(*********************************************************************** 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). 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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[ 146639, 4604]*) (*NotebookOutlinePosition[ 147389, 4630]*) (* CellTagsIndexPosition[ 147345, 4626]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(Mathematica\ Project\ on\ Wever' s\ Differential\ Equation\ Model\ for\ Sleep/Wake\ Cycles\)], "Input", TextAlignment->Center], Cell[TextData[ "Let A(t)= Activity of the person at time t. Then the second order model \ proposed by Wever based on the Van der Pol Oscillator reads:\n\nA\"(t)+ \ \[Epsilon](A^2-a)A'+A+gA^2=Bcos(t+\[Gamma])+Csin(t+\[Gamma])\n\nwhere Bcos(t+\ \[Gamma])+Csin(t+\[Gamma]) is considered as the external influence of light. \ Whenever A(t) is less than zero the person is considered to be asleep and \ whenever A(t) is greater than zero the person is considered to be awake.\n\n\ Change the second order Wever model to a system of two first order equations \ by setting \nx(t) = A(t) and y(t) = A'(t). The numerical simulations using \ NDSolve have been set up for you. You will need to vary different parameters \ within the NDSolve command and write up your solutions.\n\n"], "Text", TextAlignment->Left], Cell[TextData[StyleBox["The Uncontrolled System", FontWeight->"Bold"]], "Text", TextAlignment->Center], Cell["\<\ Set B = C + 0. This means there is no external influence of light. 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