(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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[ 441853, 13253]*) (*NotebookOutlinePosition[ 442546, 13277]*) (* CellTagsIndexPosition[ 442502, 13273]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox["Biophysics 101, Fall 2002\nProblem Set 1, part 3", \ "DisplayFormula"]], "Text", TextAlignment->Right, TextJustification->0], Cell["Warm-up Notebook", "Subtitle", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["warmup_answers", "Title", FontFamily->"Garamond", FontWeight->"Bold", FontColor->RGBColor[1, 0.501961, 0.752941]]], "Subtitle", Evaluatable->False, TextAlignment->Center, TextJustification->0, FormatType->TextForm], Cell[TextData[{ "This notebook is for beginners that have done the ", StyleBox["Mathematica", FontSlant->"Italic"], " tutorial. Most of it is no harder than a coloring book It starts with \ cell manipulation and formatting, and it works its way up to discrete \ recursive functions that operate on ", StyleBox["Mathematica", FontSlant->"Italic"], " tables and arrays. Many of us are new to ", StyleBox["Mathematica", FontSlant->"Italic"], ", so ", StyleBox["if you discover anything useful, please include them in your \ finished notebook.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["1: Saving notebooks", "Subsection"]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ Your first challenge is to save a copy of this notebook, and you get to name \ it after you. On the menu bar at the top of the screen its \"File-> Save \ As.... \". If your name is Jack Flapjack then name it Jack_Flapjack.nb. If \ your name is not Jack Flapjack then name it after your own name instead. Put \ your answers in your self-named copy and turn it in to your TF. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["2: Inserting and deleting ", "Subsection"], StyleBox["Mathematica", "Subsection", FontSlant->"Italic"], StyleBox[" cells", "Subsection"] }], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ If you click in the thin horizontal space between cells, a thin black line \ appears. Hit the enter key, and voila, a new cell. Insert a cell between the \ second and third cells in this notebook, and type your name in it. If you are \ feeling stylish, format your name text with the many options available under \ the format menu. On the menu, check out \"Format->Show Toolbar\" Though short \ on features, the formatting toolbar may be worth it's screen real estate. To \ delete a cell, click to highlight it's blue bracket on the right, and press \ the delete key. To copy/paste a cell - highlight the blue bracket of the cell \ you want to copy, ->copy,then make the thin horizontal black \ line appear where you want it to go, and ->paste.\ \>", "Text", TextAlignment->Left, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["3: ", "Subsection"], StyleBox["Mathematica", "Subsection", FontSlant->"Italic"], StyleBox[" cell types", "Subsection"] }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Notice the subtle differences in the cell bracket shapes. They tell you \ what type of cell it is. This type, with the double line at the top, means \ its a non-evaluatable text cell. If you press while the cursor \ is in this type of cell, ", StyleBox["Mathematica", FontSlant->"Italic"], " will beep at you.", StyleBox[" \n", FontColor->RGBColor[1, 0, 0]], StyleBox["(2 pts)", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox["Try it on the cell below. Now to make it evaluatable, highlight \ the cell bracket, and then its \"Cell->Cell Properties->Cell Evaluatable\"", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[StyleBox["2+2", "Subsubsection"]], "Text", Evaluatable->True, TextAlignment->Center, TextJustification->0], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell["\<\ The default cell type, for example the one you just typed your name in, is \ called \"Input Form\", and its cell bracket has a small triangle at the top. \ Change the property of the cell containing your name to non-evaluatable, and \ convert its display type to Text thru the menu via \"Cell->Display \ As->Text\".\ \>", "Text", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell["\<\ This is an input type cell in text display mode. If you press \ in this cell, you will see what this sentence means to the Mathematica kernel \ (not much). \ \>", "Input", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(input\)\" is similar to \ existing symbol \"\!\(Input\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(cell\)\" is similar to \ existing symbol \"\!\(Cell\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(text\)\" is similar to \ existing symbol \"\!\(Text\)\"."\)], "Message"], Cell[BoxData[ \(General::"stop" \(\(:\)\(\ \)\) "Further output of \!\(General :: \"spell1\"\) will be suppressed \ during this calculation."\)], "Message"], Cell[BoxData[ \(General::"spell" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(enter\)\" is similar to \ existing symbols \!\({Center, Enter}\)."\)], "Message"], Cell["\<\ Syntax::sntxf: \"cell in text display mode. If you press in \ this cell\" cannot be followed by \", you will see what this sentence means to \ <<11>>tica kernel (not much).\".\ \>", "Message"] }, Open ]], Cell[BoxData[ FormBox[ StyleBox[\(This\ is\ a\ traditional\ form\ cell . \ \ It\ has\ the\ \ coolest\ bracket\ shape, \ but\ it' s\ a\ relic\ of\ old\ Mathematica\ versions\ that' s\ \ still\ \ compatable\ with\ version\ \ \(\(4.2\)\(.\)\)\), "Text"], TraditionalForm]], "Input"] }, Open ]], Cell[TextData[StyleBox["4: Cell grouping", "Subsection"]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ Notice how most of these cells are nested, but the two cells in this section \ are not. To group them, first go on the menu to \"Cell->Cell Grouping\" and \ make sure \"Manual Grouping\" is checked. Then select both brackets by \ dragging the mouse along the right margin, and its \"Cell->Cell \ Grouping->Group Cells\". Double click on the outermost bracket to make the \ nest expand and collapse.\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["5: Named lists", "Subsection"]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "In the next cell there are some named lists and a name based list \ operation. ", StyleBox["(2 pts)", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["Evaluate the expressions by placing the cursor to the right of \ the last line, and . 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FontColor->RGBColor[0.682353, 0.32549, 0.317647]], StyleBox["Notice how the arrays are printed out when you evaluate the cell. \ This can be annoying when you want to look at a complicated graph but don't \ want to see all of it's underlying points printed out. Put a \";\" at the end \ of the expressions that you don't want to display the results of.\n\nNote 2) \ It's considered good ", FontWeight->"Bold", FontColor->RGBColor[0.682353, 0.32549, 0.317647], FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.682353, 0.32549, 0.317647], FontVariations->{"CompatibilityType"->0}], StyleBox[" practice to begin your custom variable names with a lower case \ letter, which will prevent name collisions with the ", FontWeight->"Bold", FontColor->RGBColor[0.682353, 0.32549, 0.317647], FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematicas ", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.682353, 0.32549, 0.317647], FontVariations->{"CompatibilityType"->0}], StyleBox["names, all of which begin with uppercase letters.\n\nNote 3) \ Don't forget to put the colon in front of the equals sign \":=\" when you \ define a function. This can be very frustrating, because leaving it off won't \ cause an error. So if your function is giving you unexpected results, don't \ forget to re-read this note!", FontWeight->"Bold", FontColor->RGBColor[0.682353, 0.32549, 0.317647], FontVariations->{"CompatibilityType"->0}] }], "Text", TextAlignment->Left, TextJustification->0] }, Open ]], Cell[TextData[StyleBox["8: Using lists to graph a built in function", \ "Subsection"]], "Text", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[{ "You can compose your own custom functions with ", StyleBox["Mathematica", FontSlant->"Italic"], " built in functions. Remember N from the tutorial? -", StyleBox[" evaluate these expressions.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell[BoxData[{ \(\(k\ = \ 3;\)\), "\[IndentingNewLine]", \(\(f[x_]\ = k*Tanh[x];\)\), "\[IndentingNewLine]", \(\(f[3];\)\), "\[IndentingNewLine]", \(N[f[3]]\)}], "Input"], Cell[BoxData[ \(2.9851642610601914`\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["(2 pts) ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Make a plot of Tanh[x], with the y values between -0.9 and 0.9. \ Bonus: Make it exactly between -0.9 and 0.9 (hint: look in Help->Master \ Index->Built-in Functions->Mathematical Functions).", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text", TextAlignment->Left, TextJustification->0] }, Open ]], Cell[TextData[{ StyleBox["Its possible to do this using functions with plot or using a \ lists with listplot. 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resource constraints, like bacterial cultures in \ \"mid log phase\", and people until recently. ", Evaluatable->False, FontColor->RGBColor[1, 0.501961, 1]] }], "Input", FormatType->TextForm], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[StyleBox["12: A more complex model", "Subsection"]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "The logistic model is presented below. It is considered to be the simplest \ dynamical system that exhibits chaotic behavior. It is implemented here with \ three parameters: r, k, and g. Set these three numbers to different values, \ evaluate, and observe how the graph changes. In ", StyleBox["Mathematica", FontSlant->"Italic"], " it is necessary to clear a function before you change the parameters that \ are part of it's definition, which explains the first command: Clear[pop]. \ Note: use values for r, 0 < r < 4, or you will get overflow errors. Also, \ when setting r to a whole number, such as 2, put a decimal point after the \ number: write \"r = 2.\" instead of \"r=2\" . The latter will cause ", StyleBox["Mathematica", FontSlant->"Italic"], " to hang. ", StyleBox["\n(12 pts) Match the four parameters to the following \ definitions:\n1) The population at time zero.\n2) The number of generations.\n\ 3) The capacity of the enviroment to sustain the population.\n4) The \ fertility of the organisims.\nCreate three copies of this cell. In one cell \ choose parameters that produce a graph that shows unchecked exponential \ growth, similar to the graph you created in (11). \nIn the second cell, set \ parameters that produce a graph that shows periodic behavior (i.e. the \ population changes from generation to generation in a regular way.\nIn the \ thrid cell, set parameters that result in chaotic behavior. Which parameter \ controls the chaotic/non-chaotic behavior of the system? ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ StyleBox["r - A measure of the fecundity of the population.\nk - The \ carrying capacity of the enviroment: (1-pop[t]/k) becomes negative when the \ population exceeds k.\ng - The number of generations plotted.\ns - The \ population at time 0.\n\n", FontColor->RGBColor[1, 0.501961, 1]], StyleBox["The parameters below show unrestrained population growth. 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These cycles show up better by selecting large g and by \ setting PlotJoined->False. Starting with s near k eliminates pre-convergence \ noise as well.", FontColor->RGBColor[1, 0.501961, 1]]], "Input", Evaluatable->False, FormatType->TextForm], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(Clear\ [pop]\n \(r\ = \ 3.51;\)\n \(k\ = \ 1000;\)\n \(g\ = \ 1000;\)\n \(s\ = \ 1000;\)\n \(pop[t_]\ := \ \(pop[t]\ = \ r\ pop[t - 1] \((1 - \ pop[t - 1]/k)\)\);\)\n \(pop[0]\ = \ 5;\)\n \(dt\ = Table[pop[x], \ {x, 0, g, 1}];\)\n ListPlot[dt, \ PlotJoined\ \[Rule] False, \ PlotRange \[Rule] {k/4, \ k}]\)\)\)], "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 %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.00095143 -0.206011 0.000824045 [ [.2141 .0287 -9 -9 ] [.2141 .0287 9 0 ] [.40438 .0287 -9 -9 ] [.40438 .0287 9 0 ] [.59467 .0287 -9 -9 ] [.59467 .0287 9 0 ] [.78495 .0287 -9 -9 ] [.78495 .0287 9 0 ] [.97524 .0287 -12 -9 ] [.97524 .0287 12 0 ] [.01131 .12361 -18 -4.5 ] [.01131 .12361 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .28842 -18 -4.5 ] [.01131 .28842 0 4.5 ] [.01131 .37082 -18 -4.5 ] [.01131 .37082 0 4.5 ] [.01131 .45322 -18 -4.5 ] [.01131 .45322 0 4.5 ] [.01131 .53563 -18 -4.5 ] [.01131 .53563 0 4.5 ] [.01131 .61803 -24 -4.5 ] [.01131 .61803 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .2141 .0412 m .2141 .04745 L s [(200)] .2141 .0287 0 1 Mshowa .40438 .0412 m .40438 .04745 L s [(400)] .40438 .0287 0 1 Mshowa .59467 .0412 m .59467 .04745 L s [(600)] .59467 .0287 0 1 Mshowa .78495 .0412 m .78495 .04745 L s [(800)] .78495 .0287 0 1 Mshowa .97524 .0412 m .97524 .04745 L s [(1000)] .97524 .0287 0 1 Mshowa .125 Mabswid .07138 .0412 m .07138 .04495 L s .11895 .0412 m .11895 .04495 L s .16652 .0412 m .16652 .04495 L s .26167 .0412 m .26167 .04495 L s .30924 .0412 m .30924 .04495 L s .35681 .0412 m .35681 .04495 L s .45195 .0412 m .45195 .04495 L s .49952 .0412 m .49952 .04495 L s .5471 .0412 m .5471 .04495 L s .64224 .0412 m .64224 .04495 L s .68981 .0412 m .68981 .04495 L s .73738 .0412 m .73738 .04495 L s .83252 .0412 m .83252 .04495 L s .8801 .0412 m .8801 .04495 L s .92767 .0412 m .92767 .04495 L s .25 Mabswid 0 .0412 m 1 .0412 L s .02381 .12361 m .03006 .12361 L s [(400)] .01131 .12361 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(500)] .01131 .20601 1 0 Mshowa .02381 .28842 m .03006 .28842 L s [(600)] .01131 .28842 1 0 Mshowa .02381 .37082 m .03006 .37082 L s [(700)] .01131 .37082 1 0 Mshowa .02381 .45322 m .03006 .45322 L s [(800)] .01131 .45322 1 0 Mshowa .02381 .53563 m .03006 .53563 L s [(900)] .01131 .53563 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(1000)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .05768 m .02756 .05768 L s .02381 .07416 m .02756 .07416 L s .02381 .09064 m .02756 .09064 L s .02381 .10713 m .02756 .10713 L s .02381 .14009 m .02756 .14009 L s .02381 .15657 m .02756 .15657 L s .02381 .17305 m .02756 .17305 L s .02381 .18953 m .02756 .18953 L s .02381 .22249 m .02756 .22249 L s .02381 .23897 m .02756 .23897 L s .02381 .25545 m .02756 .25545 L s .02381 .27193 m .02756 .27193 L s .02381 .3049 m .02756 .3049 L s .02381 .32138 m .02756 .32138 L s .02381 .33786 m .02756 .33786 L s .02381 .35434 m .02756 .35434 L s .02381 .3873 m .02756 .3873 L s .02381 .40378 m .02756 .40378 L s .02381 .42026 m .02756 .42026 L s .02381 .43674 m .02756 .43674 L s .02381 .46971 m .02756 .46971 L s .02381 .48619 m .02756 .48619 L s .02381 .50267 m .02756 .50267 L s .02381 .51915 m .02756 .51915 L s .02381 .55211 m .02756 .55211 L s .02381 .56859 m .02756 .56859 L s .02381 .58507 m .02756 .58507 L s .02381 .60155 m .02756 .60155 L s .02381 .02472 m .02756 .02472 L s .02381 .00824 m .02756 .00824 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .008 w .02857 .25442 Mdot .02952 .5071 Mdot .03047 .13094 Mdot .03142 .49308 Mdot .03237 .16607 Mdot .03332 .51029 Mdot .03428 .12271 Mdot .03523 .48753 Mdot .03618 .1795 Mdot .03713 .5141 Mdot .03808 .1128 Mdot .03903 .48008 Mdot .03998 .19715 Mdot .04094 .51675 Mdot .04189 .10579 Mdot .04284 .4743 Mdot .04379 .21049 Mdot .04474 .517 Mdot .04569 .10513 Mdot .04664 .47374 Mdot .0476 .21178 Mdot .04855 .51695 Mdot .0495 .10528 Mdot .05045 .47387 Mdot .0514 .21148 Mdot .05235 .51696 Mdot .0533 .10524 Mdot .05426 .47384 Mdot .05521 .21156 Mdot .05616 .51696 Mdot .05711 .10525 Mdot .05806 .47384 Mdot .05901 .21154 Mdot .05996 .51696 Mdot .06092 .10525 Mdot .06187 .47384 Mdot .06282 .21154 Mdot .06377 .51696 Mdot .06472 .10525 Mdot .06567 .47384 Mdot .06662 .21154 Mdot .06758 .51696 Mdot .06853 .10525 Mdot .06948 .47384 Mdot .07043 .21154 Mdot .07138 .51696 Mdot .07233 .10525 Mdot .07328 .47384 Mdot .07424 .21154 Mdot .07519 .51696 Mdot .07614 .10525 Mdot .07709 .47384 Mdot .07804 .21154 Mdot .07899 .51696 Mdot .07994 .10525 Mdot .0809 .47384 Mdot .08185 .21154 Mdot .0828 .51696 Mdot .08375 .10525 Mdot .0847 .47384 Mdot .08565 .21154 Mdot .0866 .51696 Mdot .08756 .10525 Mdot .08851 .47384 Mdot .08946 .21154 Mdot .09041 .51696 Mdot .09136 .10525 Mdot .09231 .47384 Mdot .09326 .21154 Mdot .09422 .51696 Mdot .09517 .10525 Mdot .09612 .47384 Mdot .09707 .21154 Mdot .09802 .51696 Mdot .09897 .10525 Mdot .09992 .47384 Mdot .10088 .21154 Mdot .10183 .51696 Mdot .10278 .10525 Mdot .10373 .47384 Mdot .10468 .21154 Mdot .10563 .51696 Mdot .10658 .10525 Mdot .10754 .47384 Mdot .10849 .21154 Mdot .10944 .51696 Mdot .11039 .10525 Mdot .11134 .47384 Mdot .11229 .21154 Mdot .11324 .51696 Mdot .1142 .10525 Mdot .11515 .47384 Mdot .1161 .21154 Mdot .11705 .51696 Mdot .118 .10525 Mdot .11895 .47384 Mdot .1199 .21154 Mdot .12086 .51696 Mdot .12181 .10525 Mdot .12276 .47384 Mdot .12371 .21154 Mdot .12466 .51696 Mdot .12561 .10525 Mdot .12656 .47384 Mdot .12752 .21154 Mdot .12847 .51696 Mdot .12942 .10525 Mdot .13037 .47384 Mdot .13132 .21154 Mdot .13227 .51696 Mdot .13322 .10525 Mdot .13418 .47384 Mdot .13513 .21154 Mdot .13608 .51696 Mdot .13703 .10525 Mdot .13798 .47384 Mdot .13893 .21154 Mdot .13988 .51696 Mdot .14084 .10525 Mdot .14179 .47384 Mdot .14274 .21154 Mdot .14369 .51696 Mdot .14464 .10525 Mdot .14559 .47384 Mdot .14654 .21154 Mdot .1475 .51696 Mdot .14845 .10525 Mdot .1494 .47384 Mdot .15035 .21154 Mdot .1513 .51696 Mdot .15225 .10525 Mdot .1532 .47384 Mdot .15416 .21154 Mdot .15511 .51696 Mdot .15606 .10525 Mdot .15701 .47384 Mdot .15796 .21154 Mdot .15891 .51696 Mdot .15986 .10525 Mdot .16082 .47384 Mdot .16177 .21154 Mdot .16272 .51696 Mdot .16367 .10525 Mdot .16462 .47384 Mdot .16557 .21154 Mdot .16652 .51696 Mdot .16748 .10525 Mdot .16843 .47384 Mdot .16938 .21154 Mdot .17033 .51696 Mdot .17128 .10525 Mdot .17223 .47384 Mdot .17318 .21154 Mdot .17414 .51696 Mdot .17509 .10525 Mdot .17604 .47384 Mdot .17699 .21154 Mdot .17794 .51696 Mdot .17889 .10525 Mdot .17984 .47384 Mdot .1808 .21154 Mdot .18175 .51696 Mdot .1827 .10525 Mdot .18365 .47384 Mdot .1846 .21154 Mdot .18555 .51696 Mdot .1865 .10525 Mdot .18746 .47384 Mdot .18841 .21154 Mdot .18936 .51696 Mdot .19031 .10525 Mdot .19126 .47384 Mdot .19221 .21154 Mdot .19316 .51696 Mdot .19412 .10525 Mdot .19507 .47384 Mdot .19602 .21154 Mdot .19697 .51696 Mdot .19792 .10525 Mdot .19887 .47384 Mdot .19982 .21154 Mdot .20078 .51696 Mdot .20173 .10525 Mdot .20268 .47384 Mdot .20363 .21154 Mdot .20458 .51696 Mdot .20553 .10525 Mdot .20648 .47384 Mdot .20744 .21154 Mdot .20839 .51696 Mdot .20934 .10525 Mdot .21029 .47384 Mdot .21124 .21154 Mdot .21219 .51696 Mdot .21314 .10525 Mdot .2141 .47384 Mdot .21505 .21154 Mdot .216 .51696 Mdot .21695 .10525 Mdot .2179 .47384 Mdot .21885 .21154 Mdot .2198 .51696 Mdot .22076 .10525 Mdot .22171 .47384 Mdot .22266 .21154 Mdot .22361 .51696 Mdot .22456 .10525 Mdot 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Mdot .27594 .21154 Mdot .27689 .51696 Mdot .27784 .10525 Mdot .27879 .47384 Mdot .27974 .21154 Mdot .2807 .51696 Mdot .28165 .10525 Mdot .2826 .47384 Mdot .28355 .21154 Mdot .2845 .51696 Mdot .28545 .10525 Mdot .2864 .47384 Mdot .28736 .21154 Mdot .28831 .51696 Mdot .28926 .10525 Mdot .29021 .47384 Mdot .29116 .21154 Mdot .29211 .51696 Mdot .29306 .10525 Mdot .29402 .47384 Mdot .29497 .21154 Mdot .29592 .51696 Mdot .29687 .10525 Mdot .29782 .47384 Mdot .29877 .21154 Mdot .29972 .51696 Mdot .30068 .10525 Mdot .30163 .47384 Mdot .30258 .21154 Mdot .30353 .51696 Mdot .30448 .10525 Mdot .30543 .47384 Mdot .30638 .21154 Mdot .30734 .51696 Mdot .30829 .10525 Mdot .30924 .47384 Mdot .31019 .21154 Mdot .31114 .51696 Mdot .31209 .10525 Mdot .31304 .47384 Mdot .314 .21154 Mdot .31495 .51696 Mdot .3159 .10525 Mdot .31685 .47384 Mdot .3178 .21154 Mdot .31875 .51696 Mdot .3197 .10525 Mdot .32066 .47384 Mdot .32161 .21154 Mdot .32256 .51696 Mdot .32351 .10525 Mdot .32446 .47384 Mdot .32541 .21154 Mdot .32636 .51696 Mdot .32732 .10525 Mdot .32827 .47384 Mdot .32922 .21154 Mdot .33017 .51696 Mdot .33112 .10525 Mdot .33207 .47384 Mdot .33302 .21154 Mdot .33398 .51696 Mdot .33493 .10525 Mdot .33588 .47384 Mdot .33683 .21154 Mdot .33778 .51696 Mdot .33873 .10525 Mdot .33968 .47384 Mdot .34064 .21154 Mdot .34159 .51696 Mdot .34254 .10525 Mdot .34349 .47384 Mdot .34444 .21154 Mdot .34539 .51696 Mdot .34634 .10525 Mdot .3473 .47384 Mdot .34825 .21154 Mdot .3492 .51696 Mdot .35015 .10525 Mdot .3511 .47384 Mdot .35205 .21154 Mdot .353 .51696 Mdot .35396 .10525 Mdot .35491 .47384 Mdot .35586 .21154 Mdot .35681 .51696 Mdot .35776 .10525 Mdot .35871 .47384 Mdot .35966 .21154 Mdot .36062 .51696 Mdot .36157 .10525 Mdot .36252 .47384 Mdot .36347 .21154 Mdot .36442 .51696 Mdot .36537 .10525 Mdot .36632 .47384 Mdot .36728 .21154 Mdot .36823 .51696 Mdot .36918 .10525 Mdot .37013 .47384 Mdot .37108 .21154 Mdot .37203 .51696 Mdot .37298 .10525 Mdot .37394 .47384 Mdot .37489 .21154 Mdot .37584 .51696 Mdot .37679 .10525 Mdot .37774 .47384 Mdot .37869 .21154 Mdot .37964 .51696 Mdot .3806 .10525 Mdot .38155 .47384 Mdot .3825 .21154 Mdot .38345 .51696 Mdot .3844 .10525 Mdot .38535 .47384 Mdot .3863 .21154 Mdot .38726 .51696 Mdot .38821 .10525 Mdot .38916 .47384 Mdot .39011 .21154 Mdot .39106 .51696 Mdot .39201 .10525 Mdot .39296 .47384 Mdot .39392 .21154 Mdot .39487 .51696 Mdot .39582 .10525 Mdot .39677 .47384 Mdot .39772 .21154 Mdot .39867 .51696 Mdot .39962 .10525 Mdot .40058 .47384 Mdot .40153 .21154 Mdot .40248 .51696 Mdot .40343 .10525 Mdot .40438 .47384 Mdot .40533 .21154 Mdot .40628 .51696 Mdot .40724 .10525 Mdot .40819 .47384 Mdot .40914 .21154 Mdot .41009 .51696 Mdot .41104 .10525 Mdot .41199 .47384 Mdot .41294 .21154 Mdot .4139 .51696 Mdot .41485 .10525 Mdot .4158 .47384 Mdot .41675 .21154 Mdot .4177 .51696 Mdot .41865 .10525 Mdot .4196 .47384 Mdot .42056 .21154 Mdot .42151 .51696 Mdot .42246 .10525 Mdot .42341 .47384 Mdot .42436 .21154 Mdot .42531 .51696 Mdot .42626 .10525 Mdot .42722 .47384 Mdot .42817 .21154 Mdot .42912 .51696 Mdot .43007 .10525 Mdot .43102 .47384 Mdot .43197 .21154 Mdot .43292 .51696 Mdot .43388 .10525 Mdot .43483 .47384 Mdot .43578 .21154 Mdot .43673 .51696 Mdot .43768 .10525 Mdot .43863 .47384 Mdot .43958 .21154 Mdot .44054 .51696 Mdot .44149 .10525 Mdot .44244 .47384 Mdot .44339 .21154 Mdot .44434 .51696 Mdot .44529 .10525 Mdot .44624 .47384 Mdot .4472 .21154 Mdot .44815 .51696 Mdot .4491 .10525 Mdot .45005 .47384 Mdot .451 .21154 Mdot .45195 .51696 Mdot .4529 .10525 Mdot .45386 .47384 Mdot .45481 .21154 Mdot .45576 .51696 Mdot .45671 .10525 Mdot .45766 .47384 Mdot .45861 .21154 Mdot .45956 .51696 Mdot .46052 .10525 Mdot .46147 .47384 Mdot .46242 .21154 Mdot .46337 .51696 Mdot .46432 .10525 Mdot .46527 .47384 Mdot .46622 .21154 Mdot .46718 .51696 Mdot .46813 .10525 Mdot .46908 .47384 Mdot .47003 .21154 Mdot .47098 .51696 Mdot .47193 .10525 Mdot .47288 .47384 Mdot .47384 .21154 Mdot .47479 .51696 Mdot .47574 .10525 Mdot 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Mdot .57754 .51696 Mdot .57849 .10525 Mdot .57944 .47384 Mdot .5804 .21154 Mdot .58135 .51696 Mdot .5823 .10525 Mdot .58325 .47384 Mdot .5842 .21154 Mdot .58515 .51696 Mdot .5861 .10525 Mdot .58706 .47384 Mdot .58801 .21154 Mdot .58896 .51696 Mdot .58991 .10525 Mdot .59086 .47384 Mdot .59181 .21154 Mdot .59276 .51696 Mdot .59372 .10525 Mdot .59467 .47384 Mdot .59562 .21154 Mdot .59657 .51696 Mdot .59752 .10525 Mdot .59847 .47384 Mdot .59942 .21154 Mdot .60038 .51696 Mdot .60133 .10525 Mdot .60228 .47384 Mdot .60323 .21154 Mdot .60418 .51696 Mdot .60513 .10525 Mdot .60608 .47384 Mdot .60704 .21154 Mdot .60799 .51696 Mdot .60894 .10525 Mdot .60989 .47384 Mdot .61084 .21154 Mdot .61179 .51696 Mdot .61274 .10525 Mdot .6137 .47384 Mdot .61465 .21154 Mdot .6156 .51696 Mdot .61655 .10525 Mdot .6175 .47384 Mdot .61845 .21154 Mdot .6194 .51696 Mdot .62036 .10525 Mdot .62131 .47384 Mdot .62226 .21154 Mdot .62321 .51696 Mdot .62416 .10525 Mdot .62511 .47384 Mdot .62606 .21154 Mdot .62702 .51696 Mdot .62797 .10525 Mdot .62892 .47384 Mdot .62987 .21154 Mdot .63082 .51696 Mdot .63177 .10525 Mdot .63272 .47384 Mdot .63368 .21154 Mdot .63463 .51696 Mdot .63558 .10525 Mdot .63653 .47384 Mdot .63748 .21154 Mdot .63843 .51696 Mdot .63938 .10525 Mdot .64034 .47384 Mdot .64129 .21154 Mdot .64224 .51696 Mdot .64319 .10525 Mdot .64414 .47384 Mdot .64509 .21154 Mdot .64604 .51696 Mdot .647 .10525 Mdot .64795 .47384 Mdot .6489 .21154 Mdot .64985 .51696 Mdot .6508 .10525 Mdot .65175 .47384 Mdot .6527 .21154 Mdot .65366 .51696 Mdot .65461 .10525 Mdot .65556 .47384 Mdot .65651 .21154 Mdot .65746 .51696 Mdot .65841 .10525 Mdot .65936 .47384 Mdot .66032 .21154 Mdot .66127 .51696 Mdot .66222 .10525 Mdot .66317 .47384 Mdot .66412 .21154 Mdot .66507 .51696 Mdot .66602 .10525 Mdot .66698 .47384 Mdot .66793 .21154 Mdot .66888 .51696 Mdot .66983 .10525 Mdot .67078 .47384 Mdot .67173 .21154 Mdot .67268 .51696 Mdot .67364 .10525 Mdot .67459 .47384 Mdot .67554 .21154 Mdot .67649 .51696 Mdot .67744 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The NestList \ second arg is the inital value of the recursive variable, and the third arg \ is the number of recursive iterations. Notice how NestList can be used with \ multivariate functions, provided the variable to be treated recursively is \ listed last." }], "Text", TextAlignment->Left, TextJustification->0], Cell[BoxData[{ \(\(Clear[g1];\)\), "\[IndentingNewLine]", \(\(g1[z_]\ := \ N[z^2];\)\), "\[IndentingNewLine]", \(NestList[g1, \ 3, \ 4]\), "\[IndentingNewLine]", \(\(Clear[g2];\)\), "\[IndentingNewLine]", \(\(\(\(g2[x_]\)[y_]\)[z_]\ := \ x*y*z;\)\), "\[IndentingNewLine]", \(NestList[\(g2[b]\)[c], \ a, \ \ 5]\)}], "Input"], Cell[TextData[StyleBox["14: Sensitivity analysis with list differences", \ "Subsection"]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "The code in the cell below offers a technique on how to compare two \ alternative histories. ", StyleBox["Apply this technique to the Logistic map. Suggested steps: 1) \ Define a non-recursive version of the pop[] function above, with all of the \ appropriate parameters. 2) Use NestList to populate two lists, that are each \ seeded with nearly identical starting populations. 3) Plot the history \ difference. Compare results of the history differences in the chaotic and \ non-chaotic regimes.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text", TextAlignment->Left, TextJustification->0], Cell[BoxData[{ \(\(Clear[browny];\)\), "\[IndentingNewLine]", \(\(browny[x_]\ := \ x\ + \ Random[Real, \ {0, 1}]\ - 0.5\ ;\)\), "\[IndentingNewLine]", \(\(hist1\ = \ NestList[browny, \ 0, \ 10];\)\), "\[IndentingNewLine]", \(\(hist2\ = \ NestList[browny, \ 0, \ 10];\)\), "\[IndentingNewLine]", \(ListPlot[hist2 - hist1, \ PlotJoined\ \[Rule] True]\)}], "Input"], Cell[TextData[StyleBox["The function fg below is the stripped down logistic \ map, where the carrying capacity is normalized to 1. Here, if we start with \ initial conditions far enough apart, in this case 0.5 and 0.75 each history \ will be at a different step in the period 4 limit cycle. 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