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Copyright (C) 2005 Chad E. Brassil; Last modified May 18, 2006\
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This compact code can be used to analyze differential equations.  It can \
handle from 1 to many variables, just keep them in the same order for deqs, \
vars, and start.\
\>", "Text"],

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Cell["\<\
deqs = a list of the right-hand side of the differential equations
vars = the variables, in the same order as the deqs
start = initial conditions
tmax = maximum time to plot time series\
\>", "Text"],

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Use the following code to specify dashing, thickness, and colors for the vars\
\
\>", "Text"],

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Cell["\<\
This code can be used to create a legend which can be called throughout the \
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\>", "Text"],

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This section uses a special package that can be downloaded from \
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Will start initial values and run them for tmax or until they go off the \
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["Attempt Analytical Stability", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(\(numberOfTerms[x_] := 
        If[TrueQ[Head[x] \[Equal] Plus], Length[x], 
          1];\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(i = firstPositivePosition; \  (*\ 
      choose\ which\ equilibrium\ to\ examine\ *) \[IndentingNewLine]jacSub = 
      Outer[D, deqs, vars] /. equilib[\([i]\)];\), "\[IndentingNewLine]", 
    \(Simplify[MatrixForm[jacSub]]\), "\[IndentingNewLine]", 
    \(\(jacNoSub = Outer[D, deqs, vars];\)\), "\[IndentingNewLine]", 
    \(Simplify[MatrixForm[jacNoSub]]\)}], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(r1\ \((k1 - 
                        k2\ \[Alpha])\)\)\/\(k1\ \((\(-1\) + \[Alpha]\ \
\[Beta])\)\)\), \(\(r1\ \[Alpha]\ \((k1 - 
                        k2\ \[Alpha])\)\)\/\(k1\ \((\(-1\) + \[Alpha]\ \
\[Beta])\)\)\)},
            {\(\(r2\ \[Beta]\ \((k2 - 
                        k1\ \[Beta])\)\)\/\(k2\ \((\(-1\) + \[Alpha]\ \
\[Beta])\)\)\), \(\(r2\ \((k2 - 
                        k1\ \[Beta])\)\)\/\(k2\ \((\(-1\) + \[Alpha]\ \
\[Beta])\)\)\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(r1\ \((k1 - 2\ n1 - 
                        n2\ \[Alpha])\)\)\/k1\), \(-\(\(n1\ r1\ \
\[Alpha]\)\/k1\)\)},
            {\(-\(\(n2\ r2\ \[Beta]\)\/k2\)\), \(\(r2\ \((k2 - 2\ n2 - 
                        n1\ \[Beta])\)\)\/k2\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["\<\
Above is the Jacobian with substitutions and without substitutions.  Below is \
attempt at the simpliest combination of the above.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(jac = 
        Table[\[IndentingNewLine]If[
            numberOfTerms[Expand[jacSub[\([ijac, jjac]\)]]] < 
              numberOfTerms[Expand[jacNoSub[\([ijac, jjac]\)]]], 
            Simplify[jacSub[\([ijac, jjac]\)]], 
            Simplify[jacNoSub[\([ijac, jjac]\)]]]\[IndentingNewLine], {ijac, 
            Length[vars]}, {jjac, 
            Length[vars]}];\)\), "\[IndentingNewLine]", 
    \(\(jac = 
        jac /. Thread[\((vars /. posEquilib)\) \[Rule] 
              vars];\)\), "\[IndentingNewLine]", 
    \(MatrixForm[jac]\)}], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(r1\ \((k1 - 2\ n1 - 
                        n2\ \[Alpha])\)\)\/k1\), \(-\(\(n1\ r1\ \
\[Alpha]\)\/k1\)\)},
            {\(-\(\(n2\ r2\ \[Beta]\)\/k2\)\), \(\(r2\ \((k2 - 2\ n2 - 
                        n1\ \[Beta])\)\)\/k2\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["\<\
1) Brute force
-stable if the real part of the largest eigenvalue is negative
-may be able to simplify be hand knowing equilibrium values need to be \
positive\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Eigenvalues[jac]\)], "Input"],

Cell[BoxData[
    \({\(-\(\(1\/\(2\ k1\ k2\)\)\((\(-k1\)\ k2\ r1 + 2\ k2\ n1\ r1 - 
            k1\ k2\ r2 + 2\ k1\ n2\ r2 + k2\ n2\ r1\ \[Alpha] + 
            k1\ n1\ r2\ \[Beta] - \[Sqrt]\((\((k1\ k2\ r1 - 2\ k2\ n1\ r1 + \
k1\ k2\ r2 - 2\ k1\ n2\ r2 - k2\ n2\ r1\ \[Alpha] - k1\ n1\ r2\ \[Beta])\)\^2 \
+ 4\ k1\ k2\ \((\(-k1\)\ k2\ r1\ r2 + 2\ k2\ n1\ r1\ r2 + 2\ k1\ n2\ r1\ r2 - 
                        4\ n1\ n2\ r1\ r2 + k2\ n2\ r1\ r2\ \[Alpha] - 
                        2\ n2\^2\ r1\ r2\ \[Alpha] + 
                        k1\ n1\ r1\ r2\ \[Beta] - 
                        2\ n1\^2\ r1\ r2\ \[Beta])\))\))\)\)\), \(-\(\(1\/\(2\
\ k1\ k2\)\)\((\(-k1\)\ k2\ r1 + 2\ k2\ n1\ r1 - k1\ k2\ r2 + 2\ k1\ n2\ r2 + 
            k2\ n2\ r1\ \[Alpha] + 
            k1\ n1\ r2\ \[Beta] + \[Sqrt]\((\((k1\ k2\ r1 - 2\ k2\ n1\ r1 + \
k1\ k2\ r2 - 2\ k1\ n2\ r2 - k2\ n2\ r1\ \[Alpha] - k1\ n1\ r2\ \[Beta])\)\^2 \
+ 4\ k1\ k2\ \((\(-k1\)\ k2\ r1\ r2 + 2\ k2\ n1\ r1\ r2 + 2\ k1\ n2\ r1\ r2 - 
                        4\ n1\ n2\ r1\ r2 + k2\ n2\ r1\ r2\ \[Alpha] - 
                        2\ n2\^2\ r1\ r2\ \[Alpha] + 
                        k1\ n1\ r1\ r2\ \[Beta] - 
                        2\ n1\^2\ r1\ r2\ \[Beta])\))\))\)\)\)}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "2)  Special rule for 2 x 2 matrix: stable if trace < 0 and det > 0\nFor \
stable or unstable, oscillatory if ",
  Cell[BoxData[
      \(TraditionalForm\`det > trace\^2\/4\)]]
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ for\ a\ 2  x2\ matrix, \ 
      the\ determinant\  > \ 
        0\ for\ stability\ *) \)\(\[IndentingNewLine]\)\(Det[jac] // 
      Simplify\)\)\)], "Input"],

Cell[BoxData[
    \(\(r1\ r2\ \((\(-2\)\ k2\ n1 + 4\ n1\ n2 - k2\ n2\ \[Alpha] + 2\ n2\^2\ \
\[Alpha] + 2\ n1\^2\ \[Beta] + k1\ \((k2 - 2\ n2 - n1\ \[Beta])\))\)\)\/\(k1\ \
k2\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ for\ a\ 2  x2\ matrix, \ 
      the\ trace\  < \ 0\ for\ stability\ *) \)\(\[IndentingNewLine]\)\(Tr[
        jac] // Simplify\)\)\)], "Input"],

Cell[BoxData[
    \(\(r1\ \((k1 - 2\ n1 - n2\ \[Alpha])\)\)\/k1 + \(r2\ \((k2 - 2\ n2 - n1\ \
\[Beta])\)\)\/k2\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "3) Routh-Hurrwitz criteria for a 3 x 3 matrix.\ncharacteristic equation: \
",
  Cell[BoxData[
      \(TraditionalForm\`\[Lambda]\^3 + \(A\_1\) \[Lambda]\^2 + \(A\_2\) \
\[Lambda] + A\_3 = 0\)]],
  "\nstable if:\n",
  "\t1) ",
  Cell[BoxData[
      \(TraditionalForm\`A\_1 > 0\)]],
  "\n\t2) ",
  Cell[BoxData[
      \(TraditionalForm\`A\_3 > 0\)]],
  "\n\t3) ",
  Cell[BoxData[
      \(TraditionalForm\`\(A\_1\) A\_2 > A\_3\)]]
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ 
      characterisic\ polynomial\ *) \)\(\[IndentingNewLine]\)\(Collect[
      Det[jac - \[Lambda]\ IdentityMatrix[
              Length[jac]]], \[Lambda]]\)\)\)], "Input"],

Cell[BoxData[
    \(\(-a\)\ c\ e + \((\(-b\)\ d - c\ e + a\ f)\)\ \[Lambda] + \((\(-a\) + 
            f)\)\ \[Lambda]\^2 - \[Lambda]\^3\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "4) Some matrices can be reduced to a 2x2 matrix.  \nA matrix in the form \
",
  Cell[BoxData[
      FormBox[
        RowBox[{"(", GridBox[{
              {"a", "x"},
              {"0", "b"}
              }], ")"}], TraditionalForm]]],
  " is called an upper triangular matrix and has the eigenvalues {a,b}.  I \
can also replace each entry in the matrix with another matrix, represented \
here by a capital letter, M = ",
  Cell[BoxData[
      FormBox[
        RowBox[{"(", GridBox[{
              {"A", "X"},
              {"0", "B"}
              }], ")"}], TraditionalForm]]],
  ".  If the matrix A and B are each square matrices, then the eigenvalues \
for the matrix M will be the eigenvalues for both A and B.  Here is an \
example.  The matrix ",
  Cell[BoxData[
      TagBox[
        RowBox[{"(", "\[NoBreak]", GridBox[{
              {"a", "b", "c"},
              {"d", "e", "f"},
              {"0", "0", "g"}
              }], "\[NoBreak]", ")"}],
        Function[ BoxForm`e$, 
          MatrixForm[ BoxForm`e$]]]]],
  "can put into the form of matrix M with A = ",
  Cell[BoxData[
      FormBox[
        RowBox[{"(", GridBox[{
              {"a", "b"},
              {"d", "e"}
              }], ")"}], TraditionalForm]]],
  ", X = ",
  Cell[BoxData[
      FormBox[
        RowBox[{"(", GridBox[{
              {"c"},
              {"f"}
              }], ")"}], TraditionalForm]]],
  ", B = (",
  StyleBox["g",
    FontSlant->"Italic"],
  "), and 0 = ",
  Cell[BoxData[
      FormBox[
        RowBox[{"(", GridBox[{
              {"0", "0"}
              }], ")"}], TraditionalForm]]],
  ".  I can now examine A as a 2x2 matrix and ",
  StyleBox["g",
    FontSlant->"Italic"],
  " is the third eigenvalue.  This technique can be useful when looking at \
invasion analysis."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Numerical Stability", "Subsection"],

Cell["\<\
Numerically we can always examine the eigenvalues of the Jacobian matrix at \
the equilibrium point.  Sometimes these can involve complex numbers.  If the \
real part of the largest eigenvalue is negative, the equilibrium is stable. \
\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(jacSolved = jacSub /. paramset;\)\), "\[IndentingNewLine]", 
    \(MatrixForm[jacSolved]\)}], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(-0.8750000000000001`\), \(-1.5750000000000002`\)},
            {\(-1.0227272727272725`\), \(-0.568181818181818`\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Eigenvalues[jacSolved]\)], "Input"],

Cell[BoxData[
    \({\(-2.`\), 0.5568181818181818`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ stable\ if\  < \ 0\ *) \)\(\[IndentingNewLine]\)\(Max[
      Re[Eigenvalues[jacSolved]]]\)\)\)], "Input"],

Cell[BoxData[
    \(0.5568181818181818`\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Lyapunov Exponent", "Subsection"],

Cell["\<\
The largest Lyapunov exponent can be used to characterize a system as \
chaotic.  If the exponent is > 0, then the system is chaotic. \
\>", "Text"],

Cell[CellGroupData[{

Cell["Example Equation", "Subsubsection"],

Cell["Same format as above", "Text"],

Cell[BoxData[{
    \(\(deqs = {\[IndentingNewLine]\(\[Alpha]\ p\ n1\)\/\(1 + \[Beta]\ n1 + \
\[Beta]\ n2\) + \(\[Alpha]\ p\ n2\)\/\(1 + \[Beta]\ n1 + \[Beta]\ n2\) - 
            d\ p, \[IndentingNewLine]n1 \((1 - 
                  n1)\) - \(n1\ p\)\/\(1 + \[Beta]\ n1 + \[Beta]\ n2\), \
\[IndentingNewLine]n2\ \((1 - 
                  n2)\) - \(n2\ p\)\/\(1 + \[Beta]\ n1 + \[Beta]\ n2\)\
\[IndentingNewLine]};\)\), "\[IndentingNewLine]", 
    \(\(vars = {p, n1, n2};\)\), "\[IndentingNewLine]", 
    \(\(start = {0.1, 0.05, 0.02};\)\), "\[IndentingNewLine]", 
    \(\(paramset = {\[Alpha] \[Rule] 
            4.5 + 4.05\ Sin[0.490987\ t], \[Beta] \[Rule] 0.02, 
          d \[Rule] 0.2};\)\), "\[IndentingNewLine]", 
    \(\(tmax = 10000;\)\)}], "Input"],

Cell["\<\
transtime is the the time units to ignore so that the long-term dynamics are \
examined\
\>", "Text"],

Cell[BoxData[
    \(\(transtime = 3000;\)\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Find largest Lyapunov using built-in integrator", "Subsubsection"],

Cell["\<\
Use the same time-solution code from \"Plot of Density vs. Time\" section\
\>", "Text"],

Cell[BoxData[
    \(\(\( (*\ 
      find\ time - solution\ *) \)\(\[IndentingNewLine]\)\(\(addT = 
        Table[vars[\([i]\)] \[Rule] \(Map[vars[\([i]\)], {t}]\)[\([1]\)], {i, 
            Length[vars]}];\)\[IndentingNewLine]
    \(outset = 
        NDSolve[Join[
              Thread[D[vars /. addT, 
                    t] \[Equal] \((deqs /. addT)\)], \(Thread[
                    vars \[Equal] start] /. addT\) /. t \[Rule] 0] /. 
            paramset, vars, {t, 0, tmax}, MaxSteps \[Rule] 50*tmax, 
          AccuracyGoal \[Rule] Infinity];\)\)\)\)], "Input"],

Cell["\<\
The following code assumes you have already defined the equations and found \
the time-solution, as illustrated just above\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Module[{n, Phi0, DPhi, PhiNoT}, \[IndentingNewLine]n = 
        Length[start]; \[IndentingNewLine]PhiNoT = 
        Transpose[Table[Unique["\<b\>"], {n}, {n}]]; \[IndentingNewLine]Phi = 
        Table[\(PhiNoT[\([i, j]\)]\)[t], {i, n}, {j, 
            n}]; \[IndentingNewLine]Phi0 = 
        IdentityMatrix[n]; \[IndentingNewLine]DPhi = 
        Outer[D, deqs /. addT, vars /. addT] . 
          Phi; \[IndentingNewLine]derivhead = 
        Flatten[D[Phi, t]]; \[IndentingNewLine]initialconditions = 
        Thread[Flatten[Phi] \[Equal] Flatten[Phi0]] /. 
          t \[Rule] transtime; \[IndentingNewLine]eqns = 
        Union[Thread[derivhead \[Equal] Flatten[DPhi]] /. outset, 
          initialconditions]; \[IndentingNewLine]solLyap = \(NDSolve[
            eqns /. paramset, Flatten[PhiNoT], {t, transtime, tmax}, 
            MaxSteps \[Rule] 100000]\)[\([1]\)]; \[IndentingNewLine]PhiT[
          tsub_] := \(Phi /. solLyap\) /. 
          t \[Rule] tsub; \[IndentingNewLine]Log[
          Norm[PhiT[tmax]]]/\((tmax - 
            transtime)\)\[IndentingNewLine]]\)], "Input"],

Cell[BoxData[
    \(\(-0.0020940103889407753`\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ 
      convergence\ of\ estimate\ of\ Lyapunov\ exponent\ with\ time\ *) \)\(\
\[IndentingNewLine]\)\(Plot[
        Log[Norm[PhiT[t]]]/\((t - transtime)\), {t, transtime + 1, tmax}, 
        PlotRange \[Rule] {\(-0.01\), 0.01}];\)\)\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
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Cell["Lyapunovs using Runga-Kuta and LCE package", "Subsubsection"],

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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
Mfstroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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    \(\(\( (*\ 
      This\ version\ may\ have\ problems\ with\ solutions\ in\ which\ the\ \
starting\ point\ happens\ to\ sit\ at\ the\ bottom\ of\ the\ cycle\ and\ the\ \
difference\ n