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Estos par\[AAcute]metros se \ entienden como funciones de g1 y g2 en tanto que son los que hacen que se \ verifique la bifurcaci\[OAcute]n.\ \>", "Section"], Cell[BoxData[{ \(\(b1[g1_, \ g2_] := \ b11\ g1\ + \ b12\ g2\ + \ b111\ g1^2\ + \ b122\ g2^2\ + \ b112\ g1\ g2\ + \ b1111\ g1^3\ + \ b1222\ g2^3\ + \ b1112\ g1^2\ g2\ + \ b1122\ g1\ g2^2\ + \ b11111\ g1^4\ + \ b11112\ g1^3\ g2\ + \ b11122\ g1^2\ g2^2\ + \ b11222\ g1\ g2^3\ + \ b12222\ g2^4;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(b2[g1_, \ g2_] := \ b21\ g1\ + \ b22\ g2\ + \ b211\ g1^2\ + \ b222\ g2^2\ + \ b212\ g1\ g2\ + \ b2111\ g1^3\ + \ b2222\ g2^3\ + \ b2112\ g1^2\ g2\ + \ b2122\ g1\ g2^2\ + \ \ b21111\ g1^4\ + \ b21112\ g1^3\ g2\ + \ b21122\ g1^2\ g2^2\ + \ b21222\ g1\ g2^3\ + \ b22222\ g2^4;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(L[g1_, \ g2_] := \ c0\ + \ c1\ g1\ + \ c2\ g2\ + \ c3\ g1^2\ + \ c4\ g2^2\ + \ c5\ g1\ g2\ + \ c6\ g1^3\ + \ c7\ g2^3\ + \ c8\ g1^2\ g2\ + \ c9\ g1\ g2^2\ + \ c10\ g1^4\ + \ c11\ g1^3\ g2\ + \ c12\ g1^2\ g2^2\ + \ c13\ g1\ g2^3\ + \ c14\ g2^4;\)\[IndentingNewLine]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["La funci\[OAcute]n \"cosa\" es la ecuaci\[OAcute]n de autovalores", \ "Section"], Cell[BoxData[ \(cosa[g1_, \ g2_] := Det[J[g1, \ g2]\ - \ L[g1, \ g2]\ IdentityMatrix[4]]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "El sistema de ecuaciones se plantea t\[EAcute]rmino a t\[EAcute]rmino en \ el desarrollo en series de g1 y g2. ", StyleBox["Mathematica", FontSlant->"Italic"], " se toma alg\[UAcute]n tiempo en simplificar el sistema. " }], "Section"], Cell[BoxData[ \(ecs\ = \ Simplify[{cosa[0, \ 0]\ \[Equal] \ 0, \ \(\(Derivative[1, \ 0]\)[cosa]\)[0, \ 0]\ \[Equal] \ 0, \ \(\(Derivative[0, \ 1]\)[cosa]\)[0, \ 0]\ \[Equal] \ 0, \ \(\(Derivative[2, \ 0]\)[cosa]\)[0, \ 0]\ \[Equal] \ 0, \ \(\(Derivative[0, \ 2]\)[cosa]\)[0, \ 0]\ \[Equal] \ 0, \ \(\(Derivative[1, \ 1]\)[cosa]\)[0, \ 0]\ \[Equal] \ 0, \ \(\(Derivative[3, \ 0]\)[cosa]\)[0, \ 0] \[Equal] \ 0, \ \(\(Derivative[0, \ 3]\)[cosa]\)[0, \ 0] \[Equal] \ 0, \ \(\(Derivative[2, \ 1]\)[cosa]\)[0, \ 0] \[Equal] \ 0, \ \(\(Derivative[1, \ 2]\)[cosa]\)[0, \ 0] \[Equal] \ 0, \ \(\(Derivative[4, \ 0]\)[cosa]\)[0, \ 0] \[Equal] 0, \ \(\(Derivative[3, 1]\)[cosa]\)[0, \ 0] \[Equal] 0, \ \(\(Derivative[2, \ 2]\)[cosa]\)[0, \ 0] \[Equal] 0, \ \(\(Derivative[1, \ 3]\)[cosa]\)[0, \ 0] \[Equal] 0, \ \(\(Derivative[0, \ 4]\)[cosa]\)[0, \ 0] \[Equal] 0}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Para hacerle m\[AAcute]s sencillas las cosas al ", StyleBox["Mathematica,", FontSlant->"Italic"], " fui resolviendo un orden por vez, y usando las soluciones de unas \ ecuaciones en otras, subiendo en los ordenes del desarrollo. Aqu\[IAcute] \ abajo lo que se ve es el estado del sistema de ecuaciones luego de haber \ resuelto los coeficientes hasta \"c9\", con la entrada particular de c0 =\ \[ImaginaryI]", Cell[BoxData[ \(\@k1\)]], ". ( Luego est\[AAcute] hecho con c0 = \[ImaginaryI]", Cell[BoxData[ \(\@k2\)]], ". ) Adem\[AAcute]s ya he fijado los primeros coeficientes de b1 y b2 en \ los valores que los c\[AAcute]lculos de menor orden requieren para que haya \ bifurcaci\[OAcute]n: partes reales de los autovalores iguales a cero. " }], "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(ss1\ = Simplify[Solve[ ReplaceAll[ ecs, \ {c0 \[Rule] \ I\ Sqrt[k1], c1 \[Rule] b11\/2, \ c2 \[Rule] b12\/2, c3 \[Rule] \(4\ \[ImaginaryI] - \[ImaginaryI]\ b11\^2\ \((k1 - \ k2)\) + 4\ b111\ \@k1\ \((k1 - k2)\)\)\/\(8\ \@k1\ \((k1 - k2)\)\), c4 \[Rule] \(\(-\[ImaginaryI]\)\ \((4 + b12\^2)\)\ k1 + 4\ b122\ \ k1\^\(3/2\) + \[ImaginaryI]\ b12\^2\ k2 - 4\ b122\ \@k1\ k2\)\/\(8\ \@k1\ \ \((k1 - k2)\)\), c5 \[Rule] \(4 - 2\ b112\ k1 + \(\[ImaginaryI]\ b11\ b12\ \((k1 \ - k2)\)\)\/\@k1 + 2\ b112\ k2\)\/\(\(-4\)\ k1 + 4\ k2\), \ c6 \[Rule] \(2\ \@k1\ \((b21 + b1111\ \((k1 - k2)\)\^2)\) - \ \[ImaginaryI]\ b11\ \((\(-2\)\ \[ImaginaryI]\ \@k1 + b111\ k1\^2 - 2\ b111\ \ k1\ k2 + b111\ k2\^2)\)\)\/\(4\ \@k1\ \((k1 - k2)\)\^2\), c7 \[Rule] \(\(-2\)\ b22\ k1\^\(3/2\) + 2\ b1222\ \@k1\ \((k1 - \ k2)\)\^2 - \[ImaginaryI]\ b12\ \((b122\ \((k1 - k2)\)\^2 + 2\ \[ImaginaryI]\ \ \@k1\ k2)\)\)\/\(4\ \@k1\ \((k1 - k2)\)\^2\), c8 \[Rule] \(-\(\(1\/\(4\ \@k1\ \((k1 - k2)\)\^2\)\) \((\ \[ImaginaryI]\ \((2\ \[ImaginaryI]\ b22\ \@k1 + 2\ b11\ k1 - 4\ b21\ k1 + b11\ b112\ k1\^2 + 2\ \[ImaginaryI]\ b1112\ k1\^\(5/2\) + 2\ b11\ k2 - 2\ b11\ b112\ k1\ k2 - 4\ \[ImaginaryI]\ b1112\ k1\^\(3/2\)\ k2 + b11\ b112\ k2\^2 + 2\ \[ImaginaryI]\ b1112\ \@k1\ k2\^2 + b12\ \((\(-2\)\ \[ImaginaryI]\ \@k1 + b111\ k1\^2 - 2\ b111\ k1\ k2 + b111\ k2\^2)\))\))\)\)\), c9 \[Rule] \(-\(1\/\(4\ \@k1\ \((k1 - k2)\)\^2\)\)\) \((\ \[ImaginaryI]\ \((\(-4\)\ b22\ k1 - 2\ \[ImaginaryI]\ b21\ k1\^\(3/2\) + b11\ b122\ k1\^2 + 2\ \[ImaginaryI]\ b1122\ k1\^\(5/2\) + 2\ \[ImaginaryI]\ b11\ \@k1\ k2 - 2\ b11\ b122\ k1\ k2 - 4\ \[ImaginaryI]\ b1122\ k1\^\(3/2\)\ k2 + b11\ b122\ k2\^2 + 2\ \[ImaginaryI]\ b1122\ \@k1\ k2\^2 + b12\ \((b112\ k1\^2 + k1\ \((2 - 2\ b112\ k2)\) + k2\ \((2 + b112\ k2)\))\))\))\), \ b11\ \[Rule] \ 0, \ b12\ \[Rule] \ 0, \ b21\ \[Rule] \ 0, \ b22\ \[Rule] \ 0, \ b111 \[Rule] \ 0, \ b211\ \[Rule] \ 0, \ b122\ \[Rule] \ 0, \ b222 \[Rule] \ 0, \ b112\ \[Rule] \ \(\(2\)\(\ \)\)\/\(k1 - k2\), \ b212\ \[Rule] \ \(-2\)\/\(k1 - k2\)}], \ {c10, \ c11, \ c12, \ c13, \ c14}]]\)], "Input"], Cell[BoxData[ \({{c10 \[Rule] 1\/8\ \((4\ b11111 - \(\[ImaginaryI]\ \((5\ k1 - \ k2)\)\)\/\(k1\^\(3/2\)\ \((k1 - k2)\)\^3\))\), c12 \[Rule] 1\/4\ \((2\ b11122 - \(\[ImaginaryI]\ \((5\ k1 - k2)\)\)\/\(\@k1\ \ \((k1 - k2)\)\^3\))\), c14 \[Rule] 1\/8\ \((4\ b12222 - \(\[ImaginaryI]\ \@k1\ \((k1 + 3\ \ k2)\)\)\/\((k1 - k2)\)\^3)\), c11 \[Rule] b11112\/2, c13 \[Rule] b11222\/2}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ El vector sol1 contiene los coeficientes {c0, c1, ..., c9, [c10, c11, c12, \ c13, 14]}, los \[UAcute]ltimos 5 est\[AAcute]n sacados de la \ soluci\[OAcute]n anterior. (Todav\[IAcute]a aparecen algunos coeficientes del \ desarollo de los b's que en realidad ya fueron fijados para hacer el c\ \[AAcute]lculo anterior, pero as\[IAcute] puede verse por qu\[EAcute] han \ sido tomados como se tomaron: b11 = b12 = 0, etc.)\ \>", "Section"], Cell[BoxData[ \(\(sol1\ = \ {I\ Sqrt[k1], \ b11\/2, b12\/2\ , \ \(4\ \[ImaginaryI] - \[ImaginaryI]\ b11\^2\ \((k1 - k2)\ \) + 4\ b111\ \@k1\ \((k1 - k2)\)\)\/\(8\ \@k1\ \((k1 - k2)\)\), \(\(-\ \[ImaginaryI]\)\ \((4 + b12\^2)\)\ k1 + 4\ b122\ k1\^\(3/2\) + \[ImaginaryI]\ \ b12\^2\ k2 - 4\ b122\ \@k1\ k2\)\/\(8\ \@k1\ \((k1 - k2)\)\), \(4 - 2\ b112\ \ k1 + \(\[ImaginaryI]\ b11\ b12\ \((k1 - k2)\)\)\/\@k1 + 2\ b112\ k2\)\/\(\(-4\ \)\ k1 + 4\ k2\), \ \ \(2\ \@k1\ \((b21 + b1111\ \((k1 - k2)\)\^2)\) - \ \[ImaginaryI]\ b11\ \((\(-2\)\ \[ImaginaryI]\ \@k1 + b111\ k1\^2 - 2\ b111\ \ k1\ k2 + b111\ k2\^2)\)\)\/\(4\ \@k1\ \((k1 - k2)\)\^2\), \(\(-2\)\ b22\ k1\^\ \(3/2\) + 2\ b1222\ \@k1\ \((k1 - k2)\)\^2 - \[ImaginaryI]\ b12\ \((b122\ \ \((k1 - k2)\)\^2 + 2\ \[ImaginaryI]\ \@k1\ k2)\)\)\/\(4\ \@k1\ \((k1 - \ k2)\)\^2\), \(-\(1\/\(4\ \@k1\ \((k1 - k2)\)\^2\)\)\) \((\[ImaginaryI]\ \((2\ \ \[ImaginaryI]\ b22\ \@k1 + 2\ b11\ k1 - 4\ b21\ k1 + b11\ b112\ k1\^2 + 2\ \[ImaginaryI]\ b1112\ k1\^\(5/2\) + 2\ b11\ k2 - 2\ b11\ b112\ k1\ k2 - 4\ \[ImaginaryI]\ b1112\ k1\^\(3/2\)\ k2 + b11\ b112\ k2\^2 + 2\ \[ImaginaryI]\ b1112\ \@k1\ k2\^2 + b12\ \((\(-2\)\ \[ImaginaryI]\ \@k1 + b111\ k1\^2 - 2\ b111\ k1\ k2 + b111\ k2\^2)\))\))\), \(-\(1\/\(4\ \@k1\ \((k1 - \ k2)\)\^2\)\)\) \((\[ImaginaryI]\ \((\(-4\)\ b22\ k1 - 2\ \[ImaginaryI]\ b21\ k1\^\(3/2\) + b11\ b122\ k1\^2 + 2\ \[ImaginaryI]\ b1122\ k1\^\(5/2\) + 2\ \[ImaginaryI]\ b11\ \@k1\ k2 - 2\ b11\ b122\ k1\ k2 - 4\ \[ImaginaryI]\ b1122\ k1\^\(3/2\)\ k2 + b11\ b122\ k2\^2 + 2\ \[ImaginaryI]\ b1122\ \@k1\ k2\^2 + b12\ \((b112\ k1\^2 + k1\ \((2 - 2\ b112\ k2)\) + k2\ \((2 + b112\ k2)\))\))\))\), \ \ \(\(ss1[\([1]\)]\)[\([1]\)]\)[\([2]\)], \ \(\(ss1[\([1]\)]\)[\([4]\)]\)[\([2]\ \)], \ \(\(ss1[\([1]\)]\)[\([2]\)]\)[\([2]\)], \ \ \(\(ss1[\([1]\)]\)[\([5]\)]\)[\([2]\)], \ \(\(ss1[\([1]\)]\)[\([3]\)]\)[\([2]\ \)]};\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "El mismo juego para el otro autovalor: \[ImaginaryI] ", Cell[BoxData[ \(\@k2\)]] }], "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(ss2\ = Simplify[Solve[ ReplaceAll[ ecs, \ {c0 \[Rule] \ I\ Sqrt[k2], c1 \[Rule] \ b21\/2, c2 \[Rule] \ b22\/2, c3 \[Rule] \ \(4\ \[ImaginaryI] + \[ImaginaryI]\ b21\^2\ \((k1 \ - k2)\) + 4\ b211\ \@k2\ \((\(-k1\) + k2)\)\)\/\(8\ \@k2\ \((\(-k1\) + \ k2)\)\), c4 \[Rule] \ \(\[ImaginaryI]\ b22\^2\ \((k1 - k2)\) - 4\ \ \[ImaginaryI]\ k2 + 4\ b222\ \@k2\ \((\(-k1\) + k2)\)\)\/\(8\ \@k2\ \ \((\(-k1\) + k2)\)\), c5 \[Rule] \ \(4 + 2\ b212\ \((k1 - k2)\) + \(\[ImaginaryI]\ \ b21\ b22\ \((\(-k1\) + k2)\)\)\/\@k2\)\/\(4\ \((k1 - k2)\)\), \ c6 \[Rule] \ \(\(-\[ImaginaryI]\)\ b21\ \((b211\ \((k1 - \ k2)\)\^2 - 2\ \[ImaginaryI]\ \@k2)\) + 2\ \((b11 + b2111\ \((k1 - k2)\)\^2)\)\ \ \@k2\)\/\(4\ \((k1 - k2)\)\^2\ \@k2\), c7 \[Rule] \ \(\(-\[ImaginaryI]\)\ b22\ \((b222\ \((k1 - \ k2)\)\^2 + 2\ \[ImaginaryI]\ k1\ \@k2)\) + 2\ \@k2\ \((b2222\ \((k1 - \ k2)\)\^2 - b12\ k2)\)\)\/\(4\ \((k1 - k2)\)\^2\ \@k2\), c8 \[Rule] \ \(-\(1\/\(4\ \((k1 - k2)\)\^2\ \@k2\)\)\) \((\ \[ImaginaryI]\ \((b211\ b22\ \((k1 - k2)\)\^2 + 2\ \[ImaginaryI]\ \@k2\ \((b12 - b22 + b2112\ k1\^2 + 2\ \[ImaginaryI]\ b11\ \@k2 - 2\ b2112\ k1\ k2 + b2112\ k2\^2)\) + b21\ \((b212\ k1\^2 + k1\ \((2 - 2\ b212\ k2)\) + k2\ \((2 + b212\ k2)\))\))\))\), c9 \[Rule] \ \(-\(1\/\(4\ \((k1 - k2)\)\^2\ \@k2\)\)\) \((\ \[ImaginaryI]\ \((b21\ \((b222\ \((k1 - k2)\)\^2 + 2\ \[ImaginaryI]\ k1\ \@k2)\) + 2\ \[ImaginaryI]\ \@k2\ \((b2122\ \((k1 - k2)\)\^2 \ + 2\ \[ImaginaryI]\ b12\ \@k2 - b11\ k2)\) + b22\ \((b212\ k1\^2 + k1\ \((2 - 2\ b212\ k2)\) + k2\ \((2 + b212\ k2)\))\))\))\), \ b11\ \[Rule] \ 0, \ b12\ \[Rule] \ 0, \ b21\ \[Rule] \ 0, \ b22\ \[Rule] \ 0, \ b111 \[Rule] \ 0, \ b211\ \[Rule] \ 0, \ b122\ \[Rule] \ 0, \ b222 \[Rule] \ 0, \ b112\ \[Rule] \ \(\(2\)\(\ \)\)\/\(k1 - k2\), \ b212\ \[Rule] \ \(-2\)\/\(k1 - k2\)}], \ {\ c10, \ c11, \ c12, \ c13, \ c14}]]\)], "Input"], Cell[BoxData[ \({{c10 \[Rule] 1\/8\ \((4\ b21111 + \(\[ImaginaryI]\ \((k1 - 5\ \ k2)\)\)\/\(k2\^\(3/2\)\ \((\(-k1\) + k2)\)\^3\))\), c12 \[Rule] 1\/4\ \((2\ b21122 + \(\[ImaginaryI]\ \((k1 - 5\ k2)\)\)\/\(\@k2\ \ \((\(-k1\) + k2)\)\^3\))\), c14 \[Rule] \(4\ b22222\ \((k1 - k2)\)\^3 + \[ImaginaryI]\ \@k2\ \((3\ \ k1 + k2)\)\)\/\(8\ \((k1 - k2)\)\^3\), c11 \[Rule] b21112\/2, c13 \[Rule] b21222\/2}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(sol2\ = Simplify[{\[ImaginaryI]\ \@k2, b21\/2, b22\/2, \(4\ \[ImaginaryI] + \[ImaginaryI]\ b21\^2\ \((k1 - k2)\) \ + 4\ b211\ \@k2\ \((\(-k1\) + k2)\)\)\/\(8\ \@k2\ \((\(-k1\) + k2)\)\), \(\ \[ImaginaryI]\ b22\^2\ \((k1 - k2)\) - 4\ \[ImaginaryI]\ k2 + 4\ b222\ \@k2\ \ \((\(-k1\) + k2)\)\)\/\(8\ \@k2\ \((\(-k1\) + k2)\)\), \(4 + 2\ b212\ \((k1 - \ k2)\) + \(\[ImaginaryI]\ b21\ b22\ \((\(-k1\) + k2)\)\)\/\@k2\)\/\(4\ \((k1 - \ k2)\)\), \(\(-\[ImaginaryI]\)\ b21\ \((b211\ \((k1 - k2)\)\^2 - 2\ \ \[ImaginaryI]\ \@k2)\) + 2\ \((b11 + b2111\ \((k1 - k2)\)\^2)\)\ \@k2\)\/\(4\ \ \((k1 - k2)\)\^2\ \@k2\), \(\(-\[ImaginaryI]\)\ b22\ \((b222\ \((k1 - \ k2)\)\^2 + 2\ \[ImaginaryI]\ k1\ \@k2)\) + 2\ \@k2\ \((b2222\ \((k1 - \ k2)\)\^2 - b12\ k2)\)\)\/\(4\ \((k1 - k2)\)\^2\ \@k2\), \(-\(\(1\/\(4\ \((k1 \ - k2)\)\^2\ \@k2\)\) \((\[ImaginaryI]\ \((b211\ b22\ \((k1 - k2)\)\^2 + 2\ \[ImaginaryI]\ \@k2\ \((b12 - b22 + b2112\ k1\^2 + 2\ \[ImaginaryI]\ b11\ \@k2 - 2\ b2112\ k1\ k2 + b2112\ k2\^2)\) + b21\ \((b212\ k1\^2 + k1\ \((2 - 2\ b212\ k2)\) + k2\ \((2 + b212\ k2)\))\))\))\)\)\), \(-\(1\/\(4\ \ \((k1 - k2)\)\^2\ \@k2\)\)\) \((\[ImaginaryI]\ \((b21\ \((b222\ \((k1 - k2)\)\ \^2 + 2\ \[ImaginaryI]\ k1\ \@k2)\) + 2\ \[ImaginaryI]\ \@k2\ \((b2122\ \((k1 - k2)\)\^2 + 2\ \[ImaginaryI]\ b12\ \@k2 - b11\ k2)\) + b22\ \((b212\ k1\^2 + k1\ \((2 - 2\ b212\ k2)\) + k2\ \((2 + b212\ k2)\))\))\))\), \ \ \ \(\(ss2[\([1]\)]\)[\([1]\)]\)[\([2]\)], \ \(\(ss2[\([1]\)]\)[\([4]\)]\)[\([2]\ \)], \ \(\(ss2[\([1]\)]\)[\([2]\)]\)[\([2]\)], \ \ \(\(ss2[\([1]\)]\)[\([5]\)]\)[\([2]\)], \ \(\(ss2[\([1]\)]\)[\([3]\)]\)[\([2]\ \)]}];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Para anular las partes reales de los autovalores, empiezo por identificar los \ valores de los coeficientes de los b's que surgen a ojo desnudo, los \ reemplazo en las soluciones y veo que queda para los coeficientes de m\ \[AAcute]s alto orden. Recordar que sol1 y sol2 son vectores que tienen por \ componentes los t\[EAcute]rminos del desarrollo en g1 y g2 de los \ autovectores L.\ \>", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ ReplaceAll[{sol1, \ sol2}, \ {b11\ \[Rule] \ 0, \ b12\ \[Rule] \ 0, \ b21\ \[Rule] \ 0, \ b22\ \[Rule] \ 0, \ b111 \[Rule] \ 0, \ b211\ \[Rule] \ 0, \ b122\ \[Rule] \ 0, \ b222 \[Rule] \ 0, \ b112\ \[Rule] \ \(\(2\)\(\ \)\)\/\(k1 - k2\), \ b212\ \[Rule] \ \(-2\)\/\(k1 - k2\)}]]\)], "Input"], Cell[BoxData[ \({{\[ImaginaryI]\ \@k1, 0, 0, \[ImaginaryI]\/\(2\ \@k1\ \((k1 - k2)\)\), \(\[ImaginaryI]\ \@k1\)\ \/\(\(-2\)\ k1 + 2\ k2\), 0, b1111\/2, b1222\/2, b1112\/2, b1122\/2, 1\/8\ \((4\ b11111 - \(\[ImaginaryI]\ \((5\ k1 - k2)\)\)\/\(k1\^\(3/2\ \)\ \((k1 - k2)\)\^3\))\), b11112\/2, 1\/4\ \((2\ b11122 - \(\[ImaginaryI]\ \((5\ k1 - k2)\)\)\/\(\@k1\ \ \((k1 - k2)\)\^3\))\), b11222\/2, 1\/8\ \((4\ b12222 - \(\[ImaginaryI]\ \@k1\ \((k1 + 3\ k2)\)\)\/\((k1 \ - k2)\)\^3)\)}, {\[ImaginaryI]\ \@k2, 0, 0, \[ImaginaryI]\/\(2\ \@k2\ \((\(-k1\) + k2)\)\), \(\[ImaginaryI]\ \ \@k2\)\/\(2\ \((k1 - k2)\)\), 0, b2111\/2, b2222\/2, b2112\/2, b2122\/2, 1\/8\ \((4\ b21111 + \(\[ImaginaryI]\ \((k1 - 5\ k2)\)\)\/\(k2\^\(3/2\ \)\ \((\(-k1\) + k2)\)\^3\))\), b21112\/2, 1\/4\ \((2\ b21122 + \(\[ImaginaryI]\ \((k1 - 5\ k2)\)\)\/\(\@k2\ \((\ \(-k1\) + k2)\)\^3\))\), b21222\/2, \(4\ b22222\ \((k1 - k2)\)\^3 + \[ImaginaryI]\ \@k2\ \((3\ \ k1 + k2)\)\)\/\(8\ \((k1 - k2)\)\^3\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Se pueden tachar a mano los t\[EAcute]rminos imaginarios puros, y queda \ \>", "Section"], Cell[BoxData[ \({{0, 0, 0, 0, 0, 0, b1111\/2, b1222\/2, b1112\/2, b1122\/2, \ b11111\/2, b11112\/2, \ \ b11122\/2, b11222\/2, b12222\/2}, {0, 0, 0, 0, 0, 0, b2111\/2, b2222\/2, b2112\/2, b2122\/2, \ b21111\/2, b21112\/2, \ \(\(\ \)\(b21122\)\)\/2, b21222\/2, \(\(\ \)\(b22222\)\(\ \)\)\/2}}\)], "Input"], Cell["\<\ Esto implica que los t\[EAcute]rminos de orden 3 y 4 en el desarrollo de los \ b's tienen que ser cero para que haya bifurcaci\[OAcute]n. \ \>", "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Resumiendo, hasta orden ", Cell[BoxData[ \(g\^4\)]], " quedan los dos autovalores" }], "Section"], Cell[BoxData[ \(L1\ = \ \[ImaginaryI]\ \@k1 + \(\[ImaginaryI]\ g1\^2\)\/\(2\ \@k1\ \ \((k1 - k2)\)\) + \(\[ImaginaryI]\ \@k1\ g2\^2\)\/\(\(-2\)\ k1 + 2\ k2\) + 1\/8\ \((\(-\(\(\[ImaginaryI]\ \((5\ k1 - k2)\)\)\/\(k1\^\(3/2\)\ \((k1 - k2)\)\^3\)\)\))\)\ g1\ \^4\ + 1\/4\ \((\(-\(\(\[ImaginaryI]\ \((5\ k1 - k2)\)\)\/\(\@k1\ \((k1 - k2)\)\^3\)\)\))\)\ \ g1\^2\ \ g2\^2\ + \ 1\/8\ \((\(-\(\(\[ImaginaryI]\ \@k1\ \((k1 + 3\ k2)\)\)\/\((k1 - k2)\)\^3\)\))\)\ g2\^4\)], "Input"], Cell[BoxData[ \(L2 = \ \[ImaginaryI]\ \@k2\ + \(\[ImaginaryI]\ g1\^2\)\/\(2\ \@k2\ \((\ \(-k1\) + k2)\)\) + \(\[ImaginaryI]\ \@k2\ g2\^2\)\/\(2\ \((k1 - k2)\)\) + 1\/8\ \((\(\[ImaginaryI]\ \((k1 - 5\ k2)\)\)\/\(k2\^\(3/2\)\ \((\(-k1\ \) + k2)\)\^3\))\)\ g1\^4 + 1\/4\ \((\(\[ImaginaryI]\ \((k1 - 5\ k2)\)\)\/\(\@k2\ \((\(-k1\) + \ k2)\)\^3\))\)\ g1\^2\ g2\^2 + \(1\/8\) \((\(\[ImaginaryI]\ \@k2\ \((3\ k1 + \ k2)\)\)\/\((k1 - k2)\)\^3)\) g2\^4\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Donde b1 y b2 se han tomado iguales a ", "Section"], Cell[BoxData[{ \(b1\ = \ \(+\(\(2\ g1\ g2\)\/\(k1 - k2\)\)\) + \ O \((5)\)\), "\[IndentingNewLine]", \(b2\ = \ \(-\ \(\(2\ g1\ g2\)\/\(k1 - k2\)\)\) + \ O \((5)\)\)}], "Input"], Cell[CellGroupData[{ Cell["\<\ Hay otra forma de calcular los los autovalores, que consiste en no \ desarrollar los par\[AAcute]metros b1 y b2 hasta que se han obtenido los \ desarrollos de los autovalores. Cuando se proponen desarrollos para b1 y b2 \ hay que reorganizar todo consistentemente orden por orden. \ \>", "Subsection"], Cell["Juan Zanella,DNL 2004", "Subsubtitle"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{1156, 849}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, StyleDefinitions -> "Classroom.nb" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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