(************** 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[ 948508, 25109]*) (*NotebookOutlinePosition[ 949279, 25134]*) (* CellTagsIndexPosition[ 949235, 25130]*) (*WindowFrame->Normal*) Notebook[{ Cell["Lorenz Equations", "Title"], Cell[TextData[{ "This notebook has been written in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by \n\n", StyleBox["Mark J. McCready\nProfessor and Chair of Chemical and \ Biomolecular Engineering\nUniversity of Notre Dame\nNotre Dame IN 46556\nUSA", FontSize->14], "\n\n", ButtonBox["Mark.J.McCready.1@nd.edu", ButtonData:>{ URL[ "mailto:mccready.1@nd.edu"], None}, ButtonStyle->"Hyperlink"], "\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy that this notice remain visible to other users. \nThere is no charge \ for copying and dissemination \n\nVersion: 02/04/04\n " }], "Text"], Cell["\<\ The Lorenz equations are a set of 3 coupled nonlinear equations \ that are a (major) simplification of natural convection in the atmosphere. \ Lorenz derived them and solved them in 1963, to study formation of weather \ patterns. The problem would have had little interest or impact, except that \ the solutions displayed interesting irregular behavior. Chaos from a simple \ model! Here are the equations x' \[Equal]-\[Sigma] (x-y), y' \[Equal] rr x - y - x z, z' \[Equal] - b z + x y, x is the velocity amplitude in the most simple convection mode y is the temperature difference between rising and falling fluid z is the departure from the linear vertical temperature profile x' = - \[Sigma] x is the viscous damping y' = -y and z' = -b z account for thermal conduction damping \[Sigma] y is the convection forcing term. Nonlinear terms correspond to heat transport by the flow. \[Sigma] is the \ Prandtl and rr is the Rayleigh number.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\[Sigma] = 10;\)\), "\[IndentingNewLine]", \(\(rr = 28;\)\), "\[IndentingNewLine]", \(\(b = \ 8/3;\)\), "\[IndentingNewLine]", \(\(tfinal = 80;\)\)}], "Input", CellLabel->"In[1]:="], Cell[CellGroupData[{ Cell[BoxData[ \(ans = NDSolve[{\(x'\)[ t]\ \[Equal] \(-\[Sigma]\)\ \((x[t] - y[t])\), \ \[IndentingNewLine]\t\ \ \ \ \ \ \ \ \(y'\)[ t]\ \[Equal] \ rr\ x[t]\ - \ y[t]\ - \ x[t]\ \ z[ t], \ \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(z'\)[t]\ \[Equal] \ x[t]\ y[t]\ - \ b\ z[t], x[0] \[Equal] 20, y[0] \[Equal] 5, \ z[0]\ \[Equal] \(-5\)}, \ {x[t], \ y[t], \ z[t]}, \[IndentingNewLine]{t, 0, tfinal}, MaxSteps \[Rule] 100000]\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{\(x[t]\), "\[Rule]", RowBox[{ TagBox[\(InterpolatingFunction[{{0.`, 80.`}}, "<>"]\), 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