambda] entre ", Cell[BoxData[ FormBox["0.72", TraditionalForm]],ExpressionUUID-> "642d8f56-bb27-43c1-ad24-1ea06fc8cd10"], " (in\[IAcute]cio dos pontos, \[AGrave] esquerda) e ", Cell[BoxData[ FormBox["0.862372", TraditionalForm]],ExpressionUUID-> "742f9b74-8a66-4ba3-91e0-759cfd3fa85c"], " segue o que mostramos em um gr\[AAcute]fico anterior, a menos do fato de \ que o ponto fixo que se torna inst\[AAcute]vel a partir de ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", "0.75"}], TraditionalForm]],ExpressionUUID-> "b3fd4d44-8ba5-4209-bb1c-28a337e650cc"], " n\[ATilde]o mais aparece, por n\[ATilde]o ser alcan\[CCedilla]\[AAcute]vel \ exceto partindo dele pr\[OAcute]prio. A \[OpenCurlyDoubleQuote]cascata\ \[CloseCurlyDoubleQuote] de bifurca\[CCedilla]\[OTilde]es de per\[IAcute]odo \ em ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Lambda]", "1"], "=", "0.75"}], TraditionalForm]], ExpressionUUID->"4daec9aa-9fb1-40f4-b8ab-1124bb7671c5"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Lambda]", "2"], "=", "0.862372"}], TraditionalForm]], ExpressionUUID->"28d4366b-cbe7-4823-bca6-6d832dd067bc"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Lambda]", "3"], "=", "0.886023"}], TraditionalForm]], ExpressionUUID->"79ced117-2dc3-4d20-b133-7acb2231e847"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Lambda]", "4"], "=", "0.891102"}], TraditionalForm]], ExpressionUUID->"1e6842ef-8263-47aa-be1a-86da5b32a32f"], " etc. \[EAcute] percept\[IAcute]vel, mas a partir da\[IAcute] fica dif\ \[IAcute]cil visualizar as bifurca\[CCedilla]\[OTilde]es, embora elas estejam \ l\[AAcute] e possam ser identificadas por uma investiga\[CCedilla]\[ATilde]o \ cuidadosa. 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Vamos investigar melhor essa \ regi\[ATilde]o, produzindo uma segunda figura (que novamente vamos ler de um \ arquivo), mostrando uma amplia\[CCedilla]\[ATilde]o do ramo central naquele \ intervalo:" }], "Text", CellChangeTimes->{ 3.9086436618225684`*^9},ExpressionUUID->"ef683ebd-03d8-4d44-9d0e-\ df698d2ebd21"], Cell[BoxData[ RowBox[{"(*", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"grafico", "=", RowBox[{ RowBox[{"Show", "[", RowBox[{ RowBox[{"graflambda", "[", RowBox[{"0.96", ",", "0.963", ",", "1000", ",", "1024", ",", "2048"}], "]"}], ",", " ", RowBox[{"Evaluate", "[", "optam", "]"}], ",", " ", RowBox[{"FrameLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\[Lambda]\>\"", ",", "\"\<\!\(\*StyleBox[\"x\",FontSlant->\"Italic\"]\)\>\""}], "}"}]}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"0.455", ",", "0.525"}], "}"}]}], ",", " ", RowBox[{"AspectRatio", "\[Rule]", "1"}]}], "]"}], " ", "\[IndentingNewLine]", RowBox[{"Export", "[", RowBox[{"\"\\"", ",", "grafico"}], "]"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"Clear", "[", "grafico", "]"}]}], " ", "\[IndentingNewLine]", "*)"}]], "Input",ExpressionUUID->"8048d789-3e5c-4908-8452-9bfb095a89ee"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Import", "[", "\"\\"", "]"}]], "Input",ExpressionUUID->"c41517e9-1dd0-4799-912a-adeaac0fe8d8"], Cell[TextData[{ "Note que, exceto pela mudan\[CCedilla]a na escala e por uma \ invers\[ATilde]o em rela\[CCedilla]\[ATilde]o ao eixo horizontal, o aspecto \ desse gr\[AAcute]fico \[EAcute] indistingu\[IAcute]vel daquele observado no \ slide anterior. 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