(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 386283, 7953] NotebookOptionsPosition[ 380812, 7774] NotebookOutlinePosition[ 381328, 7796] CellTagsIndexPosition[ 381241, 7791] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[StyleBox["Linear Differential Equations (continued):", FontColor->GrayLevel[0]]], "Section", CellChangeTimes->{{3.622973133683782*^9, 3.622973152291214*^9}, { 3.6550907148775053`*^9, 3.6550907364464083`*^9}, {3.69985370934711*^9, 3.6998537293547783`*^9}}, FontSize->36], Cell[CellGroupData[{ Cell["General Strategy from last time:", "Section", CellChangeTimes->{{3.6550988036100883`*^9, 3.655098815790513*^9}, { 3.699855662028679*^9, 3.699855663747609*^9}}], Cell[TextData[{ "Example: Solve the system ", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", RowBox[{"x", "+", "y"}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", RowBox[{ RowBox[{"4", "x"}], "+", "y"}]}], 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]]], " using DSolve[..]." }], "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}, {3.69985452413451*^9, 3.699854533133728*^9}}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["By the analysis done in class last time, we know that (0,0) is a \ saddle. 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The two solutions are ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { RowBox[{"x", "(", "t", ")"}]}, { RowBox[{"y", "(", "t", ")"}]} }], ")"}], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox[" = ", FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SubscriptBox["c", "2"], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox[" ", FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{"-", "t"}]], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { RowBox[{"-", "1"}]}, {"2"} }], ")"}], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], " ", StyleBox["and", FontColor->GrayLevel[0]], " ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { RowBox[{"x", "(", "t", ")"}]}, { RowBox[{"y", "(", "t", ")"}]} }], ")"}], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox[" = ", FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SubscriptBox["c", "1"], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox[" ", FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{"3", "t"}]], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1"}, {"2"} }], ")"}], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], ". 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Notice also that \ we defined the functions g2 and h2 using \[OpenCurlyDoubleQuote]=\ \[CloseCurlyDoubleQuote] and not \[OpenCurlyDoubleQuote]:=\ \[CloseCurlyDoubleQuote].\ \>", "Subsubsection", CellChangeTimes->{{3.622979589139554*^9, 3.622979663376666*^9}, { 3.622979782824671*^9, 3.62297979995275*^9}, {3.622980381235146*^9, 3.6229804063982573`*^9}, {3.622982776195292*^9, 3.622982789169695*^9}, { 3.699856731392005*^9, 3.699856745532298*^9}}, TextJustification->1.], Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", "t", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"g2", "[", "t_", "]"}], "=", " ", RowBox[{ RowBox[{"x", "[", "t", "]"}], " ", "/.", " ", RowBox[{"Flatten", "[", "sol2", "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"h2", "[", "t_", "]"}], "=", " ", RowBox[{ RowBox[{"y", "[", "t", "]"}], " ", "/.", " ", RowBox[{"Flatten", "[", "sol2", "]"}]}]}]}], "Input", CellChangeTimes->{{3.62297957388767*^9, 3.622979574259205*^9}, { 3.622979851019341*^9, 3.622979881370195*^9}, 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Editable->False, SelectWithContents->True, Selectable->False], TraditionalForm]]], " where f and g are functions of two variables x and y. Assume that f and g \ are not both linear. " }], "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}, {3.6550907440986423`*^9, 3.655090840713463*^9}, { 3.655090876806201*^9, 3.655090892386242*^9}, {3.655090928173422*^9, 3.6550909555006523`*^9}, {3.655091007363132*^9, 3.655091054753298*^9}}], Cell[TextData[{ "The functions x and y are functions of time t and the derivatives are also \ taken with respect to the variable t. 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" }], "Subsection", CellChangeTimes->{{3.622806089809124*^9, 3.6228061016956873`*^9}, { 3.622806146675913*^9, 3.622806280122282*^9}, {3.6228063317840443`*^9, 3.622806352340406*^9}, {3.622973262660453*^9, 3.62297331713972*^9}, 3.6229751603254747`*^9, 3.622986200393735*^9, {3.655090980795087*^9, 3.655090997361874*^9}}, FontColor->GrayLevel[0]], Cell[TextData[{ "A linear system ", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", " ", RowBox[{"ax", "+", "by"}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", " ", RowBox[{"cx", "+", "dy"}]}], 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]]], " whose coefficient matrix ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " is invertible has only one equilibrium point (critical point, steady state \ solution) at (x,y)=(0,0). Analyzing the stability of this equilibrium \ solutions gives complete information about the Phase Portrait of the ODE (the \ trajectories)." }], "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}, {3.6550907440986423`*^9, 3.655090840713463*^9}, { 3.655090876806201*^9, 3.655090892386242*^9}, {3.655090928173422*^9, 3.6550909555006523`*^9}, {3.655091007363132*^9, 3.655091054753298*^9}, { 3.655091161628648*^9, 3.655091273316389*^9}, {3.6550914828617163`*^9, 3.6550914872083073`*^9}, {3.655096889323382*^9, 3.65509689613396*^9}, { 3.699853752941098*^9, 3.6998538003090982`*^9}}], Cell["\<\ Unlike a linear system, a nonlinear system could have none, one, two, three, \ or any number of critical points. Like a linear system, however, the critical \ points are found by setting x\[CloseCurlyQuote] = y\[CloseCurlyQuote] = 0, \ and solve the resulting system f(x,y)=0 and g(x,y)=0. Since there might be multiple critical points present on the phase portrait, \ each trajectory could be influenced by more than one critical point. This \ results in a much more chaotic appearance of the phase portrait. \ Consequently, the type and stability of each critical point need to be \ determined locally (in a small neighborhood on the phase plane around the \ critical point in question) on a case-by-case basis. The trajectories will \ appear very differently than those of the linear systems, except very near \ the critical points.\ \>", "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}, {3.6550907440986423`*^9, 3.655090840713463*^9}, { 3.655090876806201*^9, 3.655090892386242*^9}, {3.655090928173422*^9, 3.6550909555006523`*^9}, {3.655091007363132*^9, 3.655091054753298*^9}, { 3.655091161628648*^9, 3.655091273316389*^9}, {3.655091327338723*^9, 3.6550913629766006`*^9}, {3.655091460514735*^9, 3.655091465173214*^9}, { 3.6551030276398363`*^9, 3.655103073955945*^9}}, FontColor->GrayLevel[0]] }, Open ]], Cell[CellGroupData[{ Cell["General Strategy :", "Section", CellChangeTimes->{{3.6550988036100883`*^9, 3.655098815790513*^9}}], Cell["\<\ We approximate the nonlinear system by a linear system near the equilibrium \ points. Suppose that (a, b) is an equilibrium point, that is f(a, b) = g(a, b) = 0.\ \>", "Subsection", CellChangeTimes->{{3.655098916287817*^9, 3.655098954285157*^9}, { 3.6550992601285563`*^9, 3.655099270260988*^9}, {3.655099309498228*^9, 3.655099317123205*^9}, {3.6998580461646147`*^9, 3.6998580492288303`*^9}}], Cell[TextData[{ "\n", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", " ", RowBox[{ RowBox[{ RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}], " ", "\[TildeEqual]", " ", RowBox[{ RowBox[{"f", "(", RowBox[{"a", ".", "b"}], ")"}], " ", "+", " ", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], RowBox[{"(", RowBox[{"x", "-", "a"}], ")"}]}], " ", "+", " ", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], RowBox[{"(", RowBox[{"y", "-", "b"}], ")"}]}]}]}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], RowBox[{"(", RowBox[{"x", "-", "a"}], ")"}]}], " ", "+", " ", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], RowBox[{"(", RowBox[{"y", "-", "b"}], ")"}], " "}]}]}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", " ", RowBox[{ RowBox[{ RowBox[{"g", "(", RowBox[{"x", ",", "y"}], ")"}], " ", "\[TildeEqual]", " ", RowBox[{ RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}], " ", "+", " ", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], RowBox[{"(", RowBox[{"x", "-", "a"}], ")"}]}], " ", "+", " ", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], RowBox[{"(", RowBox[{"y", "-", "b"}], ")"}]}]}]}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], RowBox[{"(", RowBox[{"x", "-", "a"}], ")"}]}], " ", "+", " ", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], RowBox[{"(", RowBox[{"y", "-", "b"}], ")"}], " "}]}]}]}], 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]], FontColor->GrayLevel[0]], "\n" }], "Subsection", CellChangeTimes->{{3.655098983375843*^9, 3.65509907446028*^9}, { 3.655099108064087*^9, 3.655099287581564*^9}, {3.655099402678459*^9, 3.655099409171327*^9}}], Cell[TextData[{ "The critical point can be translated to 0 via the coordinate change x := x \ - a and y := y - b and still retains all of its properties. The approximated \ system becomes a linear system:\n\n", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", " ", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], "x"}], " ", "+", " ", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], "y"}]}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], "x"}], " ", "+", " ", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], "y", " "}]}]}], 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]], FontColor->GrayLevel[0]], " with coefficient matrix ", Cell[BoxData[ RowBox[{" ", StyleBox[ RowBox[{ SubscriptBox["J", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]], "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { FormBox[ FormBox[ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm], TraditionalForm], FormBox[ RowBox[{" ", RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}]}], TraditionalForm]}, { FormBox[ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm], FormBox[ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm]} }, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}], ")"}]}], FontColor->GrayLevel[0]]}]], FontWeight->"Bold"], StyleBox[" .", FontWeight->"Bold"] }], "Subsection", CellChangeTimes->{{3.655099367558625*^9, 3.655099723802659*^9}, 3.655099761582212*^9, 3.655099939536467*^9}], Cell[TextData[{ "The matrix ", Cell[BoxData[ StyleBox[ RowBox[{ SubscriptBox["J", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]], "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { FormBox[ FormBox[ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}]}], TraditionalForm], TraditionalForm], FormBox[ RowBox[{" ", RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}]}]}], TraditionalForm]}, { FormBox[ RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"g", "(", RowBox[{"x", ",", "y"}], ")"}]}], TraditionalForm], FormBox[ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"g", "(", RowBox[{"x", ",", "y"}], ")"}]}], TraditionalForm]} }, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}], ")"}]}], FontWeight->"Bold", FontColor->GrayLevel[0]]], CellChangeTimes->{{3.655099764839529*^9, 3.65509982016846*^9}}], "is called the Jacobian matrix of the system. It contains the first order \ partial derivatives of f and g evaluated at the point (x,y)." }], "Subsection", CellChangeTimes->{{3.655098916287817*^9, 3.655098954285157*^9}, { 3.6550992601285563`*^9, 3.655099270260988*^9}, {3.655099309498228*^9, 3.655099317123205*^9}, {3.655099836560462*^9, 3.655099898399392*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Stability of equilibrium points:", "Section", CellChangeTimes->{{3.6550988036100883`*^9, 3.655098815790513*^9}, { 3.6998583867172403`*^9, 3.699858434621813*^9}}], Cell[CellGroupData[{ Cell["\<\ We can use the eigenvalues of the matrix Jacobian matrix to decide the type \ and stability of the critical point (a,b). Let \[Lambda] and \[Mu] be the two \ eigenvalues. 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Set x' = 0 and y' = 0. In this case, it is easy to solve the equation by \ hand: x(1-y)=0 and y(x-1)=0 implies (x,y)=(0,0) or (x,y)=(1,1). For more \ complicated cases, we will use Solve[..] or NSolve[..]\ \>", "Subsection", CellChangeTimes->{{3.655087200675926*^9, 3.65508735076502*^9}, { 3.655089281146304*^9, 3.655089288108397*^9}, {3.6551078210711737`*^9, 3.655107847059272*^9}, {3.655107925249029*^9, 3.6551079260887337`*^9}}], Cell[BoxData[ RowBox[{"EqPoints", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", RowBox[{"(", RowBox[{"1", "-", "y"}], ")"}]}], "\[Equal]", "0"}], ",", " ", RowBox[{ RowBox[{"y", RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}]}], "\[Equal]", "0"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.6550873544106283`*^9, 3.655087403633822*^9}, { 3.6550891198578043`*^9, 3.655089137130628*^9}, 3.655089290679175*^9}, FontSize->16, Background->RGBColor[0.87, 0.94, 1]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "2. Next we find the Jacobian matrix of the system. In this case, it is easy \ to compute it by hand: \n ", Cell[BoxData[ RowBox[{ SubscriptBox["J", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]], "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ SubscriptBox["\[PartialD]", "x"], " ", RowBox[{"(", RowBox[{"x", RowBox[{"(", RowBox[{"1", "-", "y"}], ")"}]}], ")"}]}], RowBox[{ SubscriptBox["\[PartialD]", "y"], " ", RowBox[{"(", RowBox[{"x", RowBox[{"(", RowBox[{"1", "-", "y"}], ")"}]}], ")"}]}]}, { RowBox[{ SubscriptBox["\[PartialD]", "x"], " ", RowBox[{"(", RowBox[{"y", RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}]}], ")"}]}], RowBox[{ SubscriptBox["\[PartialD]", "y"], " ", RowBox[{"(", RowBox[{"y", RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}]}], ")"}]}]} }, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}]}]], FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox["=", FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"1", "-", "y"}], RowBox[{"-", "x"}]}, {"y", RowBox[{"x", "-", "1"}]} }, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}]], FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[0.85]] }], "Subsection", CellChangeTimes->{{3.6550874467335443`*^9, 3.65508755989434*^9}, { 3.655087599288155*^9, 3.655087738073269*^9}, {3.655087782176153*^9, 3.655087805036108*^9}, 3.655087870537085*^9, {3.655090135622225*^9, 3.655090188415666*^9}, {3.65509023090219*^9, 3.655090254936325*^9}, { 3.655107824407999*^9, 3.6551078442176933`*^9}, {3.655107929498901*^9, 3.655107930330934*^9}, {3.655107965701727*^9, 3.65510796792603*^9}}], Cell["\<\ In more complicated cases, we will ask Mathematica to compute the Jacobian \ matrix for us, as follows :\ \>", "Subsubsection", CellChangeTimes->{{3.655087818690167*^9, 3.6550878857838182`*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "The commands ", StyleBox["D[f[x], {x}]", FontWeight->"Bold"], " or ", StyleBox["f'[x] ", FontWeight->"Bold"], " give the derivative of f with respect to x. 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We evaluate the Jacobian matrix at the two equilibrium points and find its \ eigenvalues:\ \>", "Subsection", CellChangeTimes->{{3.655087746495585*^9, 3.655087777852461*^9}, 3.6550888656120377`*^9, {3.6551079378226547`*^9, 3.6551079386710777`*^9}, { 3.655107995507238*^9, 3.655108004679715*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"B", "=", RowBox[{ RowBox[{"J", "[", "f", "]"}], " ", "/.", " ", RowBox[{"Flatten", "[", RowBox[{"EqPoints", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", " ", RowBox[{"MatrixForm", "[", "B", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", " ", RowBox[{"Eigenvalues", "[", "B", "]"}]}], "]"}]}], "Input", CellChangeTimes->{{3.6550888851365128`*^9, 3.6550888961578627`*^9}, { 3.655089005102582*^9, 3.6550890198854237`*^9}, {3.655089143982304*^9, 3.655089168057946*^9}, {3.6550892433017063`*^9, 3.6550892556352377`*^9}, { 3.6550893118651857`*^9, 3.6550893594792*^9}, {3.655089811420371*^9, 3.655089815851158*^9}, {3.655108017175014*^9, 3.655108020943777*^9}}, FontSize->16, Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[{ RowBox[{ RowBox[{"T", "=", RowBox[{ RowBox[{"J", "[", "f", "]"}], " ", "/.", " ", RowBox[{"Flatten", "[", RowBox[{"EqPoints", "[", RowBox[{"[", "2", "]"}], "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", RowBox[{"MatrixForm", "[", " ", "T", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", " ", RowBox[{"Eigenvalues", "[", "T", "]"}]}], "]"}]}], "Input", CellChangeTimes->{{3.655089189472192*^9, 3.655089228426024*^9}, { 3.655089655102026*^9, 3.655089678846499*^9}, {3.655089731063439*^9, 3.655089755740191*^9}, {3.655108023368781*^9, 3.6551080288596153`*^9}}, FontSize->16, Background->RGBColor[0.87, 0.94, 1]] }, Open ]], Cell[CellGroupData[{ Cell["4. Conclusions:", "Subsection", CellChangeTimes->{{3.655087746495585*^9, 3.655087777852461*^9}, 3.6550888656120377`*^9, {3.6551079378226547`*^9, 3.6551079386710777`*^9}, { 3.655107995507238*^9, 3.655108004679715*^9}, {3.699858666485015*^9, 3.699858674284013*^9}}], Cell[TextData[StyleBox["By analyzing the Real and Imaginary Part of the \ Eigenvalues, we conclude that the point (0, 0) is a saddle (the Jacobian \ matrix has two real eigenvalues, one smaller than 0, one greater than 0) \ whereas the point (1, 1) is a center (the Jacobian matrix has two complex \ conjugate eigenvalues with Real part equal to 0, and the Phase Portrait of \ the nonlinear equation depicts closed circular trajectories). This gives us \ information about the shape of trajectories in a vicinity of the equilibrium \ solutions.", FontColor->GrayLevel[0]]], "Subsubsection", CellChangeTimes->{{3.655089824616383*^9, 3.655090099994589*^9}, { 3.655105710731675*^9, 3.655105743755711*^9}, 3.699858536674307*^9, { 3.699858637900161*^9, 3.699858645475404*^9}}, Background->GrayLevel[0.85]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Exercise : ", "Section", CellChangeTimes->{{3.655091701850492*^9, 3.6550917048336153`*^9}, 3.655105639643676*^9, {3.655108393211068*^9, 3.6551083967965918`*^9}}], Cell[TextData[{ "Consider the system ", 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]]], ". Study the nature of the critical points (x, y) of the system for which \ -2\[Pi] < x,y < 2\[Pi]. 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