(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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[ 645868, 21725]*) (*NotebookOutlinePosition[ 646838, 21761]*) (* CellTagsIndexPosition[ 646733, 21754]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox[ "Kadera, K., Crescenzi, M. and M. Shannon, 2003, \"Democratic Survival, \ Peace, and War in the International System,\" ", "Title", FontFamily->"Times New Roman", FontSize->16], StyleBox["AJPS", "Title", FontFamily->"Times New Roman", FontSize->16, FontSlant->"Italic"], StyleBox[" 47:234-47", "Title", FontFamily->"Times New Roman", FontSize->16] }], "Text", Evaluatable->False, CellHorizontalScrolling->False, FontSize->10], Cell[CellGroupData[{ Cell["\<\ Load the add on packages we need & set the font we want to use.\ \>", "Section"], Cell[BoxData[ \(<< "\"\)], "Input"], Cell[BoxData[ \(\($DefaultFont = {"\", 12}; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Set the parameter values for the model . 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\ i . e . , \ the\ systemic\ effect\ of\ democracies\ is\ uniformly\ felt\ by\ all\ dyads\)], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({\[Kappa]\_12, \[Kappa]\_13, \[Kappa]\_23, \[Eta]\_12, \[Eta]\_13, \[Eta]\_23} = \t{ .2, .15, .09, .52, .52, .52}\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox["0.2`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", StyleBox["0.15`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.09`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", StyleBox["0.520000000000000017`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.520000000000000017`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.520000000000000017`", StyleBoxAutoDelete->True, PrintPrecision->2]}], "}"}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Switching functions for the dyadic regime effect in the conflict equations. \ \ \>", "Section", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ \(In\ the\ article\ we\ name\ the\ function\ f\ \((see\ equation\ 3, \ pg . \ 237)\) . \ Here\ we\ give\ the\ function\ 3\ different\ names - f\), ",", "g", ",", \(and\ h\), ",", "\n", \(to\ help\ distinguish\ which\ dyad' s\ conflict\ equation\ it\ is\ in . \n\nNote\ that\ we' re\ using\ the\ MULTIPLICATIVE\ effect\ of \ two\ states'\ regime\ scores\), ",", " ", \(i . e . \), ",", " ", RowBox[{ RowBox[{ RowBox[{ FormBox[\(r\_1\), "TraditionalForm"], FormBox[\(r\_2\), "TraditionalForm"], " ", "not", " ", FormBox[\(r\_1\), "TraditionalForm"]}], "+", " ", RowBox[{ RowBox[{ FormBox[\(r\_2\), "TraditionalForm"], ".", "\n", "\t\t", "\n", "We"}], " ", "consider", " ", "rone", " ", "to", " ", "reflect", " ", "democracy", " ", "at", " ", "any", " ", "point", " ", "past", " ", "the", " ", "inflection", " ", \(point . \ The\), " ", "inflection", " ", "point", " ", "occurs", " ", "at", " ", "rone"}]}], "=", ".5"}], ",", " ", \(where\ the\ second\ derivative\ on\ the\ s\ curve\ is\ 0. \)}], TextForm]], "Text"], Cell["For the 1,2 dyad:", "Text"], Cell[BoxData[ \(f[rone_, rtwo_, c12_] := \(-\[Delta]\_DD\)\ c12\ rone\ rtwo /; N[rone > .5] \[And] N[rtwo > .5]\)], "Input"], Cell[BoxData[ \(f[rone_, rtwo_, c12_] := \(\[Delta]\_NN\) \((1 - \ rone)\) \((1 - \ rtwo)\) /; N[rone \[LessEqual] .5] \[And] N[rtwo \[LessEqual] .5]\)], "Input"], Cell[BoxData[ \(f[rone_, rtwo_, c12_] := \(\[Delta]\_ND\) \((1 - \ rone)\) \((1 - \ rtwo)\) /; \n\t\t \((N[rone > .5] \[And] N[rtwo \[LessEqual] .5])\) \[Or] \((N[rone \[LessEqual] .5] \[And] N[rtwo > .5])\)\)], "Input"], Cell["For the 1,3 dyad:", "Text"], Cell[BoxData[ \(g[rone_, rthree_, c13_] := \(-\[Delta]\_DD\) c13\ rone\ rthree /; N[rone > .5] \[And] N[rthree > .5]\)], "Input"], Cell[BoxData[ \(g[rone_, rthree_, c13_] := \(\[Delta]\_NN\) \((1 - rone)\) \((1 - \ rthree)\) /; \n\t\t N[rone \[LessEqual] .5] \[And] N[rthree \[LessEqual] .5]\)], "Input"], Cell[BoxData[ \(g[rone_, rthree_, c13_] := \(\[Delta]\_ND\) \((1 - rone)\) \((1 - \ rthree)\) /; \n \((N[rone > .5]\ \[And] \ N[rthree \[LessEqual] .5])\) \[Or] \((N[rone \[LessEqual] .5] \[And] N[rthree > .5])\)\)], "Input"], Cell["For the 2,3 dyad:", "Text"], Cell[BoxData[ \(h[rtwo_, rthree_, c23_] := \(-\[Delta]\_DD\) c23\ rtwo\ rthree /; \n\t\t N[rtwo > .5] \[And] N[rthree > .5]\)], "Input"], Cell[BoxData[ \(h[rtwo_, rthree_, c23_] := \(\[Delta]\_NN\) \((1 - rtwo)\) \((1 - \ rthree)\) /; \n\t\t N[rtwo \[LessEqual] .5] \[And] N[rthree \[LessEqual] .5]\)], "Input"], Cell[BoxData[ \(h[rtwo_, rthree_, c23_] := \(\[Delta]\_ND\) \((1 - rtwo)\) \((1 - \ rthree)\) /; \n\t\t \((N[rtwo > .5] \[And] N[rthree \[LessEqual] .5])\) \[Or] \((N[rtwo \[LessEqual] .5] \[And] N[rthree > .5])\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Switching function for the systemic regime effect in the conflict equations. \ \ \>", "Section"], Cell[BoxData[ \(TextForm\`\(In\ the\ article, \ this\ function\ is\ called\ g\ \((see\ equation\ 4, \ pg . \ 238)\) . \ Here\ we\ call\ it\ bigR\ \(\((because\ we\ had\ to\ \nuse\ g\ \ to\ help\ ease\ the\ representation\ of\ the\ f\ function\ in\ the\ previous\ section) \) . \)\ \)\)], "Text"], Cell[BoxData[ \(bigR[rone_, rtwo_, rthree_] := \((pow\_1\ rone + pow\_2\ rtwo\ + pow\_3\ rthree)\) /; N[rone > .5] \[And] N[rtwo > .5] \[And] N[rthree > .5]\)], "Input"], Cell[BoxData[ \(bigR[rone_, rtwo_, rthree_] := 0 /; N[rone \[LessEqual] .5] \[And] N[rtwo \[LessEqual] .5] \[And] N[rthree \[LessEqual] .5]\)], "Input"], Cell[BoxData[ \(bigR[rone_, rtwo_, rthree_] := pow\_1\ rone /; N[rone > .5] \[And] N[rtwo \[LessEqual] .5] \[And] N[rthree \[LessEqual] .5]\)], "Input"], Cell[BoxData[ \(bigR[rone_, rtwo_, rthree_] := pow\_2\ rtwo /; N[rone \[LessEqual] .5] \[And] N[rtwo > .5] \[And] N[rthree \[LessEqual] .5]\)], "Input"], Cell[BoxData[ \(bigR[rone_, rtwo_, rthree_] := pow\_3\ rthree /; N[rone \[LessEqual] .5] \[And] N[rtwo \[LessEqual] .5] \[And] N[rthree > .5]\)], "Input"], Cell[BoxData[ \(bigR[rone_, rtwo_, rthree_] := \((pow\_1\ rone + pow\_2\ rtwo)\) /; N[rone > .5] \[And] N[rtwo > .5] \[And] N[rthree \[LessEqual] .5]\)], "Input"], Cell[BoxData[ \(bigR[rone_, rtwo_, rthree_] := \((pow\_1\ rone + \(pow\_3\) rthree)\) /; N[rone > .5] \[And] N[rtwo \[LessEqual] .5] \[And] N[rthree > .5]\)], "Input"], Cell[BoxData[ \(bigR[rone_, rtwo_, rthree_] := \((pow\_2\ rtwo + pow\_3\ rthree)\) /; N[rone \[LessEqual] .5] \[And] N[rtwo > .5] \[And] N[rthree > .5]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Six equations that make up our system. ", "Section"], Cell["\<\ Naming your equations is a good idea because it allows you to reference them \ later on. It doesn't much matter what you name your equations, so these are \ named after Kelly's kids (no, she does not have 6 of them, just 3 with first \ and middle names). Equation 1 (see pg. 237) is represented by maddy, lily, \ and maxx. Equation 2 (see pg. 237) is represented by mae, june, and thomas. One assumption that is not completely clear in the article is that the \ deleterious effect of conflict on a nation's regime intensity is only \ affected by conflict in which it is involved. That is, in the last term of \ the rone'[t] equation, only c12[t] and c13[t] play a role, not c23[t]. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(maddy = \(rone'\)[t] == \(\[Alpha]\_1\) rone[t] - \(\[Alpha]\_1\) rone[t]\^2 - \[Gamma]\_1\ rone[t]\ \((1 - rone[t])\) \((c12[t] + c13[t])\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["rone", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ RowBox[{ StyleBox["0.3`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \(rone[t]\)}], "-", \(0.005`\ \((c12[t] + c13[t])\)\ \((1 - rone[t])\)\ rone[t]\), "-", \(0.3`\ rone[t]\^2\)}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(lily = \(rtwo'\)[t] == \(\[Alpha]\_2\) rtwo[t] - \(\[Alpha]\_2\) rtwo[t]\^2 - \[Gamma]\_2\ rtwo[t]\ \((1 - rtwo[t])\)\ \((c12[t] + c23[t])\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["rtwo", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ RowBox[{ StyleBox["0.4`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \(rtwo[t]\)}], "-", \(0.02`\ \((c12[t] + c23[t])\)\ \((1 - rtwo[t])\)\ rtwo[t]\), "-", \(0.4`\ rtwo[t]\^2\)}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(maxx = \(rthree'\)[t] == \(\[Alpha]\_3\) rthree[t] - \(\[Alpha]\_3\) rthree[t]\^2 - \[Gamma]\_3\ rthree[t]\ \((1 - rthree[t])\)\ \((c23[t] + c13[t])\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["rthree", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ RowBox[{ StyleBox["0.55`", StyleBoxAutoDelete->True, PrintPrecision->2], " ", \(rthree[t]\)}], "-", \(0.01`\ \((c13[t] + c23[t])\)\ \((1 - rthree[t])\)\ rthree[t]\), "-", \(0.55`\ rthree[t]\^2\)}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mae = \(c12'\)[t] == \(\[Kappa]\_12\) \((c12[t] + c13[t] + c23[t])\) - \(\[Eta]\_12\) c12[t]\ bigR[rone[t], rtwo[t], rthree[t]] + \ f[rone[t], rtwo[t], \ c12[t]]\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["c12", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ \(\(-0.520000000000000017`\)\ bigR[rone[t], rtwo[t], rthree[t]]\ c12[t]\), "+", RowBox[{ StyleBox["0.2`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \((c12[t] + c13[t] + c23[t])\)}], "+", \(f[rone[t], rtwo[t], c12[t]]\)}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(june = \(c13'\)[t] == \(\[Kappa]\_13\) \((c12[t] + c13[t] + c23[t])\) - \(\[Eta]\_13\) c13[t]\ bigR[rone[t], rtwo[t], rthree[t]] + \ g[rone[t], rthree[t], c13[t]]\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["c13", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ \(\(-0.520000000000000017`\)\ bigR[rone[t], rtwo[t], rthree[t]]\ c13[t]\), "+", RowBox[{ StyleBox["0.15`", StyleBoxAutoDelete->True, PrintPrecision->2], " ", \((c12[t] + c13[t] + c23[t])\)}], "+", \(g[rone[t], rthree[t], c13[t]]\)}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(thomas = \(c23'\)[t] == \(\[Kappa]\_23\) \((c12[t] + c13[t] + c23[t])\) - \(\[Eta]\_23\) c23[t]\ bigR[rone[t], rtwo[t], rthree[t]] + \ h[rtwo[t], rthree[t], c23[t]]\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["c23", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", RowBox[{ \(\(-0.520000000000000017`\)\ bigR[rone[t], rtwo[t], rthree[t]]\ c23[t]\), "+", RowBox[{ StyleBox["0.09`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \((c12[t] + c13[t] + c23[t])\)}], "+", \(h[rtwo[t], rthree[t], c23[t]]\)}]}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["An NDSolve routine for solving the system numerically.", "Section"], Cell[TextData[{ "Before running this NDSolve, enter a set of initial conditions. They are \ to be entered on the right hand side of these equations: \n\n", Cell[BoxData[ \(rone[0] == .4, rtwo[0] == .4, rthree[0] == .6, \nc12[0] == 1, c13[0] == .4, c23[0] == .2\)]], "\n\nEach time you use NDSolve, you will have to carefully choose your time \ interval. I always go some negative & some positive. Watch for error messages \ that say you've reached the MaxSteps at some positive or negative value of t. \ Then choose a t a bit smaller than that for the min or max (whichever is \ relevant). If you don't reach the MaxSteps, then check to see if one of the \ rs reaches 0 or 1 before increasing the time interval in either direction. \ This is explained in the plotting directions." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(modelrun = NDSolve[{maddy, lily, maxx, \ mae, june, thomas, \n\trone[0] == .77, rtwo[0] == .075, rthree[0] == .647, c12[0] == 3.144, c13[0] == 1.288, c23[0] == 2.63}, \n \t{rone, rtwo, rthree, c12, c13, c23}, {t, \(-2.5\), 20}, \ \ MaxSteps \[Rule] 500, AccuracyGoal \[Rule] 3, PrecisionGoal \[Rule] 3, \n\tWorkingPrecision \[Rule] 5]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"rone", "\[Rule]", TagBox[\(InterpolatingFunction[{{\(-2.5`\), 20.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"rtwo", "\[Rule]", TagBox[\(InterpolatingFunction[{{\(-2.5`\), 20.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"rthree", "\[Rule]", TagBox[\(InterpolatingFunction[{{\(-2.5`\), 20.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"c12", "\[Rule]", TagBox[\(InterpolatingFunction[{{\(-2.5`\), 20.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"c13", "\[Rule]", TagBox[\(InterpolatingFunction[{{\(-2.5`\), 20.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"c23", "\[Rule]", TagBox[\(InterpolatingFunction[{{\(-2.5`\), 20.`}}, "<>"]\), False, Editable->False]}]}], "}"}], "}"}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Plotting Routines", "Section"], Cell["\<\ First, we need to plot all three regime values & all three conflict values \ over time. After plotting the 3 regime variables, we visually inspect the \ graph to see where one of the rs becomes 0 or 1. Take the approximate time \ value at that point and use the FindRoot function to tell us the precise \ value of t at which the first r becomes 0 or 1.\ \>", "Text"], Cell["\<\ This defines 6 new functions as the result of the NDSolve routine. 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Cell[CellGroupData[{ Cell["Initial Conditions", "Section"], Cell[TextData[{ "Here are 15 sets of random initial conditions. The first 3 of each list \ are for regime 1, 2, and 3. The last 3 are for ", Cell[BoxData[ \(TraditionalForm\`c\_12\)]], ", ", Cell[BoxData[ \(TraditionalForm\`c\_13\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`c\_23\)]], ". Be careful to always use them in that exact order. 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