(************** 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). 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Consider a series of points:\n\n\t\t\t\t\t", Cell[BoxData[ \(TraditionalForm\`x\_\(n + 1\)\)], FontSize->16], StyleBox["= \[Alpha] ", FontSize->16], Cell[BoxData[ \(TraditionalForm\`x\_n\)], FontSize->16], StyleBox["(1-", FontSize->16], Cell[BoxData[ \(TraditionalForm\`x\_n\)], FontSize->16], StyleBox[")\n", FontSize->16], "\nThis is a completely deterministic equation. If we know x at \"time\" ", Cell[BoxData[ \(TraditionalForm\`t\_n\)]], ", ", Cell[BoxData[ FormBox[ FormBox[\(x\_n\), "TraditionalForm"], TraditionalForm]]], ", then we can find x at the next time ", Cell[BoxData[ \(TraditionalForm\`t\_\(n + 1\)\)]], ".\n\nBefore proceeding, discuss what you believe will happen. What is the \ effect of changing ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], "? Does it matter what value is chosen for \[Alpha]?" }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ In order to explore this equation, we can define a set of initial values, \ give \[Alpha] a value, compute the series for some finite number of terms, \ and look at it. Your assignment is to use the commands below, modified in any \ way you wish, to explore the logistic map. 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