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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[ 44058, 1003]*) (*NotebookOutlinePosition[ 44888, 1030]*) (* CellTagsIndexPosition[ 44844, 1026]*) (*WindowFrame->Normal*) Notebook[{ Cell["Ejercicio de yapa: Hopf-Hopf", "Subtitle"], Cell["\<\ En este notebook figura el esquema de c\[AAcute]lculo de la forma normal de \ un sistema formado por dos sistemas de VDP acoplados linealmente. Los \ verdaderos c\[AAcute]lculos figuran en dos notebooks auxiliares, \ \"autovalores.nb\" y \"autovectores.nb\". \ \>", "Subsection"], Cell[CellGroupData[{ Cell["Definici\[OAcute]n del sistema de ecuaciones", "Section"], Cell[BoxData[{ \(f11[y1_] := \ y1\), "\[IndentingNewLine]", \(f12[x1_, \ y1_, \ x2_, \ y2_] := \ \(-k1\)\ x1\ + \ b1\ y1 - \ c\ x1^2\ y1\ + \ g1\ x2\ + \ g2\ y2\), "\[IndentingNewLine]", \(f21[y2_] := \ y2\), "\[IndentingNewLine]", \(f22[x1_, \ y1_, \ x2_, \ y2_] := \ \(-k2\)\ x2\ + \ b2\ y2\ - \ c\ x2^2\ y2\ + \ g1\ x1\ + \ g2\ y1\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Las constantes de acoplamiento entre los sistemas son g1 y g2, llamadas en \ clase gamma1 y gamma2. Los par\[AAcute]metros b1 y b2 se eligen, c\[AAcute]lculo auxiliar mediante, \ para que exista bifurcaci\[OAcute]n de Hopf en el sistema completo.\ \>", "Section"], Cell[CellGroupData[{ Cell["La parte lineal es", "Subsection"], Cell[BoxData[ \(\(J[g1_, \ g2_] := \ {{0, \ 1, \ 0, \ 0}, \ {\(-k1\), \ 2 \( g1\ g2\)\/\((k1 - k2)\), \ g1, \ g2}, \ {0, \ 0, \ 0, 1}, \ {g1, \ g2, \ \(-k2\), \ 2 \( g1\ g2\)\/\((k2 - k1)\)}};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(J[g1, \ g2]\ // \ MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "0", "0"}, {\(-k1\), \(\(2\ g1\ g2\)\/\(k1 - k2\)\), "g1", "g2"}, {"0", "0", "0", "1"}, {"g1", "g2", \(-k2\), \(\(2\ g1\ g2\)\/\(\(-k1\) + k2\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell["\<\ En el notebook auxiliar \"autovalores.nb\" se calculan los autovalores hasta \ \[OAcute]rdenes dados de g\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " A orden ", Cell[BoxData[ \(g\^3\)]], " los autovalores resultan" }], "Subsection"], Cell[BoxData[ \(\(eval[g1_, \ g2_] := {\(-\[ImaginaryI]\)\ \@k1\ - \(\[ImaginaryI]\ g1\^2\)\/\(2\ \ \@k1\ \((k1 - k2)\)\)\ + \ \(\(\[ImaginaryI]\ \@k1\)\/\(2\ \((k1 - k2)\)\)\) g2\^2, \[ImaginaryI]\ \@k1\ + \(\[ImaginaryI]\ g1\^2\)\/\(2\ \ \@k1\ \((k1 - k2)\)\) - \(\(\[ImaginaryI]\ \@k1\)\/\(2\ \((k1 - k2)\)\)\) g2\^2, \ \(-\[ImaginaryI]\)\ \@k2\ - \(\[ImaginaryI]\ \ g1\^2\)\/\(2\ \@k2\ \((\(-k1\) + k2)\)\) - \(\(\[ImaginaryI]\ \@k2\)\/\(2\ \ \((k1 - k2)\)\)\) g2\^2, \ \[ImaginaryI]\ \@k2 + \(\[ImaginaryI]\ g1\^2\)\/\(2\ \ \@k2\ \((\(-k1\) + k2)\)\) + \(\(\[ImaginaryI]\ \@k2\)\/\(2\ \((k1 - k2)\)\)\) g2\^2};\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ A las ecuaciones para los dos primeros autovectores los saco a mano alzada \ de J[g1, g2]. \ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(ReplaceAll[ Simplify[Solve[{\(-L\)\ u1\ + \ u2\ \[Equal] \ 0, \ \(-L\)\ u3\ + \ u4\ \[Equal] \ 0, \ \(-k1\)\ u1\ + \((2 \( g1\ g2\)\/\((k1 - k2)\) - \ L)\)\ u2\ + \ g1\ u3\ + \ g2\ u4\ \[Equal] \ 0}, \ {u2, \ u3, \ u4}]]\ , \ {u1 \[Rule] 1}]\)], "Input"], Cell[BoxData[ \({{u3 \[Rule] \(k1\^2 - L\ \((2\ g1\ g2 + k2\ L)\) + k1\ \((\(-k2\) + \ L\^2)\)\)\/\(\((k1 - k2)\)\ \((g1 + g2\ L)\)\), u4 \[Rule] \(L\ \((k1\^2 - L\ \((2\ g1\ g2 + k2\ L)\) + k1\ \ \((\(-k2\) + L\^2)\))\)\)\/\(\((k1 - k2)\)\ \((g1 + g2\ L)\)\), u2 \[Rule] L}}\)], "Output"] }, Open ]], Cell["\<\ De estas relaciones, se ve que se necesita un orden m\[AAcute]s en los \ autovalores que el que se desea obtener para los autovectores.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "El otro par de autovectores a orden ", Cell[BoxData[ \(g\^2\)]], " sale de aqu\[IAcute]. " }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(ReplaceAll[ Simplify[Solve[{\(-L\)\ u1\ + \ u2\ \[Equal] \ 0, \ \(-L\)\ u3\ + \ u4\ \[Equal] \ 0, \ g1\ u1\ + \ g2\ u2\ - k2\ u3\ + \((2 \( g1\ g2\)\/\((k2 - k1)\) - L)\)\ u4\ \[Equal] \ 0}, \ {u1, \ u2, \ u4}]]\ , \ {u3 \[Rule] 1}]\)], "Input"], Cell[BoxData[ \({{u1 \[Rule] \(\(-k2\^2\) + 2\ g1\ g2\ L - k2\ L\^2 + k1\ \((k2 + L\^2)\ \)\)\/\(\((k1 - k2)\)\ \((g1 + g2\ L)\)\), u2 \[Rule] \(L\ \((\(-k2\^2\) + 2\ g1\ g2\ L - k2\ L\^2 + k1\ \((k2 + \ L\^2)\))\)\)\/\(\((k1 - k2)\)\ \((g1 + g2\ L)\)\), u4 \[Rule] L}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Estas ecuaciones se resuelven a orden 2 en el \ notebook auxiliar llamado \"autovectores.nb\".", "Subsubsection", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Con los autovalores calculados a orden 3 se obtienen entonces los \ autovectores hasta orden 2\ \>", "Subsection"], Cell[BoxData[ \(evec[g1_, \ g2_] := \ {{1, \ \(-\[ImaginaryI]\)\ \@k1\ - 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k2)\)\) + \(\[ImaginaryI]\ g1\^2\)\/\(2\ \@k2\ \ \((\(-k1\) + k2)\)\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " Hasta orden g", Cell[BoxData[ \(\^2\)]], ", la matriz inversa de la Bt es " }], "Subsection"], Cell[BoxData[ \(BtInv[g1_, \ g2_] := \ {{\(\(-g1\^2\)\ \@k1 + \@k1\ \((g2\^2\ k1 + \((k1 - \ k2)\)\^2)\) + \[ImaginaryI]\ g1\ g2\ \((k1 + k2)\)\)\/\(2\ \@k1\ \((k1 - \ k2)\)\^2\), \(\[ImaginaryI]\ \((4\ \[ImaginaryI]\ g1\ g2\ k1\^\(3/2\) + g1\^2\ \ \((\(-3\)\ k1 + k2)\) + k1\ \((2\ \((k1 - k2)\)\^2 + g2\^2\ \((k1 + \ k2)\))\))\)\)\/\(4\ k1\^\(3/2\)\ \((k1 - k2)\)\^2\), \(g1 - \(\[ImaginaryI]\ \ g2\ k2\)\/\@k1\)\/\(\(-2\)\ k1 + 2\ k2\), \(g2 + \(\[ImaginaryI]\ \ g1\)\/\@k1\)\/\(\(-2\)\ k1 + 2\ k2\)}, {\(\(-g1\^2\)\ \@k1 + \@k1\ \((g2\^2\ \ k1 + \((k1 - k2)\)\^2)\) - \[ImaginaryI]\ g1\ g2\ \((k1 + k2)\)\)\/\(2\ \@k1\ \ \((k1 - k2)\)\^2\), \(-\(\(\[ImaginaryI]\ \((\(-4\)\ \[ImaginaryI]\ g1\ g2\ \ k1\^\(3/2\) + g1\^2\ \((\(-3\)\ k1 + k2)\) + k1\ \((2\ \((k1 - k2)\)\^2 + g2\^2\ \((k1 + k2)\))\))\)\)\/\(4\ k1\^\(3/2\)\ \((k1 - \ k2)\)\^2\)\)\), \(g1 + \(\[ImaginaryI]\ g2\ k2\)\/\@k1\)\/\(\(-2\)\ k1 + 2\ \ k2\), \(g2 - \(\[ImaginaryI]\ g1\)\/\@k1\)\/\(\(-2\)\ k1 + 2\ k2\)}, {\(g1 - \ \(\[ImaginaryI]\ g2\ k1\)\/\@k2\)\/\(2\ k1 - 2\ k2\), \(g2 + \(\[ImaginaryI]\ \ g1\)\/\@k2\)\/\(2\ k1 - 2\ k2\), \(\(-g1\^2\)\ \@k2 + \[ImaginaryI]\ g1\ g2\ \ \((k1 + k2)\) + \@k2\ \((k1\^2 - 2\ k1\ k2 + k2\ \((g2\^2 + k2)\))\)\)\/\(2\ \ \((k1 - k2)\)\^2\ \@k2\), \(\[ImaginaryI]\ \((g1\^2\ \((k1 - 3\ k2)\) + 4\ \ \[ImaginaryI]\ g1\ g2\ k2\^\(3/2\) + k2\ \((2\ \((k1 - k2)\)\^2 + g2\^2\ \ \((k1 + k2)\))\))\)\)\/\(4\ \((k1 - k2)\)\^2\ k2\^\(3/2\)\)}, {\(g1 + \(\ \[ImaginaryI]\ g2\ k1\)\/\@k2\)\/\(2\ k1 - 2\ k2\), \(g2 - \(\[ImaginaryI]\ \ g1\)\/\@k2\)\/\(2\ k1 - 2\ k2\), \(\(-g1\^2\)\ \@k2 - \[ImaginaryI]\ g1\ g2\ \ \((k1 + k2)\) + \@k2\ \((k1\^2 - 2\ k1\ k2 + k2\ \((g2\^2 + k2)\))\)\)\/\(2\ \ \((k1 - k2)\)\^2\ \@k2\), \(-\(\(\[ImaginaryI]\ \((g1\^2\ \((k1 - 3\ k2)\) - 4\ \[ImaginaryI]\ g1\ g2\ k2\^\(3/2\) + k2\ \((2\ \((k1 - k2)\)\^2 + g2\^2\ \((k1 + k2)\))\))\)\)\/\(4\ \((k1 - k2)\)\^2\ \ k2\^\(3/2\)\)\)\)}}\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ En t\[EAcute]rminos de las antiguas coordenadas {x1, y1, x2, y2}, las nuevas \ {z1, z2, z3, z4} se escriben como\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[BtInv[g1, \ g2]] . \ {x1, \ y1, \ x2, \ y2}\)], "Input"], Cell[BoxData[ \({\(\((\(-g1\^2\)\ \@k1 + \@k1\ \((g2\^2\ k1 + \((k1 - k2)\)\^2)\) + \ \[ImaginaryI]\ g1\ g2\ \((k1 + k2)\))\)\ x1\)\/\(2\ \@k1\ \((k1 - k2)\)\^2\) \ + \(\((g1 - \(\[ImaginaryI]\ g2\ k2\)\/\@k1)\)\ x2\)\/\(\(-2\)\ k1 + 2\ k2\) \ + \(\[ImaginaryI]\ \((4\ \[ImaginaryI]\ g1\ g2\ k1\^\(3/2\) + g1\^2\ \ \((\(-3\)\ k1 + k2)\) + k1\ \((2\ \((k1 - k2)\)\^2 + g2\^2\ \((k1 + k2)\))\))\ \)\ y1\)\/\(4\ k1\^\(3/2\)\ \((k1 - k2)\)\^2\) + \(\((g2 + \(\[ImaginaryI]\ \ g1\)\/\@k1)\)\ y2\)\/\(\(-2\)\ k1 + 2\ k2\), \(\((\(-g1\^2\)\ \@k1 + \@k1\ \ \((g2\^2\ k1 + \((k1 - 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2\ k2\) + \(\((\(-g1\^2\)\ \@k2 - \[ImaginaryI]\ \ g1\ g2\ \((k1 + k2)\) + \@k2\ \((k1\^2 - 2\ k1\ k2 + k2\ \((g2\^2 + k2)\))\))\ \)\ x2\)\/\(2\ \((k1 - k2)\)\^2\ \@k2\) + \(\((g2 - \(\[ImaginaryI]\ \ g1\)\/\@k2)\)\ y1\)\/\(2\ k1 - 2\ k2\) - \(\[ImaginaryI]\ \((g1\^2\ \((k1 - 3\ \ k2)\) - 4\ \[ImaginaryI]\ g1\ g2\ k2\^\(3/2\) + k2\ \((2\ \((k1 - k2)\)\^2 \ + g2\^2\ \((k1 + k2)\))\))\)\ y2\)\/\(4\ \((k1 - k2)\)\^2\ k2\^\(3/2\)\)}\)], \ "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Se puede comprobar que z1 y z2, por un lado, y z3 y z4 por otro, con \ complejos conjugados\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(\((\(-g1\^2\)\ \@k1 + \@k1\ \((g2\^2\ k1 + \((k1 - k2)\)\^2)\ \) + \[ImaginaryI]\ g1\ g2\ \((k1 + k2)\))\)\ x1\)\/\(2\ \@k1\ \((k1 - \ k2)\)\^2\) + \(\((g1 - \(\[ImaginaryI]\ g2\ k2\)\/\@k1)\)\ x2\)\/\(\(-2\)\ k1 \ + 2\ k2\) + \(\[ImaginaryI]\ \((4\ \[ImaginaryI]\ g1\ g2\ k1\^\(3/2\) + g1\^2\ \ \((\(-3\)\ k1 + k2)\) + k1\ \((2\ \((k1 - k2)\)\^2 + g2\^2\ \((k1 + \ k2)\))\))\)\ y1\)\/\(4\ k1\^\(3/2\)\ \((k1 - k2)\)\^2\) + \(\((g2 + \(\ \[ImaginaryI]\ g1\)\/\@k1)\)\ y2\)\/\(\(-2\)\ k1 + 2\ k2\) - \((\(\((\(-g1\^2\ \)\ \@k1 + \@k1\ \((g2\^2\ k1 + \((k1 - k2)\)\^2)\) + \[ImaginaryI]\ g1\ g2\ \ \((k1 + k2)\))\)\ x1\)\/\(2\ \@k1\ \((k1 - k2)\)\^2\) + \(\((g1 + \(\(-\ \[ImaginaryI]\)\ g2\ k2\)\/\@k1)\)\ x2\)\/\(\(-2\)\ k1 + 2\ k2\) - \(\(-\ \[ImaginaryI]\)\ \((4\ \[ImaginaryI]\ g1\ g2\ k1\^\(3/2\) + g1\^2\ \((\(-3\)\ \ k1 + k2)\) + k1\ \((2\ \((k1 - k2)\)\^2 + g2\^2\ \((k1 + k2)\))\))\)\ \ y1\)\/\(4\ k1\^\(3/2\)\ \((k1 - k2)\)\^2\) + \(\((g2 + \(\[ImaginaryI]\ \ g1\)\/\@k1)\)\ y2\)\/\(\(-2\)\ k1 + 2\ k2\))\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Como verificaci\[OAcute]n, puede verse que la transformaci\[OAcute]n de la \ parte lineal del sistema, conduce a la matriz diagonal definida por los \ autovalores\ \>", "Subsection"], Cell[BoxData[ \(\(Jprima[g1_, \ g2_] := \ {{\(-\[ImaginaryI]\)\ \@k1 + \(\[ImaginaryI]\ \((\(-g1\^2\ \) + g2\^2\ k1)\)\)\/\(2\ \@k1\ \((k1 - k2)\)\), 0, 0, 0}, {0, \[ImaginaryI]\ \@k1 + \(\[ImaginaryI]\ \((g1\^2 - g2\^2\ \ k1)\)\)\/\(2\ \@k1\ \((k1 - k2)\)\), 0, 0}, {0, 0, \(-\[ImaginaryI]\)\ \@k2 + \(\[ImaginaryI]\ \((\(-g1\^2\) + g2\ \^2\ k2)\)\)\/\(2\ \@k2\ \((\(-k1\) + k2)\)\), 0}, {0, 0, 0, \[ImaginaryI]\ \@k2 + \(\[ImaginaryI]\ \((g1\^2 - g2\^2\ k2)\)\ \)\/\(2\ \@k2\ \((\(-k1\) + k2)\)\)}};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Jprima[g1, \ g2]\ // \ MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-\[ImaginaryI]\)\ \@k1 + \(\[ImaginaryI]\ \((\(-g1\^2\) + g2\ \^2\ k1)\)\)\/\(2\ \@k1\ \((k1 - k2)\)\)\), "0", "0", "0"}, { "0", \(\[ImaginaryI]\ \@k1 + \(\[ImaginaryI]\ \((g1\^2 - g2\^2\ \ k1)\)\)\/\(2\ \@k1\ \((k1 - k2)\)\)\), "0", "0"}, {"0", "0", \(\(-\[ImaginaryI]\)\ \@k2 + \(\[ImaginaryI]\ \ \((\(-g1\^2\) + g2\^2\ k2)\)\)\/\(2\ \@k2\ \((\(-k1\) + k2)\)\)\), "0"}, {"0", "0", "0", \(\[ImaginaryI]\ \@k2 + \(\[ImaginaryI]\ \((g1\^2 - g2\^2\ \ k2)\)\)\/\(2\ \@k2\ \((\(-k1\) + k2)\)\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Transformaci\[OAcute]n de la parte NL", "Section"], Cell[CellGroupData[{ Cell["\<\ Para transformar la parte no lineal del campo a la nueva base, primero \ escribo la parte NL en la antigua base\ \>", "Subsection"], Cell[BoxData[ \(\(\(nl[x_]\)\(:=\)\(\ \)\({0, \ \(-\ c\)\ x[\([1]\)]^2\ x[\([2]\)], \ 0, \ \(-\ c\)\ x[\([3]\)]^2\ x[\([4]\)]}\)\(\ \ \)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "La parte no lineal del campo vector en las nuevas variables {z} est\ \[AAcute] dado por nlorden2 = BtInv[g1, g2] . nl[Bt[g1, g2] . {z1, z2, z3, \ z4}]. Hasta orden g", Cell[BoxData[ \(\^2\)]], " resulta " }], "Subsection"], Cell[BoxData[ \(cosa[g1_, g2_] := BtInv[g1, g2] . nl[Bt[g1, g2] . {z1, z2, z3, z4}]\)], "Input"], Cell[CellGroupData[{ Cell["Primero desarrollo a orden 1", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ cosa[0, \ 0]\ + \ \ g1\ \(\(Derivative[1, \ 0]\)[cosa]\)[0, \ 0]\ + \ g2\ \(\(Derivative[0, \ 1]\)[cosa]\)[0, \ 0]\ ]\)], "Input"], Cell[BoxData[ \({\(1\/\(2\ \@k1\ \((k1 - k2)\)\)\) \((c\ \((\(-k1\^\(3/2\)\)\ \((z1 - z2)\)\ \((z1 + z2)\)\^2 + \@k2\ \((\[ImaginaryI]\ g2\ \ \@k2\ \((z1 + z2)\)\^2\ \((z3 + z4)\) + g1\ \((z3 - z4)\)\ \((\(-z1\^2\) - 2\ z1\ z2 - z2\^2 + \((z3 + z4)\)\^2)\))\) + \@k1\ \((k2\ \ \((z1 - z2)\)\ \((z1 + z2)\)\^2 - 2\ g1\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) - \[ImaginaryI]\ g2\ \@k2\ \((z3 - z4)\)\ \((\(-2\)\ z1\^2 + 2\ z2\^2 + \((z3 + z4)\)\^2)\))\))\))\), \(1\/\(2\ \ \@k1\ \((k1 - k2)\)\)\) \((c\ \((k1\^\(3/2\)\ \((z1 - z2)\)\ \((z1 + z2)\)\^2 + \@k2\ \((\(-\[ImaginaryI]\)\ \ g2\ \@k2\ \((z1 + z2)\)\^2\ \((z3 + z4)\) + g1\ \((z3 - z4)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 + z4)\)\^2)\))\) + \@k1\ \((\(-k2\)\ \ \((z1 - z2)\)\ \((z1 + z2)\)\^2 + 2\ g1\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) - \[ImaginaryI]\ g2\ \@k2\ \((z3 - z4)\)\ \((2\ z1\^2 - 2\ z2\^2 + \((z3 + z4)\)\^2)\))\))\))\), \(1\/\(2\ \ \@k2\ \((\(-k1\) + k2)\)\)\) \((c\ \((g1\ \((\(-2\)\ \@k2\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\) + \@k1\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 + z4)\)\^2)\))\) + \[ImaginaryI]\ \ \((\[ImaginaryI]\ \@k2\ \((\(-k1\) + k2)\)\ \((z3 - z4)\)\ \((z3 + z4)\)\^2 + g2\ \@k1\ \((\@k1\ \((z1 + z2)\)\ \((z3 + z4)\)\^2 - \@k2\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - 2\ z3\^2 + 2\ z4\^2)\))\))\))\))\), \(-\(\(1\/\(2\ \ \@k2\ \((\(-k1\) + k2)\)\)\) \((c\ \((g1\ \((\(-2\)\ \@k2\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\) + \@k1\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 + z4)\)\^2)\))\) + \[ImaginaryI]\ \ \((\[ImaginaryI]\ \@k2\ \((\(-k1\) + k2)\)\ \((z3 - z4)\)\ \((z3 + z4)\)\^2 + g2\ \@k1\ \((\@k1\ \((z1 + z2)\)\ \((z3 + z4)\)\^2 + \@k2\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 + 2\ z3\^2 - 2\ z4\^2)\))\))\))\))\)\)\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Luego a orden 2", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ 1/2\ \((g1^2\ \(\(Derivative[2, \ 0]\)[cosa]\)[0, \ 0]\ + g2^2\ \(\(Derivative[0, \ 2]\)[cosa]\)[0, \ 0]\ \ + 2\ g1\ g2\ \(\(Derivative[1, \ 1]\)[cosa]\)[0, \ 0])\)]\)], "Input"], Cell[BoxData[ \({\(1\/\(2\ \@k1\ \((k1 - k2)\)\^2\)\) \((c\ \((g2\^2\ \((\(-\@k1\)\ k2\ \ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 - z4)\)\^2)\) + k1\^\(3/2\)\ \((z1 + z2)\)\ \((z3 + z4)\)\^2 + 2\ k1\ \@k2\ \((z1 - z2)\)\ \((z3\^2 - z4\^2)\) + 2\ k2\^\(3/2\)\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\))\) + g1\^2\ \((\(-4\)\ \@k2\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\) + \@k1\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - 2\ \((z3 + z4)\)\^2)\))\) - 2\ \[ImaginaryI]\ g1\ g2\ \((\(-\@k1\)\ \@k2\ \((3\ z1 - z2)\)\ \((z3\^2 - z4\^2)\) - 2\ k2\ \((z1 + z2)\)\ \((z3\^2 + z4\^2)\) + k1\ \((z1\^3 + z1\^2\ z2 - z2\^3 - z1\ \((z2\^2 + \((z3 + z4)\)\^2)\))\))\))\))\), \ \(1\/\(2\ \@k1\ \((k1 - k2)\)\^2\)\) \((c\ \((g2\^2\ \((\@k1\ k2\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 - z4)\)\^2)\) + k1\^\(3/2\)\ \((z1 + z2)\)\ \((z3 + z4)\)\^2 + 2\ k1\ \@k2\ \((z1 - z2)\)\ \((z3\^2 - z4\^2)\) - 2\ k2\^\(3/2\)\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\))\) + g1\^2\ \((4\ \@k2\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\) - \@k1\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - 2\ \((z3 + z4)\)\^2)\))\) - 2\ \[ImaginaryI]\ g1\ g2\ \((\@k1\ \@k2\ \((z1 - 3\ z2)\)\ \((z3\^2 - z4\^2)\) + 2\ k2\ \((z1 + z2)\)\ \((z3\^2 + z4\^2)\) + k1\ \((z1\^3 + z1\^2\ z2 - z1\ z2\^2 + z2\ \((\(-z2\^2\) + \((z3 + z4)\)\^2)\))\))\))\))\ \), \(1\/\(2\ \((k1 - k2)\)\^2\ \@k2\)\) \((c\ \((g1\^2\ \((\(-4\)\ \@k1\ \ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) + \@k2\ \((z3 - z4)\)\ \((\(-2\)\ z1\^2 - 4\ z1\ z2 - 2\ z2\^2 + \((z3 + z4)\)\^2)\))\) + g2\^2\ \((2\ \@k1\ k2\ \((z1\^2 - z2\^2)\)\ \((z3 - z4)\) + k2\^\(3/2\)\ \((z1 + z2)\)\^2\ \((z3 + z4)\) + 2\ k1\^\(3/2\)\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) - k1\ \@k2\ \((z3 - z4)\)\ \((\(-z1\^2\) + 2\ z1\ z2 - z2\^2 + \((z3 + z4)\)\^2)\))\) + 2\ \[ImaginaryI]\ g1\ g2\ \((\@k1\ \@k2\ \((z1\^2 - z2\^2)\)\ \((3\ z3 - z4)\) + 2\ k1\ \((z1\^2 + z2\^2)\)\ \((z3 + z4)\) + k2\ \((z1\^2\ z3 + 2\ z1\ z2\ z3 + z2\^2\ z3 - \((z3 - z4)\)\ \((z3 + z4)\)\^2)\))\))\))\), \ \(1\/\(2\ \((k1 - k2)\)\^2\ \@k2\)\) \((c\ \((g1\^2\ \((4\ \@k1\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) - \@k2\ \((z3 - z4)\)\ \((\(-2\)\ z1\^2 - 4\ z1\ z2 - 2\ z2\^2 + \((z3 + z4)\)\^2)\))\) + g2\^2\ \((2\ \@k1\ k2\ \((z1\^2 - z2\^2)\)\ \((z3 - z4)\) + k2\^\(3/2\)\ \((z1 + z2)\)\^2\ \((z3 + z4)\) - 2\ k1\^\(3/2\)\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) + k1\ \@k2\ \((z3 - z4)\)\ \((\(-z1\^2\) + 2\ z1\ z2 - z2\^2 + \((z3 + z4)\)\^2)\))\) - 2\ \[ImaginaryI]\ g1\ g2\ \((\@k1\ \@k2\ \((z1\^2 - z2\^2)\)\ \((z3 - 3\ z4)\) + 2\ k1\ \((z1\^2 + z2\^2)\)\ \((z3 + z4)\) + k2\ \((z3\^3 + z3\^2\ z4 - z3\ z4\^2 + z4\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - z4\^2)\))\))\))\))\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Y agrupo todo", "Subsubsection"], Cell[BoxData[ \(nlorden2[z1_, \ z2_, \ z3_, \ z4_] := \ {\(1\/\(2\ \@k1\ \((k1 - k2)\)\)\) \((c\ \((\(-k1\^\(3/2\)\)\ \((z1 - z2)\)\ \((z1 + z2)\)\^2 + \@k2\ \((\[ImaginaryI]\ \ g2\ \@k2\ \((z1 + z2)\)\^2\ \((z3 + z4)\) + g1\ \((z3 - z4)\)\ \((\(-z1\^2\) - 2\ z1\ z2 - z2\^2 + \((z3 + z4)\)\^2)\))\) + \@k1\ \((k2\ \ \((z1 - z2)\)\ \((z1 + z2)\)\^2 - 2\ g1\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) - \[ImaginaryI]\ g2\ \@k2\ \((z3 - z4)\)\ \((\(-2\)\ z1\^2 + 2\ z2\^2 + \((z3 + z4)\)\^2)\))\))\))\), \ \(1\/\(2\ \@k1\ \((k1 - k2)\)\)\) \((c\ \((k1\^\(3/2\)\ \((z1 - z2)\)\ \((z1 + z2)\)\^2 + \@k2\ \((\(-\[ImaginaryI]\ \)\ g2\ \@k2\ \((z1 + z2)\)\^2\ \((z3 + z4)\) + g1\ \((z3 - z4)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 + z4)\)\^2)\))\) + \@k1\ \ \((\(-k2\)\ \((z1 - z2)\)\ \((z1 + z2)\)\^2 + 2\ g1\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) - \[ImaginaryI]\ g2\ \@k2\ \((z3 - z4)\)\ \((2\ z1\^2 - 2\ z2\^2 + \((z3 + z4)\)\^2)\))\))\))\), \ \(1\/\(2\ \@k2\ \((\(-k1\) + k2)\)\)\) \((c\ \((g1\ \((\(-2\)\ \@k2\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\) + \@k1\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 + z4)\)\^2)\))\) + \ \[ImaginaryI]\ \((\[ImaginaryI]\ \@k2\ \((\(-k1\) + k2)\)\ \((z3 - z4)\)\ \((z3 + z4)\)\^2 + g2\ \@k1\ \((\@k1\ \((z1 + z2)\)\ \((z3 + z4)\)\^2 - \@k2\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - 2\ z3\^2 + 2\ z4\^2)\))\))\))\))\), \(-\(1\/\(2\ \ \@k2\ \((\(-k1\) + k2)\)\)\)\) \((c\ \((g1\ \((\(-2\)\ \@k2\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\) + \@k1\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 + z4)\)\^2)\))\) + \ \[ImaginaryI]\ \((\[ImaginaryI]\ \@k2\ \((\(-k1\) + k2)\)\ \((z3 - z4)\)\ \((z3 + z4)\)\^2 + g2\ \@k1\ \((\@k1\ \((z1 + z2)\)\ \((z3 + z4)\)\^2 + \@k2\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 + 2\ z3\^2 - 2\ z4\^2)\))\))\))\))\)} + \ {\(1\/\(2\ \ \@k1\ \((k1 - k2)\)\^2\)\) \((c\ \((g2\^2\ \((\(-\@k1\)\ k2\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 - z4)\)\^2)\) + k1\^\(3/2\)\ \((z1 + z2)\)\ \((z3 + z4)\)\^2 + 2\ k1\ \@k2\ \((z1 - z2)\)\ \((z3\^2 - z4\^2)\) + 2\ k2\^\(3/2\)\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\))\) + g1\^2\ \((\(-4\)\ \@k2\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\) + \@k1\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - 2\ \((z3 + z4)\)\^2)\))\) - 2\ \[ImaginaryI]\ g1\ g2\ \((\(-\@k1\)\ \@k2\ \((3\ z1 - z2)\)\ \((z3\^2 - z4\^2)\) - 2\ k2\ \((z1 + z2)\)\ \((z3\^2 + z4\^2)\) + k1\ \((z1\^3 + z1\^2\ z2 - z2\^3 - z1\ \((z2\^2 + \((z3 + \ z4)\)\^2)\))\))\))\))\), \(1\/\(2\ \@k1\ \((k1 - k2)\)\^2\)\) \((c\ \((g2\^2\ \ \((\@k1\ k2\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - \((z3 - z4)\)\^2)\) + k1\^\(3/2\)\ \((z1 + z2)\)\ \((z3 + z4)\)\^2 + 2\ k1\ \@k2\ \((z1 - z2)\)\ \((z3\^2 - z4\^2)\) - 2\ k2\^\(3/2\)\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\))\) + g1\^2\ \((4\ \@k2\ \((z1 + z2)\)\ \((z3\^2 - z4\^2)\) - \@k1\ \((z1 - z2)\)\ \((z1\^2 + 2\ z1\ z2 + z2\^2 - 2\ \((z3 + z4)\)\^2)\))\) - 2\ \[ImaginaryI]\ g1\ g2\ \((\@k1\ \@k2\ \((z1 - 3\ z2)\)\ \((z3\^2 - z4\^2)\) + 2\ k2\ \((z1 + z2)\)\ \((z3\^2 + z4\^2)\) + k1\ \((z1\^3 + z1\^2\ z2 - z1\ z2\^2 + z2\ \((\(-z2\^2\) + \((z3 + \ z4)\)\^2)\))\))\))\))\), \(1\/\(2\ \((k1 - k2)\)\^2\ \@k2\)\) \((c\ \((g1\^2\ \ \((\(-4\)\ \@k1\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) + \@k2\ \((z3 - z4)\)\ \((\(-2\)\ z1\^2 - 4\ z1\ z2 - 2\ z2\^2 + \((z3 + z4)\)\^2)\))\) + g2\^2\ \((2\ \@k1\ k2\ \((z1\^2 - z2\^2)\)\ \((z3 - z4)\) + k2\^\(3/2\)\ \((z1 + z2)\)\^2\ \((z3 + z4)\) + 2\ k1\^\(3/2\)\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) - k1\ \@k2\ \((z3 - z4)\)\ \((\(-z1\^2\) + 2\ z1\ z2 - z2\^2 + \((z3 + z4)\)\^2)\))\) + 2\ \[ImaginaryI]\ g1\ g2\ \((\@k1\ \@k2\ \((z1\^2 - z2\^2)\)\ \((3\ z3 - z4)\) + 2\ k1\ \((z1\^2 + z2\^2)\)\ \((z3 + z4)\) + k2\ \((z1\^2\ z3 + 2\ z1\ z2\ z3 + z2\^2\ z3 - \((z3 - z4)\)\ \((z3 + z4)\)\^2)\))\))\))\), \ \(1\/\(2\ \((k1 - k2)\)\^2\ \@k2\)\) \((c\ \((g1\^2\ \((4\ \@k1\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) - \@k2\ \((z3 - z4)\)\ \((\(-2\)\ z1\^2 - 4\ z1\ z2 - 2\ z2\^2 + \((z3 + z4)\)\^2)\))\) + g2\^2\ \((2\ \@k1\ k2\ \((z1\^2 - z2\^2)\)\ \((z3 - z4)\) + k2\^\(3/2\)\ \((z1 + z2)\)\^2\ \((z3 + z4)\) - 2\ k1\^\(3/2\)\ \((z1\^2 - z2\^2)\)\ \((z3 + z4)\) + k1\ \@k2\ \((z3 - z4)\)\ \((\(-z1\^2\) + 2\ z1\ z2 - z2\^2 + \((z3 + z4)\)\^2)\))\) - 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