(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 1398311, 25762] NotebookOptionsPosition[ 1394449, 25629] NotebookOutlinePosition[ 1394861, 25647] CellTagsIndexPosition[ 1394818, 25644] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Interactive Models with Stream Plot and Locator", "Title", CellChangeTimes->{{3.65570395008041*^9, 3.655703977428302*^9}}], Cell[CellGroupData[{ Cell["Exercise", "Section", CellChangeTimes->{{3.655707227652182*^9, 3.6557072344930964`*^9}}], Cell[TextData[{ StyleBox["In a previous lab, we studied the system", FontColor->GrayLevel[0]], " ", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", " ", "y"}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", RowBox[{ RowBox[{"-", "y"}], "-", RowBox[{"2", RowBox[{"sin", "(", "x", ")"}]}]}]}], RowBox[{"(", "2", ")"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False], TraditionalForm]]], ". \n", StyleBox["We will now focus on a generalization ", FontColor->GrayLevel[0]], "\n", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", " ", "y"}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", RowBox[{ RowBox[{ RowBox[{"-", "\[Epsilon]"}], " ", "y"}], "-", RowBox[{"2", RowBox[{"sin", "(", "x", ")"}]}]}]}], RowBox[{"(", "2", ")"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False], TraditionalForm]]], ", where \[Epsilon] is a real parameter in the interval [0,1]. This system \ defines the equation of motion of a pendulum. \n\n", StyleBox["The physical meaning of the variables and parameters of the \ system:", FontColor->GrayLevel[0]], "\nx = x(t) is the angular coordinate (the angle that the bob of the \ pendulum makes with the vertical direction, represented as a function of \ time),\ny = x\[CloseCurlyQuote](t) is the velocity of the angular motion. \n\ \[Epsilon] \[GreaterSlantEqual] 0 is the coefficient of friction (when \ \[Epsilon]=0 there is no friction). The model assumes that the friction is \ proportional to the velocity.\n\n", StyleBox["Our task for today is to make an interactive plot of the Phase \ Portrait when -3\[Pi] < x < 3\[Pi] , -2\[Pi] < y < 2\[Pi], which can \ highlight the trajectory corresponding to the initial condition chosen by the \ user. The parameter \[Epsilon] of the system of differential equations should \ be defined as a parameter of the interactive model.", FontColor->GrayLevel[0]] }], "Subsection", CellChangeTimes->{{3.6227922426259937`*^9, 3.62279231309624*^9}, { 3.6227927369105253`*^9, 3.622792744486435*^9}, {3.622805954711268*^9, 3.6228060525702353`*^9}, {3.622815694267383*^9, 3.6228156945063753`*^9}, { 3.622973160147009*^9, 3.622973167010988*^9}, {3.622973244164934*^9, 3.6229732511893044`*^9}, {3.655087026014307*^9, 3.655087111864697*^9}, { 3.655107659238628*^9, 3.655107723111965*^9}, {3.655108195181444*^9, 3.655108202311469*^9}, {3.655108433634169*^9, 3.655108452593494*^9}, { 3.655108552313053*^9, 3.655108635137816*^9}, {3.655108690544921*^9, 3.6551087036780977`*^9}, {3.6551194458193007`*^9, 3.655119467449971*^9}, { 3.655705051031673*^9, 3.655705205286602*^9}, {3.6557052913470573`*^9, 3.655705299927477*^9}, {3.655705378348617*^9, 3.65570538945216*^9}, 3.6557060261000443`*^9, 3.655706057727594*^9, {3.655707213858821*^9, 3.655707225118703*^9}, {3.7010590245732927`*^9, 3.701059056190977*^9}, { 3.701059120487578*^9, 3.701059229535728*^9}, {3.701059314464995*^9, 3.701059327560141*^9}, {3.701059389667399*^9, 3.7010594311396027`*^9}, { 3.7010594795349483`*^9, 3.701059693524678*^9}, {3.70105973520794*^9, 3.7010598520834723`*^9}, {3.7010610789542093`*^9, 3.701061098856992*^9}, { 3.701065096308284*^9, 3.7010650999776497`*^9}, {3.701065149377658*^9, 3.701065152106441*^9}, {3.701065199178872*^9, 3.701065217001707*^9}}, TextJustification->1.] }, Open ]], Cell[CellGroupData[{ Cell["Solution 1", "Section", CellChangeTimes->{{3.622798264387392*^9, 3.62279834397857*^9}, { 3.655704733400276*^9, 3.655704747704707*^9}, {3.6557050045796633`*^9, 3.655705021301585*^9}, {3.6557054050330667`*^9, 3.655705405760648*^9}, { 3.6557072558054647`*^9, 3.6557072736216383`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"StreamPlot", "[", RowBox[{ RowBox[{"{", RowBox[{"y", ",", RowBox[{ RowBox[{ RowBox[{"-", "2"}], RowBox[{"Sin", "[", "x", "]"}]}], "-", RowBox[{"\[Epsilon]", StyleBox["*", FontColor->GrayLevel[0]], "y"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{ RowBox[{"-", "3"}], "\[Pi]"}], ",", RowBox[{"3", "\[Pi]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{ RowBox[{"-", "2"}], "\[Pi]"}], ",", RowBox[{"2", "\[Pi]"}]}], "}"}], ",", " ", RowBox[{"ImageSize", "\[Rule]", "Large"}], ",", "\[IndentingNewLine]", RowBox[{"StreamPoints", "\[Rule]", RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"a", ",", "b"}], "}"}], ",", "Red"}], "}"}], ",", "Automatic"}], "}"}], "}"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"a", ",", " ", RowBox[{ RowBox[{"-", "3"}], "\[Pi]"}], ",", RowBox[{"3", "\[Pi]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"b", ",", RowBox[{ RowBox[{"-", "2"}], "\[Pi]"}], ",", RowBox[{"2", "\[Pi]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", StyleBox[ RowBox[{"\[Epsilon]", ",", " ", "0", ",", "1"}], FontColor->GrayLevel[0]], StyleBox["}", FontColor->GrayLevel[0]]}]}], "\[IndentingNewLine]", "]"}]], "Input", CellChangeTimes->{{3.655705422152248*^9, 3.655705537663167*^9}, { 3.701059871218348*^9, 3.7010598952085037`*^9}, {3.701061598599634*^9, 3.701061624822537*^9}}, FontSize->16, Background->RGBColor[1, 1, 0.85]], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = (-3) Pi, $CellContext`b$$ = (-2) Pi, $CellContext`\[Epsilon]$$ = 0, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`a$$], (-3) Pi, 3 Pi}, { Hold[$CellContext`b$$], (-2) Pi, 2 Pi}, { Hold[$CellContext`\[Epsilon]$$], 0, 1}}, Typeset`size$$ = { 576., {286., 291.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$32403$$ = 0, $CellContext`b$32404$$ = 0, $CellContext`\[Epsilon]$32405$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = (-3) Pi, $CellContext`b$$ = (-2) Pi, $CellContext`\[Epsilon]$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$32403$$, 0], Hold[$CellContext`b$$, $CellContext`b$32404$$, 0], Hold[$CellContext`\[Epsilon]$$, $CellContext`\[Epsilon]$32405$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> StreamPlot[{$CellContext`y, (-2) Sin[$CellContext`x] - $CellContext`\[Epsilon]$$ $CellContext`y}, \ {$CellContext`x, (-3) Pi, 3 Pi}, {$CellContext`y, (-2) Pi, 2 Pi}, ImageSize -> Large, StreamPoints -> {{{{$CellContext`a$$, $CellContext`b$$}, Red}, Automatic}}], "Specifications" :> {{$CellContext`a$$, (-3) Pi, 3 Pi}, {$CellContext`b$$, (-2) Pi, 2 Pi}, {$CellContext`\[Epsilon]$$, 0, 1}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{621., {359., 365.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.655705542714554*^9, 3.701059905914709*^9, {3.701061604208774*^9, 3.701061627162973*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Physical Interpretation :", "Section", CellChangeTimes->{{3.701060783068432*^9, 3.7010607906291323`*^9}}], Cell[TextData[StyleBox["Case I: \[Epsilon]=0 (no friction)", FontVariations->{"Underline"->True}]], "Subsection", CellChangeTimes->{{3.70106081795328*^9, 3.701060836400304*^9}}], Cell[CellGroupData[{ Cell["\<\ There are two types of critical points: centers (stable, but not \ asymptotically stable), and saddle fixed points (unstable).\ \>", "Subsection", CellChangeTimes->{{3.7010608718621893`*^9, 3.701060883874892*^9}, { 3.7010609222879*^9, 3.701060961843237*^9}}], Cell[CellGroupData[{ Cell["\<\ The closed trajectories correspond to back and forth oscillations of the \ pendulum.\ \>", "ItemNumbered", CellChangeTimes->{{3.7010603414568663`*^9, 3.7010603505517473`*^9}, { 3.701060430761951*^9, 3.701060459738689*^9}}], Cell["\<\ The points at the centers of these closed trajectories correspond to the \ stable equilibrium of the pendulum at rest at the bottom of its motion.\ \>", "ItemNumbered", CellChangeTimes->{ 3.7010603414568663`*^9, {3.701060418161108*^9, 3.701060455571944*^9}}], Cell["\<\ The trajectories above and beneath the closed trajectories correspond to \ motions in which the pendulum has enough energy to simply go full circle, \ round and round over the top.\ \>", "ItemNumbered", CellChangeTimes->{ 3.7010603414568663`*^9, {3.701060418161108*^9, 3.7010605105244904`*^9}, { 3.701061421547632*^9, 3.7010614234902353`*^9}}], Cell["\<\ The level curves joining the saddle fixed points are those exceptional \ motions in which for t\[RightArrow]\[Infinity] and for t\[RightArrow]-\ \[Infinity] the pendulum tends to the unstable equilibrium at the very top \ of the swing. Note that the time required to go from one saddle to the next \ is infinite.\ \>", "ItemNumbered", CellChangeTimes->{ 3.7010603414568663`*^9, {3.701060418161108*^9, 3.7010606234218607`*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Case II: \[Epsilon] > 0 (friction acts as a damping factor on the \ motion of the pendulum)", FontVariations->{"Underline"->True}], "\n There are two types of critical points : spiral sinks (asymptotically \ stable), and saddle fixed points (unstable)." }], "Subsection", CellChangeTimes->{{3.701060703528297*^9, 3.701060755704681*^9}, { 3.701060983485578*^9, 3.7010610089409227`*^9}}], Cell[CellGroupData[{ Cell["\<\ The spiral sinks correspond to the stable equilibrium of the pendulum at \ rest at the bottom of its motion.\ \>", "ItemNumbered", CellChangeTimes->{ 3.7010603414568663`*^9, {3.701060418161108*^9, 3.701060455571944*^9}, { 3.701061341929648*^9, 3.701061344465839*^9}}], Cell["\<\ Almost all trajectories converge to one of the spiral sinks as \ t\[RightArrow]\[Infinity].\ \>", "ItemNumbered", CellChangeTimes->{ 3.7010603414568663`*^9, {3.701060418161108*^9, 3.7010605105244904`*^9}, { 3.70106145285078*^9, 3.701061501556839*^9}}], Cell["\<\ There are some exceptional trajectories converging to the saddle fixed \ points for t\[RightArrow]\[Infinity] OR for t\[RightArrow]-\[Infinity] \ which correspond to the exceptional motions where the pendulum tends to the \ unstable equilibrium at the very top of the swing. \ \>", "ItemNumbered", CellChangeTimes->{ 3.7010603414568663`*^9, {3.701060418161108*^9, 3.7010606234218607`*^9}, { 3.701061375525337*^9, 3.701061385403475*^9}, {3.701061520241086*^9, 3.701061579122678*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Solution 1 can be improved:", "Section", CellChangeTimes->{{3.655707289618917*^9, 3.655707301594944*^9}, 3.6561580632420683`*^9, {3.656158182290447*^9, 3.6561581826799726`*^9}, { 3.6561584445726547`*^9, 3.656158445189904*^9}, {3.7010634596151743`*^9, 3.701063461421919*^9}, {3.701063528290634*^9, 3.701063530530222*^9}}], Cell["\<\ The disadvantage of Solution 1 is that every time we drag the slider to \ change the initial condition, the entire phase portrait gets replotted, not \ just the trajectory selected by the user!\ \>", "Subsubsection", CellChangeTimes->{{3.656157314055661*^9, 3.656157400775189*^9}, { 3.701060131458043*^9, 3.701060132098003*^9}, {3.701063435982583*^9, 3.701063436510661*^9}, {3.701063468151561*^9, 3.701063468902317*^9}}], Cell[TextData[{ "\n", StyleBox["To improve Solution 1, let' s first recall how to combine \ graphical objects together:", FontColor->GrayLevel[0]] }], "Subsubsection", CellChangeTimes->{{3.701063476466667*^9, 3.701063553843872*^9}}], Cell[TextData[{ "Show[g1, g2, \[Ellipsis], gk] ", StyleBox["--- shows several graphics combined.", FontColor->GrayLevel[0]] }], "Subsubsection", CellChangeTimes->{{3.655705743712083*^9, 3.655705749851266*^9}, { 3.701063627257139*^9, 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First assume that \[Epsilon]=1, as in the examples \ above. Use Manipulate and Locator to make an interactive plot that shows the \ corresponding StreamPlot. When the user clicks somewhere in the Phase \ Portrait, we plot the trajectory that passes through that point in red.\n\n2. \ Now take the code from part 1, and make \[Epsilon] a parameter of your model \ as well. Be sure to implement it in such a way that you only draw the Phase \ Portrait when the user enter a different value for \[Epsilon]. 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