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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[ 80629, 2471]*) (*NotebookOutlinePosition[ 82655, 2539]*) (* CellTagsIndexPosition[ 82418, 2527]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ H2 Help File for Discrete Dynamical Systems in One Variable\ \>", "Subtitle", FontSize->16, FontWeight->"Bold"], Cell[CellGroupData[{ Cell["How to Use This Help File", "Subsection", Background->RGBColor[0, 1, 1]], Cell[TextData[{ "When evaluating a cell for the first time, a window with the following \ text will be displayed:\n", StyleBox[ "Do you want to automatically evaluate all the initialization cells in the \ notebook 'DDS1H.nb?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontWeight->"Bold"], "Click on ", StyleBox["Yes", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontWeight->"Bold"], " to insure that all the functions are available for this help file to \ function properly." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ \(<< DDS1`\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["Common Features of the Functions", "Subsection"], Cell[TextData[{ "The palette functions described in this help file are designed for use \ with a first-order discrete dynamical system (DDS) consisting of a single \ equation of the form\n\t\t\t\t\t", StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n+", FontSlant->"Italic"], "1)", StyleBox[" = f", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`(\)]], StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], ")).\nThe illustrations used with the specific palette functions are based \ on the following example:\t\t" }], "Text"], Cell[TextData[{ "Assume you have run up a credit card bill of $8000, and you can pay off \ $100 each month. The monthly interest rate is 1.075%. Defining ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "(", StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], ") to be the balance (in dollars) ", StyleBox["n", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"], " months from now leads to the following discrete dynamical system:\n", StyleBox["\n", FontSize->9], "\t\t new balance = old balance + interest on old balance - payment\n\t\t \ ", StyleBox[" ", FontSlant->"Italic"], StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "(", StyleBox["n+1", FontWeight->"Bold", FontSlant->"Italic"], ")", StyleBox[" = ", FontSlant->"Italic"], StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "(", StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], ")", StyleBox[" + 0.01075 ", FontSlant->"Italic"], StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "(", StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], ")", StyleBox[" - 100 \n\t\t = ", FontSlant->"Italic"], StyleBox["1.01075 x", FontWeight->"Bold", FontSlant->"Italic"], "(", StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], ")", StyleBox[" - 100", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["\n", FontSlant->"Italic"], StyleBox["\t", FontSize->9, FontSlant->"Italic"], StyleBox["\t ", FontSlant->"Italic"], "\nThus, the iterative model equation is given by ", StyleBox[" f(x) = 1.01075 x - 100 ", FontWeight->"Bold", FontSlant->"Italic"], "(replace ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "(", StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], ") by ", StyleBox["x", FontSlant->"Italic"], "). Furthermore, ", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "(0)", StyleBox[" = 8000, ", FontWeight->"Bold", FontSlant->"Italic"], "as the initial balance is $8000." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ How to Locate Specific Palette Functions within the Help File\ \>", "Subsection"], Cell["\<\ When you click on any of the function buttons, you will be linked \ to a description of the selected function. A highlighted cell bracket will \ indicate the cell containing the relevant explanation. The buttons are \ grouped vertically according to their usage. Palette functions in the first \ column compute and display system values. Functions in the second column are \ used to compute equilibrium values, and the functions in the last column show \ how the system evolves over time.\ \>", "Text"], Cell[BoxData[GridBox[{ { ButtonBox["IteratedValueSeq", ButtonData:>"ValueSeq", ButtonStyle->"Hyperlink"], ButtonBox["Equil", ButtonData:>"Equil", ButtonStyle->"Hyperlink"], ButtonBox["MapIt", ButtonData:>"MapIt", ButtonStyle->"Hyperlink"]}, { ButtonBox["ListGraph", ButtonData:>"ListGraph", ButtonStyle->"Hyperlink"], ButtonBox["SolveIt", ButtonData:>"SolveIt", ButtonStyle->"Hyperlink"], ButtonBox["LiveMap", ButtonData:>"LiveMap", ButtonStyle->"Hyperlink"]} }, ColumnLines->True]], "Input", Evaluatable->False, TextAlignment->-0.5, FontWeight->"Plain", CellTags->"top"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Description of Palette Functions Contained in the DDS1P \ Palette\ \>", "Subsection"], Cell[TextData[{ StyleBox["\nIteratedValueSeq[f, var, init, nmax, s] ", FontWeight->"Bold"], "gives the values of ", StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], ") for n = 0 to n = ", StyleBox["nmax", FontWeight->"Bold"], ", in steps of size ", StyleBox["s", FontWeight->"Bold"], ", for the discrete dynamical system described by ", StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n+", FontSlant->"Italic"], "1)", StyleBox[" = f", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`(\)]], StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], "))", StyleBox[",", FontSlant->"Italic"], " with ", StyleBox["init", FontWeight->"Bold"], " being the initial value for the system and ", StyleBox["var ", FontWeight->"Bold"], "the variable used to express the function ", StyleBox["f", FontWeight->"Bold"], ". If no value for", StyleBox[" s", FontWeight->"Bold"], " is given, the step size will be 1. To just get a single value for ", StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], "), use the same value for ", StyleBox["nmax", FontWeight->"Bold"], " and ", StyleBox["s", FontWeight->"Bold"], ". Here is an illustration of the two different uses of ", StyleBox["IteratedValueSeq", FontWeight->"Bold"], ". From the example described above we read off ", StyleBox["f = 1.01075 x - 100", FontWeight->"Bold"], ", ", StyleBox["var = x", FontWeight->"Bold"], ", and ", StyleBox["init = 8000 ", FontWeight->"Bold"], "(be careful not to use a comma after the 8!)." }], "Text", CellTags->"ValueSeq"], Cell[TextData[{ "If you just want to find out what your balance will be after 2 years ( = \ 24 months), i.e., you want to compute the value of ", StyleBox["x", FontSlant->"Italic"], "(24), then use ", StyleBox["nmax", FontWeight->"Bold"], " = ", StyleBox["s = 24. ", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(IteratedValueSeq[1.01075 x - 100, x, 8000, 24, 24]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {\("n"\), \("x(n)"\)}, {"0", "8000"}, {"24", "7618.99438698435194`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "On the other hand, to see your credit card balances over the next 2 years \ in intervals of three months, you would use ", StyleBox["nmax = 24 ", FontWeight->"Bold"], " and ", StyleBox["s = 3 ", FontWeight->"Bold"], "(every three months)." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(IteratedValueSeq[1.01075\ x - 100, x, 8000, 24, 3]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {\("n"\), \("x(n)"\)}, {"0", "8000"}, {"3", "7957.54688212499949`"}, {"6", "7913.70988049385282`"}, {"9", "7868.44388336102973`"}, {"12", "7821.70230843148317`"}, {"15", "7773.43705492378767`"}, {"18", "7723.59845407062284`"}, {"21", "7672.13521800566944`"}, {"24", "7618.99438698435194`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output"] }, Open ]], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ "\n", StyleBox["ListGraph[list, joint]", FontWeight->"Bold"], " is used to graph a ", StyleBox["list ", FontWeight->"Bold"], "of pairs of input-output values, generated, for example, by", StyleBox[" IteratedValueSeq", FontWeight->"Bold"], ". If the optional value ", StyleBox["joint", FontWeight->"Bold"], " is set to TRUE, then the data points will be connected by line segments. \ If no value is given, then the points will not be connected. As an example, \ we compute the credit card balances for one year (", StyleBox["nmax = 12", FontWeight->"Bold"], ") and name the resulting list ", StyleBox["A", FontWeight->"Bold"], ". Then we graph this list of values using", StyleBox[" ListGraph.", FontWeight->"Bold"], " Since we do not give a value for ", StyleBox["joint", FontWeight->"Bold"], ", the data points will not be connected." }], "Text", CellTags->"ListGraph"], Cell[CellGroupData[{ Cell[BoxData[ \(A = IteratedValueSeq[1.01075\ x - 100, x, 8000, 12, 1]; \n ListGraph[A]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0793651 -25.8066 0.00330124 [ [.18254 .0131 -3 -9 ] [.18254 .0131 3 0 ] [.34127 .0131 -3 -9 ] [.34127 .0131 3 0 ] [.5 .0131 -3 -9 ] [.5 .0131 3 0 ] [.65873 .0131 -3 -9 ] [.65873 .0131 3 0 ] [.81746 .0131 -6 -9 ] [.81746 .0131 6 0 ] [.97619 .0131 -6 -9 ] [.97619 .0131 6 0 ] [1.025 .0256 0 -7.5 ] [1.025 .0256 10 7.5 ] [.01131 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ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooo0ol00>koool001koool00`000?ooooooo`0>oooo0ol00>koool0 01koool00`000?ooooooo`0?oooo00?o003oooooool0kOooo`007_ooo`80003ooooo0Oooo`007_oo o`030000oooooooo0?ooool001koool00`000?ooooooo`3ooooo000Noooo00<0003oooooool0oooo o`007_ooo`80003ooooo0Oooo`007_ooo`030000oooooooo0?ooool000Woool00`000?ooooooo`02 oooo00<0003oooooool00_ooo`030000oooooooo00;oool00`000?ooooooo`03oooo00<0003ooooo ool0ooooo`002?ooo`030000oooo000000;oool00`000?ooo`000002oooo00<0003oool000000_oo o`030000oooo000000Coool00`000?ooooooo`3ooooo0008oooo00<0003oool000000_ooo`030000 oooo000000;oool00`000?ooo`000002oooo00<0003oool000000_ooo`?o00020000ooooo`002Ooo o`070000oooooooooooo0000oooo000000;oool00`000?ooo`000002oooo00<0003oool000000_oo o`?o003ooooo0_ooo`002?ooo`030000oooo000000;oool00`000?ooo`000002oooo00<0003oool0 00000_ooo`030000oooo000000?oool00ol000000?ooo`3ooooo0Oooo`002Oooo`030000oooooooo 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Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ "\n", StyleBox["Equil[c, d]", FontWeight->"Bold"], " computes the equilibrium value for the ", StyleBox["linear", FontVariations->{"Underline"->True}], " discrete dynamical system of the form\n\n\t\t\t", StyleBox["x", FontWeight->"Bold", FontSlant->"Italic"], "(", StyleBox["n+", FontWeight->"Bold", FontSlant->"Italic"], "1)", StyleBox[" = c * x", FontWeight->"Bold", FontSlant->"Italic"], "(", StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], ")", StyleBox[" + d.\n\t\t\t", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["\n", FontWeight->"Bold"], "The example of the credit card balance is a linear system with ", StyleBox["c", FontSlant->"Italic"], " = 1.01075 and ", StyleBox["d", FontSlant->"Italic"], " = -100. In this example, the equilibrium value gives the amount at which \ the credit card balance remains the same (the interest on the balance equals \ the monthly payment). " }], "Text", CellTags->"Equil"], Cell[CellGroupData[{ Cell[BoxData[ \(Equil[1.01075, \(-100\)]\)], "Input"], Cell[BoxData[ \(9302.32558139531739`\)], "Output"] }, Open ]], Cell["\<\ This result indicates that if the initial balance is $9302.33, then \ the balance will not change at all over time.\ \>", "Text"], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ "\n\nA graphical display of the changing value of ", StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], ") over time can be produced with the functions ", StyleBox["MapIt", FontWeight->"Bold"], " and ", StyleBox["LiveMap", FontWeight->"Bold"], ". " }], "Text"], Cell[TextData[{ StyleBox["MapIt[{f, x}, xinit, n, {xmin, xmax}, focus, percent]", FontWeight->"Bold"], " displays the first ", StyleBox["n ", FontWeight->"Bold"], "iterations of the DDS ", StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n+", FontSlant->"Italic"], "1)", StyleBox[" = f", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`(\)]], StyleBox["x", FontSlant->"Italic"], "(", StyleBox["n", FontSlant->"Italic"], ")), where ", StyleBox["f", FontWeight->"Bold"], " is expressed in terms of ", StyleBox["x", FontWeight->"Bold"], ". The iteration starts at", StyleBox[" x", FontSlant->"Italic"], " = ", StyleBox["xinit. ", FontWeight->"Bold"], "The horizontal axis of the graph shows values from ", StyleBox["xmin", FontWeight->"Bold"], " to ", StyleBox["xmax", FontWeight->"Bold"], ". When zooming in, the graph is centered at ", StyleBox["x", FontSlant->"Italic"], " =", StyleBox[" focus", FontWeight->"Bold"], ". The value of ", StyleBox["percent", FontWeight->"Bold"], " determines the amount of zooming: For 0 < ", StyleBox["percent", FontWeight->"Bold"], " < 1, we zoom in, for ", StyleBox["percent", FontWeight->"Bold"], " > 1, we zoom out." }], "Text", CellTags->"MapIt"], Cell[TextData[{ "For example, if we want to see what happens to the credit card balance in \ the first 6 months, we use ", StyleBox["f = ", FontWeight->"Bold"], "1.01075 ", StyleBox["x", FontSlant->"Italic"], " - 100, ", StyleBox["xinit", FontWeight->"Bold"], " = 8000, and ", StyleBox["n", FontWeight->"Bold"], " = 6. From our previous computations we know that the balance after 6 \ months is $7913.71. Thus, we could use as range for the horizontal axis ", StyleBox["{xmin,xmax}", FontWeight->"Bold"], " = {7850,8050}. 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