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NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 3636520, 130936]*) (*NotebookOutlinePosition[ 3637258, 130962]*) (* CellTagsIndexPosition[ 3637214, 130958]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Exploring the Theory of Geometric Bifurcation"]], "Title", PageWidth->PaperWidth, TextAlignment->Center], Cell["\<\ Jay H. Wolkowisky Department of Mathematics University of Colorado Boulder, Colorado 80309, USA wolkowis@euclid.colorado.edu\ \>", "Subtitle", PageWidth->PaperWidth, TextAlignment->Center], Cell[CellGroupData[{ Cell["Abstract", "SectionFirst", PageWidth->PaperWidth, TextAlignment->Center], Cell[TextData[{ "This paper will deal with the Theory of Geometric Bifurcation which the \ author developed in 1986. The theory developed in those papers is very \ general and abstract. So, until a symbolic software program such as ", StyleBox["Mathematica", FontSlant->"Italic"], " came along, it was very difficult to examine concrete examples which \ would illustrate and explain the theory. In this paper we will look at \ examples of bifurcating branches of solutions of nonlinear: algebraic \ equations, ordinary differential equations, and partial differential \ equations." }], "Text", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell["1. Introduction", "Section", PageWidth->PaperWidth], Cell[TextData[{ "This paper will investigate nonlinear eigenvalue problems of the form", StyleBox[" \nLw - \[Lambda] w + F(\[Lambda],w) = 0", FontWeight->"Bold"], ", where ", StyleBox["L", FontWeight->"Bold"], " is a linear operator in a Hilbert space ", StyleBox["H", FontWeight->"Bold"], ", with ", StyleBox["\[Lambda]", FontWeight->"Bold"], " being a scalar parameter, and", StyleBox[" F(\[Lambda],w) = o(||w||)", FontWeight->"Bold"], ". Such equations arise in bifurcation (branching) problems where a \ solution becomes non-unique and bifurcates into more than one solution. For \ example, in the buckling of rods, plates, and shells. Also in the field of \ fluid dynamics, such as the Bernard and Taylor problems.\n\nIn this paper we \ will look at examples of the following types:\n(a) ", StyleBox["L = A", FontWeight->"Bold"], ", ", StyleBox["A", FontWeight->"Bold"], " is a n x n matrix,\n(b)", StyleBox[" L =", FontWeight->"Bold"], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{" ", FormBox[\(\[DoubleStruckD]\^2\), "TraditionalForm"]}], \(\[DoubleStruckD]x\^2\)], FontSize->15], TraditionalForm]], FontSize->14, FontWeight->"Bold"], ",\n(c) ", StyleBox["L = ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["\[EmptyDownTriangle]", FontSize->14], "2"], TraditionalForm]], FontWeight->"Bold"], ", the Laplace operator.\n\nWe will investigate the existence(or \ non-existence) of global branches of solutions of the above equations. We \ also will show how to construct these branches. The method used to accomplish \ this is based on the\"Theory of Geometric Bifurcation\" developed by the \ author [", StyleBox["1", FontWeight->"Bold"], ",", StyleBox["2", FontWeight->"Bold"], "]. The above equation has ", StyleBox["w \[Congruent] 0", FontWeight->"Bold"], " as a solution for all values of ", StyleBox["\[Lambda].", FontWeight->"Bold"], " The \"linearized\" equation ", StyleBox["Lw - \[Lambda] w = 0", FontWeight->"Bold"], ", besides having this ", StyleBox["trivial solution ", FontWeight->"Bold"], "also has non-trivial solutions at the eigenvalues of ", StyleBox["L", FontWeight->"Bold"], ", ", StyleBox["\[Lambda] = ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], TraditionalForm]], FontWeight->"Bold"], ". The main question we will be interested in is whether the nonlinear \ equation ", StyleBox[" Lw - \[Lambda] w + F(\[Lambda],w) = 0 ", FontWeight->"Bold"], "has non-trivial solutions bifurcating from the trivial solution at the \ eigenvalues ", StyleBox["\[Lambda] = ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], TraditionalForm]], FontWeight->"Bold"], ". (It is easy to show that if the nonlinear equation has non-trivial \ solutions they must bifurcate from the trivial solution at the eigenvalues of \ the linearized equation). We will be restricting our attention to the case \ that the operator ", StyleBox["L", FontWeight->"Bold"], " is self-adjoint (symmetric), which is the case for most physical models. \ This will allow the results proved in [", StyleBox["1", FontWeight->"Bold"], ",", StyleBox["2", FontWeight->"Bold"], "] to be stated more simply. We will use ", StyleBox["dim ", FontWeight->"Bold"], "and", StyleBox[" null ", FontWeight->"Bold"], "for dimensiom and null space respectively. When ", StyleBox["dim null", FontWeight->"Bold"], "(L - ", Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], TraditionalForm]]], "I) = 1 (", StyleBox["multiplicity of", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], FontWeight->"Bold"], TraditionalForm]]], " ", StyleBox["= 1", FontWeight->"Bold"], "), then bifurcation from the eigenvalue ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], FontWeight->"Bold"], TraditionalForm]]], " always occurs. This case is very simple to handle and not very \ interesting. We will only be looking at the case when the ", StyleBox["multiplicity of ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SubscriptBox["\[Lambda]", StyleBox["i", FontSize->14]], TraditionalForm]], FontWeight->"Bold"], StyleBox[" is \[GreaterEqual] 2", FontWeight->"Bold"], ". If we let\n ", StyleBox[" w = \[Epsilon] (y + \[Eta])", FontWeight->"Bold"], ", (1)\nwhere \n \ ", StyleBox["y", FontWeight->"Bold"], " \[Element] ", StyleBox["N", FontWeight->"Bold"], "= ", StyleBox["null", FontWeight->"Bold"], "(L - ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], "I), \n ", StyleBox[" ||y|| = 1", FontWeight->"Bold"], ",\n and \n \ ", StyleBox[" \[Eta]", FontWeight->"Bold"], " \[Element] ", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox[ StyleBox["N", FontSlant->"Plain"], "\[UpTee]"], FontWeight->"Bold"], TraditionalForm]]], " (the orthogonal compliment of", StyleBox[" N, H =", FontWeight->"Bold"], " ", StyleBox["N ", FontWeight->"Bold"], "\[CirclePlus] ", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox[ StyleBox["N", FontSlant->"Plain"], "\[UpTee]"], FontWeight->"Bold"], TraditionalForm]]], ").\n \nWhere ", StyleBox["\[Epsilon]", FontWeight->"Bold"], " is a small parameter, and ", Cell[BoxData[ FormBox[ StyleBox[\(\[Lambda]\_0\), FontWeight->"Bold"], TraditionalForm]]], " is an eigenvalue of ", StyleBox["L", FontWeight->"Bold"], " with \n ", StyleBox["dim", FontWeight->"Bold"], " ", StyleBox["N ", FontWeight->"Bold"], " ", StyleBox["\[GreaterEqual] 2.", FontWeight->"Bold"], "\nWe can now state the above mentioned results. But first we define ", StyleBox["T", FontWeight->"Bold"], "[y], which is the projection operator of ", StyleBox["H", FontWeight->"Bold"], " onto the tangent plane to the unit sphere in ", StyleBox["N", FontWeight->"Bold"], " at y, i.e.,\n", StyleBox["T", FontWeight->"Bold"], "[y]", StyleBox[" :", FontWeight->"Bold"], " ", StyleBox["H ", FontWeight->"Bold"], "\[RightArrow] ", StyleBox["\[Tau]", FontSize->18], "[y] = {x | \[LeftAngleBracket]x , y\[RightAngleBracket] = 0, x and y \ \[Element] ", StyleBox["N", FontWeight->"Bold"], ", ", StyleBox["|", FontWeight->"Bold"], "|y", StyleBox["||", FontWeight->"Bold"], " = 1}, \nwith \[LeftAngleBracket] , \[RightAngleBracket] being the inner \ product on ", StyleBox["H", FontWeight->"Bold"], ". \n", StyleBox["Theorem 1 ", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox["\n", FontWeight->"Bold"], " ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " ", StyleBox["is a bifurcation point of", FontSlant->"Italic"], " \n Lw - \[Lambda] w + \ F(\[Lambda],w) = 0 (2)\n", StyleBox["if and only if for each", FontSlant->"Italic"], " \[Epsilon] \[Succeeds] 0 ", StyleBox["sufficiently small there exists a", FontSlant->"Italic"], " y \[Element] ", StyleBox["N", FontWeight->"Bold"], " ", StyleBox["with", FontSlant->"Italic"], " ", StyleBox["||", FontWeight->"Bold"], "y", StyleBox["||", FontWeight->"Bold"], " = 1", StyleBox[" such that it satisfies", FontSlant->"Italic"], "\n ", StyleBox["T", FontWeight->"Bold"], "[y]F( ", Cell[BoxData[ \(TraditionalForm\`\(\[Lambda]\&\[Wedge]\)\)]], "( y, \[Epsilon] ), \[Epsilon] (y + ", Cell[BoxData[ \(TraditionalForm\`\(\[Eta]\&\[Wedge]\)\)]], "( y, \[Epsilon] ))) = 0. (3) \n", StyleBox["Where", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\(\[Lambda]\&\[Wedge]\)\)]], "( y, \[Epsilon] ) ", StyleBox["and", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\(\[Eta]\&\[Wedge]\)\)]], "( y, \[Epsilon] ) ", StyleBox["are solutions of the equations", FontSlant->"Italic"], "\n L\[Eta] - ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " \[Eta] - (\[Lambda] - ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], ") (y + \[Eta]) + ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\^\(-1\)\)]], " F(\[Lambda] , \[Epsilon] (y + \[Eta])) = 0 (4)\n \ \[Lambda] - ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " - \[LeftAngleBracket] ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\^\(-1\)\)]], " F(\[Lambda] , \[Epsilon] (y + \[Eta])) , y\[RightAngleBracket] = 0 , \ (5)\n", StyleBox["for a fixed", FontSlant->"Italic"], " y ", StyleBox["and", FontSlant->"Italic"], " \[Epsilon].\n\nLoosely speaking, this theorem says that solutions of (2) \ will bifurcate from the zero solution at an eigenvalue ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " of L if and only if the tangential vector field ", StyleBox["\nT", FontWeight->"Bold"], "[y]F( ", Cell[BoxData[ \(TraditionalForm\`\(\[Lambda]\&\[Wedge]\)\)]], "( y, \[Epsilon] ), \[Epsilon] (y + ", Cell[BoxData[ \(TraditionalForm\`\(\[Eta]\&\[Wedge]\)\)]], "( y, \[Epsilon] )))", StyleBox[" ", FontWeight->"Bold"], "has a zero for each \[Epsilon] sufficiently small .\n\nIt can be shown \ that ", Cell[BoxData[ \(TraditionalForm\`\(\[Lambda]\&\[Wedge]\)\)]], "(y,0) = ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\[Eta]\&\[Wedge]\)\)]], "(y,0) = 0. In addition, if we assume that\n \ F(\[Lambda], \[Epsilon] w) = ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\^\[Alpha]\)]], " F(\[Lambda],w), with \[Alpha]\[Succeeds]1, \nthen the leading term in \ approximating Equation (3) is\n ", StyleBox["T", FontWeight->"Bold"], "[y]F( ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], ",y ) = 0 . 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A ", StyleBox["transverse zero ", FontWeight->"Bold"], "of a vector field is a zero of the vector field such that any small \ perturbation of the vector field will not eliminate the zero. For example, \ in one dimension ", Cell[BoxData[ \(TraditionalForm\`\(x\^3\ \)\)]], "has a transverse zero at x = 0, while ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " does not.\n\nIt is shown in [", StyleBox["1", FontWeight->"Bold"], "] that if the multiplicity of an eigenvalue ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " is", StyleBox[" ", FontVariations->{"Underline"->True}], StyleBox["odd", FontWeight->"Bold", FontVariations->{"Underline"->True}], ", then bifurcation from ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " will occur. In [", StyleBox["2", FontWeight->"Bold"], "] it is shown that if F(\[Lambda],w) is an ", StyleBox["even", FontWeight->"Bold", FontVariations->{"Underline"->True}], " function of the variable w, then bifurcation from ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)]], " will also occur. An even function being one that satisfies the condition \ F(\[Lambda],-w) = F(\[Lambda],w).", StyleBox[ " It is for these reasons that, in this paper, we will be looking at \ examples where the", FontWeight->"Bold"], " ", StyleBox["multiplicity of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)], FontWeight->"Bold"], StyleBox[" is ", FontWeight->"Bold"], StyleBox["even", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" and F(\[Lambda],w) is an ", FontWeight->"Bold"], StyleBox["odd", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" function of w", FontWeight->"Bold"], ". ", StyleBox["In this case bifurcation from ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_0\)], FontWeight->"Bold"], StyleBox[" may or may not occur, as the following examples will show.", FontWeight->"Bold"] }], "Text", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["2. L = A, an n \[Times]n matrix and H = ", "Section"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontSize->16], "n"], TraditionalForm]], FontSize->18, FontWeight->"Bold"], " ." }], "Text", PageWidth->PaperWidth, FontSize->13], Cell[TextData[{ "We first consider the following example for", StyleBox[" n=4.", FontWeight->"Bold"], "\n" }], "Text", PageWidth->PaperWidth, FontSize->13], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"A", "=", TagBox[ RowBox[{"(", GridBox[{ {"1", "0", "0", "0"}, {"0", "1", "0", "0"}, {"0", "0", "2", "0"}, {"0", "0", "0", "2"} }, ColumnAlignments->{Decimal}], ")"}], (MatrixForm[ #]&)]}], ";"}], TraditionalForm]], "Input", CellLabel->"In[48]:=", PageWidth->PaperWidth, FontSize->13], Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{"(", GridBox[{ {"x"}, {"y"}, {"u"}, {"v"} }], ")"}]}], TraditionalForm]], "Input", CellLabel->"In[5]:=", PageWidth->PaperWidth], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ StyleBox["F", FontSlant->"Plain"], "[", \(\[Lambda], w\), "]"}], "=", RowBox[{"(", GridBox[{ {\(x\^3 + y\^3 + u\ y\^2 + v\ y\^2\)}, {\(x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + u\ v\ y\)} }], ")"}]}], ";"}], TraditionalForm]], "Input", CellLabel->"In[19]:=", PageWidth->PaperWidth], Cell["So that equation (2) becomes", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(A . w - \[Lambda]\ w + F[\[Lambda], w] == 0\)], "Input", CellLabel->"In[20]:=", PageWidth->PaperWidth], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ {\(x\^3 + x + y\^3 + u\ y\^2 + v\ y\^2 - x\ \[Lambda]\)}, {\(x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x + y - y\ \[Lambda]\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + 2\ u - u\ \[Lambda]\)}, { \(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + 2\ v + u\ v\ y - v\ \[Lambda]\)} }], ")"}], "==", "0"}], TraditionalForm]], "Output", CellLabel->"Out[20]=", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "The eigenvalues of ", StyleBox["A", FontWeight->"Bold"], " are ", StyleBox["{1, 1, 2, 2}", FontWeight->"Bold"], ". We will be investigating the bifurcation from the double eigenvalue" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[Lambda]\_0 = \ 1\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Therefore ", StyleBox["dim", FontWeight->"Bold"], " ", StyleBox["N", FontWeight->"Bold"], " = ", StyleBox["dim", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[ StyleBox[ SuperscriptBox[ StyleBox["N", FontSlant->"Plain"], "\[UpTee]"], FontWeight->"Bold"], "TraditionalForm"]}], TraditionalForm]]], " ", StyleBox["= 2", FontWeight->"Bold"], "." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ FormBox[ RowBox[{\(In\ this\ case\), ",", " ", RowBox[{"bases", " ", "for", " ", StyleBox["N", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], " ", "and", " ", FormBox[ StyleBox[ 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\[Epsilon] - \[Epsilon]\ \[Lambda]\ \(cos(\[Alpha])\)\)}, { \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(cos\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \(cos(\[Alpha])\)\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + \(sin(\[Alpha])\)\ \[Epsilon] - \[Epsilon]\ \[Lambda]\ \(sin(\[Alpha])\)\)}, { \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + 2\ \[Eta]1\ \[Epsilon] - \[Epsilon]\ \[Eta]1\ \[Lambda]\)}, { \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \[Eta]2\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + 2\ \[Eta]2\ \[Epsilon] - \[Epsilon]\ \[Eta]2\ \[Lambda]\)} }], ")"}], "==", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"0"}, {"0"} }, ColumnAlignments->{Decimal}], ")"}]}], TraditionalForm]], "Output",\ CellLabel->"Out[34]="] }, Closed]], Cell[TextData[{ "Since ", Cell[BoxData[ RowBox[{"(", GridBox[{ {\(Sin[\[Alpha]]\)}, {\(-Cos[\[Alpha]]\)}, {"0"}, {"0"} }], ")"}]], FontWeight->"Bold"], "is orthogonal to ", Cell[BoxData[ StyleBox[ RowBox[{"(", GridBox[{ {\(Cos[\[Alpha]]\)}, {\(Sin[\[Alpha]]\)}, {"0"}, {"0"} }], ")"}], FontWeight->"Bold"]]], " then Equation (3) becomes " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Dot", "[", RowBox[{ RowBox[{"Flatten", "[", RowBox[{"(", GridBox[{ {\(Sin[\[Alpha]]\)}, {\(-Cos[\[Alpha]]\)}, {"0"}, {"0"} }], ")"}], "]"}], ",", RowBox[{"Flatten", "[", RowBox[{ RowBox[{"(", GridBox[{ {\(x\^3 + y\^3 + u\ y\^2 + v\ y\^2\)}, {\(x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + u\ v\ y\)} }], ")"}], "/.", \({x -> \[Epsilon]\ Cos[\[Alpha]], y -> \[Epsilon]\ Sin[\[Alpha]], u -> \[Epsilon]\ 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We can calculate these zeros by" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]1 = \[Alpha] /. FindRoot[t[\[Alpha]], {\[Alpha], .8}]\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \(0.785398165043810614`\)], "Output", CellLabel->"Out[21]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]2 = \[Alpha] /. FindRoot[t[\[Alpha]], {\[Alpha], 2.2}]\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(2.16956303205093758`\)], "Output", CellLabel->"Out[22]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]3 = \[Alpha] /. FindRoot[t[\[Alpha]], {\[Alpha], 3.8}]\)], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \(3.92699081701547356`\)], "Output", CellLabel->"Out[41]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Alpha]4 = \[Alpha] /. FindRoot[t[\[Alpha]], {\[Alpha], 5.3}]\)], "Input", CellLabel->"In[42]:="], Cell[BoxData[ \(5.31115568565196927`\)], "Output", CellLabel->"Out[42]="] }, Closed]], Cell[TextData[{ "We note that ", StyleBox["\[Alpha]1 ", FontWeight->"Bold"], "-", StyleBox[" \[Alpha]3 = \[Pi]", FontWeight->"Bold"], " and ", StyleBox["\[Alpha]2 - \[Alpha]4 = \[Pi]", FontWeight->"Bold"], ", which means that the solution branches bifurcate in pairs in opposite \ directions. This is expected since the ", StyleBox["nonlinearity is cubic", FontWeight->"Bold"], ", so that if ", StyleBox["(x,y,u,v)", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "is a solution then ", StyleBox["(-x,-y,-u,-v)", FontFamily->"Terminal", FontWeight->"Bold"], " is also a solution." }], "Text"], Cell["\<\ Now that we know bifurcation will occur, we can construct the bifurcating \ branches.\ \>", "Text"], Cell[CellGroupData[{ Cell["2.1 Constructing the Solution Branches", "Subsection"], Cell[TextData[{ "We first find ", StyleBox["\[Eta]1", FontWeight->"Bold"], " and ", StyleBox["\[Eta]2", FontWeight->"Bold"], " for small ", StyleBox["\[Epsilon]", FontWeight->"Bold"], "." }], "Text"], Cell[TextData[{ "Equation (4) can be written, using, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[\(\[Lambda]\_0\), FontWeight->"Bold"], " ", StyleBox["=", FontWeight->"Bold"], StyleBox["1", FontWeight->"Bold"]}], ","}], TraditionalForm]]], " as" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"A", ".", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"\[Eta]1"}, {"\[Eta]2"} }], ")"}]}], " ", "-", " ", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"\[Eta]1"}, {"\[Eta]2"} }], ")"}], " ", "-", " ", RowBox[{\((\[Lambda]\ - \ 1)\), " ", RowBox[{"(", GridBox[{ {\(Cos[\[Alpha]]\)}, {\(Sin[\[Alpha]]\)}, {"\[Eta]1"}, {"\[Eta]2"} }], ")"}]}], " ", "+", RowBox[{\(1\/\[Epsilon]\), " ", RowBox[{"(", GridBox[{ {\(x\^3 + y\^3 + u\ y\^2 + v\ y\^2\)}, {\(x\^3 + v\ x\^2 + y\^2\ x + u\ y\ x\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2\)}, {\(x\^3 + u\ y\ x + y\^3 + v\ y\^2 + u\ v\ y\)} }], ")"}]}]}], "/.", \({x -> \[Epsilon]\ Cos[\[Alpha]], y -> \[Epsilon]\ Sin[\[Alpha]], u -> \[Epsilon]\ \[Eta]1, v -> \[Epsilon]\ \[Eta]2}\)}], ")"}], "==", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"0"}, {"0"} }], ")"}]}], " ", \( (*8*) \)}]], "Input", CellLabel->"In[49]:="], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ { \(\(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3\)\/\[Epsilon] - \((\[Lambda] - 1)\)\ \(cos(\[Alpha])\)\)}, { \(\(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(cos\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \(cos(\[Alpha])\)\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3\)\/\[Epsilon] - \((\[Lambda] - 1)\)\ \(sin(\[Alpha])\)\)}, { \(\[Eta]1 - \[Eta]1\ \((\[Lambda] - 1)\) + \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3\)\/\[Epsilon]\)}, { \(\[Eta]2 - \[Eta]2\ \((\[Lambda] - 1)\) + \(\(\(cos\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \(\(sin\^3\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]2\ \(\(sin\^2\)(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \[Eta]2\ \(sin(\[Alpha])\)\ \[Epsilon]\^3 + \[Eta]1\ \(cos(\[Alpha])\)\ \(sin(\[Alpha])\)\ \[Epsilon]\^3\)\/\[Epsilon]\)} }], ")"}], "==", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"0"}, {"0"} }, ColumnAlignments->{Decimal}], ")"}]}], TraditionalForm]], "Output",\ CellLabel->"Out[49]="] }, Closed]], Cell["The first two elements in the above vector equation are ", "Text"], Cell[BoxData[ \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]1\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 - \((\[Lambda] - 1)\)\ Cos[\[Alpha]] == 0\)], "Input"], Cell["and", "Text"], Cell[BoxData[ \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]2\ Cos[\[Alpha]]\^2\ \[Epsilon]\^2 + Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]]\ \[Epsilon]\^2 - \((\[Lambda] - 1)\)\ sin\ \[Alpha] == 0\)], "Input"], Cell[TextData[{ "Solving each of these for ", StyleBox["(\[Lambda] - 1)", FontWeight->"Bold"], ", we get" }], "Text"], Cell[BoxData[ \(\((\[Lambda] - 1)\)\ == \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]1\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]2\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2\)\/Cos[\[Alpha]]\)], "Input", CellLabel->"In[4]:="], Cell["and", "Text"], Cell[BoxData[ \(\((\[Lambda] - 1)\)\ == \(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]2\ Cos[\[Alpha]]\^2\ \[Epsilon]\^2 + Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ \[Epsilon]\^2 + \[Eta]1\ Cos[\[Alpha]]\ Sin[\[Alpha]]\ \[Epsilon]\^2\)\/Sin[ \[Alpha]]\)], "Input"], Cell["Equating these we get", "Text"], Cell[BoxData[ \(\(Cos[\[Alpha]]\^3\ \[Epsilon]\^2 + Sin[\[Alpha]]\^3\ \[Epsilon]\^2 + \[Eta]1\ Sin[\[Alpha]]\^2\ \[Epsilo