(* 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[ 1148679, 21486] NotebookOptionsPosition[ 1142938, 21295] NotebookOutlinePosition[ 1143296, 21311] CellTagsIndexPosition[ 1143253, 21308] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Nonlinear Differential Equations:", FontColor->GrayLevel[0]]], "Section", CellChangeTimes->{{3.622973133683782*^9, 3.622973152291214*^9}, { 3.6550907148775053`*^9, 3.6550907364464083`*^9}, {3.69985370934711*^9, 3.6998537109538937`*^9}}, FontSize->36], Cell[TextData[{ "Consider the system ", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", " ", RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", " ", RowBox[{"g", "(", RowBox[{"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]]], " 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}, {3.700278413730899*^9, 3.700278416814308*^9}, { 3.70027853041405*^9, 3.700278544706778*^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", "'"}], "=", StyleBox[" ", FontSize->24], RowBox[{ RowBox[{ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "x"], FontSize->24], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], "x"}], " ", "+", StyleBox[" ", FontSize->24], RowBox[{ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "y"], FontSize->24], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], "y"}]}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", RowBox[{ RowBox[{ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "x"], FontSize->24], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], "x"}], " ", "+", " ", RowBox[{ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "y"], FontSize->24], 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]], FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontWeight->"Bold"], " with coefficient matrix ", Cell[BoxData[ RowBox[{" ", StyleBox[ RowBox[{ StyleBox[ SubscriptBox["J", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]], FontSize->24], "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { FormBox[ FormBox[ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "x"], FontSize->24], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm], TraditionalForm], FormBox[ RowBox[{" ", RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "y"], FontSize->24], RowBox[{"f", "(", RowBox[{"a", ",", "b"}], ")"}]}]}], TraditionalForm]}, { FormBox[ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "x"], FontSize->24], RowBox[{"g", "(", RowBox[{"a", ",", "b"}], ")"}]}], TraditionalForm], FormBox[ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "y"], FontSize->24], 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[{ StyleBox[ SubscriptBox["J", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]], FontSize->24], "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { FormBox[ FormBox[ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "x"], FontSize->24], RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}]}], TraditionalForm], TraditionalForm], FormBox[ RowBox[{" ", RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "y"], FontSize->24], RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}]}]}], TraditionalForm]}, { FormBox[ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "x"], FontSize->24], RowBox[{"g", "(", RowBox[{"x", ",", "y"}], ")"}]}], TraditionalForm], FormBox[ RowBox[{ StyleBox[ SubscriptBox["\[PartialD]", "y"], FontSize->24], 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[CellGroupData[{ 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]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"y", "\[Rule]", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "1"}], ",", RowBox[{"y", "\[Rule]", "1"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.6550873839450073`*^9, 3.6550874045247602`*^9}, 3.655089140179831*^9, 3.655089291817565*^9, 3.65510804439375*^9, 3.700279172275207*^9}] }, Open ]] }, 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. If f is a function of several \ variables, ", StyleBox["D[f[x, y], {x}]", FontWeight->"Bold"], " or ", Cell[BoxData[ StyleBox[ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"(", RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], ")"}]}], FontWeight->"Bold"]], CellChangeTimes->{{3.655090197589192*^9, 3.655090220567108*^9}, { 3.65509026422198*^9, 3.655090267317932*^9}}], " gives the partial derivative of f with respect to x. Try them out!" }], "Subsubsection", CellChangeTimes->{{3.6551052006981153`*^9, 3.655105364562283*^9}, { 3.655105415078999*^9, 3.655105419272311*^9}, 3.65510544928227*^9, 3.6551055207155*^9, 3.699858115315482*^9}, FontColor->GrayLevel[0]], Cell[BoxData[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"(", RowBox[{ RowBox[{"x", " ", "y"}], "+", " ", RowBox[{"y", "^", "2"}]}], ")"}]}], "\[IndentingNewLine]", RowBox[{"D", "[", RowBox[{ RowBox[{"y", "^", "2"}], ",", " ", RowBox[{"{", "y", "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.655090197589192*^9, 3.655090220567108*^9}, { 3.65509026422198*^9, 3.655090267317932*^9}, {3.655105345393256*^9, 3.655105345728566*^9}, {3.655105558163006*^9, 3.655105562779581*^9}, { 3.700279139255945*^9, 3.700279140175475*^9}}, Background->GrayLevel[0.85]] }, Open ]], Cell[CellGroupData[{ Cell["The following code computes the Jacobian matrix:", "Subsubsection", CellChangeTimes->{{3.655087818690167*^9, 3.6550878857838182`*^9}, { 3.655105454899314*^9, 3.655105475052891*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"f", ",", "J", ",", "x", ",", "y"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", " ", RowBox[{"{", RowBox[{ RowBox[{"x", RowBox[{"(", RowBox[{"1", "-", "y"}], ")"}]}], ",", " ", RowBox[{"y", RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}]}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"J", "[", "f_", "]"}], ":=", "\[IndentingNewLine]", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"A", "=", RowBox[{"Table", "[", RowBox[{"0", ",", "2", ",", "2"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"A", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"{", "x", "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"A", "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"{", "y", "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"A", "[", RowBox[{"[", RowBox[{"2", ",", "1"}], "]"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", "x", "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"A", "[", RowBox[{"[", RowBox[{"2", ",", "2"}], "]"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", "y", "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", "A", "]"}]}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"J", "[", "f", "]"}]}], "Input", CellChangeTimes->{{3.655087912388377*^9, 3.6550880280346317`*^9}, { 3.65508807522073*^9, 3.6550880948034363`*^9}, {3.655088143861784*^9, 3.655088316593759*^9}, {3.655088352278129*^9, 3.655088500144615*^9}, { 3.655088799529388*^9, 3.655088810349916*^9}, {3.655088971895095*^9, 3.6550889970348454`*^9}, {3.655089052477755*^9, 3.655089080034995*^9}, { 3.65509029294143*^9, 3.655090296204915*^9}, {3.655090368388937*^9, 3.655090376711759*^9}}, FontSize->16, Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"1", "-", "y"}], ",", RowBox[{"-", "x"}]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{ RowBox[{"-", "1"}], "+", "x"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.65508881116529*^9, {3.6550889771953363`*^9, 3.655088998696457*^9}, 3.655089083925418*^9, 3.655089296924486*^9, 3.655090370405978*^9, 3.655105106353671*^9, 3.655108032890057*^9, 3.699857992174074*^9, 3.7002791850272207`*^9}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3. 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The nonlinear system \ does not have a center! The origin is a spiral source. " }], "Subsubsection", CellChangeTimes->{{3.6555358449466057`*^9, 3.6555358522065372`*^9}, { 3.655535997301132*^9, 3.655536003167778*^9}, {3.700280029911862*^9, 3.700280032054591*^9}}] }, 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]]], ". \n", StyleBox["Study the nature of the critical points (x, y) of the system for \ which -2\[Pi] < x,y < 2\[Pi]. Use only ", FontColor->GrayLevel[0]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" commands, even when computing the critical points!", FontColor->GrayLevel[0]] }], "Subsubsection", 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}, { 3.699858082914884*^9, 3.6998580851523333`*^9}, 3.7002809475870123`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Exercise (predator-prey system): ", "Section", CellChangeTimes->{{3.622790466683278*^9, 3.622790476625743*^9}, { 3.622790511711554*^9, 3.6227905335575237`*^9}, {3.6227906625672903`*^9, 3.622790669390012*^9}, {3.6538711094179993`*^9, 3.65387111076064*^9}, { 3.653874243652176*^9, 3.653874255935793*^9}, {3.654463985629437*^9, 3.654464002198286*^9}, {3.654464069364059*^9, 3.654464102635427*^9}, { 3.655176007208185*^9, 3.655176007686988*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Consider the system usually described as a predator - prey model (Voltera, \ 1931):\n ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox["dx", "dt"], FontSize->36], StyleBox["=", FontSize->24], StyleBox[ RowBox[{ RowBox[{"0.2", "x"}], "-", RowBox[{"0.7", "xy"}]}], FontSize->24]}], TraditionalForm]], FontSize->36], StyleBox[",", FontSize->36], " ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["dy", "dt"], StyleBox["=", FontSize->24], StyleBox[ RowBox[{ RowBox[{ RowBox[{"-", "0.4"}], "y"}], "+", RowBox[{"0.5", "xy"}]}], FontSize->24]}], TraditionalForm]], FontSize->36], "." }], "Subsubsection", CellChangeTimes->{{3.6544642408637457`*^9, 3.654464263463875*^9}, { 3.65446429674226*^9, 3.6544644794677*^9}, {3.654464601362609*^9, 3.654464602290267*^9}, {3.654464650514975*^9, 3.654464665223145*^9}}], Cell["\<\ where you can think of the function x(t) as representing a population of \ rabbits that naturally grow at a rate proportional to their population (that \ is, exponential growth in the absence of predators) and the function y(t) as \ representing a population of foxes that naturally decline (that is, \ exponential decay in the absence of prey). The term xy in both equations is \ proportional to the number of likely encounters of the two population in a \ certain environment. Encounters are detrimental to the rabbits and beneficial \ to the foxes.\ \>", "Text", CellChangeTimes->{{3.654464605624176*^9, 3.654464628508052*^9}, { 3.654465226697503*^9, 3.6544652297351027`*^9}, {3.6544652632054167`*^9, 3.654465270382663*^9}}, TextJustification->1., FontSize->16], Cell[CellGroupData[{ Cell[TextData[{ "Plot the vector field using ", StyleBox["StreamPlot[...]", FontWeight->"Bold"], " in the region that makes sense for the given problem. On the same plot, \ plot the trajectory starting at the point (0, 1) in Red, the trajectory \ starting at the point (1, 0) in Green, the trajectory starting at (1,1) in \ Orange, and the trajectory starting at (5, 2) in Purple. What do you notice? \ How do the solutions look like?" }], "ItemNumbered", CellChangeTimes->{{3.622814562760298*^9, 3.622814564601383*^9}, 3.6228147114882507`*^9, {3.654465494368629*^9, 3.654465595364773*^9}, { 3.70028038372861*^9, 3.7002804037473516`*^9}}, TextJustification->1., FontSize->16], Cell["\<\ Find the critical points of the system and describe their nature.\ \>", "ItemNumbered", CellChangeTimes->{{3.622814562760298*^9, 3.622814588339142*^9}, { 3.6228146853418417`*^9, 3.6228146932370367`*^9}, {3.622816137363825*^9, 3.622816145483821*^9}, {3.654465777331684*^9, 3.654465791122643*^9}, { 3.7002804090977716`*^9, 3.7002804296650677`*^9}}, TextJustification->1., FontSize->16], Cell[TextData[{ "Assume that initially there are 5 rabbits and 2 foxes. Use ", StyleBox["NDSolve[...]", FontWeight->"Bold"], " or ", StyleBox["DSolve[...]", FontWeight->"Bold"], " to find the solution corresponding to this initial condition. Try plotting \ the solution that NDSolve returns. 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