ÿþ<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=Content-Type content="text/html; charset=unicode"> <meta name=ProgId content=Word.Document> <meta name=Generator content="Microsoft Word 12"> <meta name=Originator content="Microsoft Word 12"> <link rel=File-List href="Chap8_files/filelist.xml"> <link rel=Edit-Time-Data href="Chap8_files/editdata.mso"> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} </style> <![endif]--> <title>Chap8.html</title> <!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Author>Stephen Lynch</o:Author> <o:Template>Normal</o:Template> <o:LastAuthor>Stephen Lynch</o:LastAuthor> <o:Revision>2</o:Revision> <o:TotalTime>0</o:TotalTime> <o:Created>2010-06-16T11:54:00Z</o:Created> <o:LastSaved>2010-06-16T11:54:00Z</o:LastSaved> <o:Pages>5</o:Pages> <o:Words>1004</o:Words> <o:Characters>5728</o:Characters> <o:Company>MMU</o:Company> <o:Lines>47</o:Lines> <o:Paragraphs>13</o:Paragraphs> <o:CharactersWithSpaces>6719</o:CharactersWithSpaces> <o:Version>12.00</o:Version> </o:DocumentProperties> </xml><![endif]--> <link rel=themeData href="Chap8_files/themedata.thmx"> <link rel=colorSchemeMapping href="Chap8_files/colorschememapping.xml"> <!--[if gte mso 9]><xml> <w:WordDocument> <w:Zoom>200</w:Zoom> <w:TrackMoves>false</w:TrackMoves> <w:TrackFormatting/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-GB</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="&#45;-"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true" DefSemiHidden="true" DefQFormat="false" DefPriority="99" LatentStyleCount="267"> <w:LsdException Locked="false" Priority="0" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Normal"/> <w:LsdException Locked="false" Priority="9" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="heading 1"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 2"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 3"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 4"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 5"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 6"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 7"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 8"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 9"/> <w:LsdException Locked="false" Priority="39" Name="toc 1"/> <w:LsdException Locked="false" Priority="39" Name="toc 2"/> <w:LsdException Locked="false" Priority="39" Name="toc 3"/> <w:LsdException Locked="false" Priority="39" Name="toc 4"/> <w:LsdException Locked="false" Priority="39" Name="toc 5"/> <w:LsdException Locked="false" Priority="39" Name="toc 6"/> <w:LsdException Locked="false" Priority="39" Name="toc 7"/> <w:LsdException Locked="false" Priority="39" Name="toc 8"/> <w:LsdException Locked="false" Priority="39" Name="toc 9"/> <w:LsdException Locked="false" Priority="35" QFormat="true" Name="caption"/> <w:LsdException Locked="false" Priority="10" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Title"/> <w:LsdException Locked="false" Priority="1" Name="Default Paragraph Font"/> <w:LsdException Locked="false" Priority="11" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtitle"/> <w:LsdException Locked="false" Priority="22" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Strong"/> <w:LsdException Locked="false" Priority="20" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Emphasis"/> <w:LsdException Locked="false" Priority="59" SemiHidden="false" UnhideWhenUsed="false" Name="Table Grid"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Placeholder Text"/> <w:LsdException Locked="false" Priority="1" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="No Spacing"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 1"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 1"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 1"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 1"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 1"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Revision"/> <w:LsdException Locked="false" Priority="34" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="List Paragraph"/> <w:LsdException Locked="false" Priority="29" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Quote"/> <w:LsdException Locked="false" Priority="30" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Quote"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 1"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 1"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 1"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 1"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 1"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 1"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 1"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 2"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 2"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 2"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 2"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 2"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 2"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 2"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 2"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 2"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 2"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 2"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 3"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 3"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 3"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 3"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 3"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 3"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 3"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 3"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 3"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 3"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 3"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 3"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 3"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 4"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 4"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 4"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 4"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 4"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 4"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 4"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 4"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 4"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 4"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 4"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 4"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 4"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 4"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 5"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 5"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 5"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 5"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 5"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 5"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 5"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 5"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 5"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 5"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 5"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 5"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 5"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 5"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 6"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 6"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 6"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 6"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 6"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 6"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 6"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 6"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 6"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 6"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/> <w:LsdException Locked="false" Priority="19" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/> <w:LsdException Locked="false" Priority="21" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/> <w:LsdException Locked="false" Priority="31" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/> <w:LsdException Locked="false" Priority="32" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/> <w:LsdException Locked="false" Priority="33" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Book Title"/> <w:LsdException Locked="false" Priority="37" Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> </w:LatentStyles> </xml><![endif]--> <style> <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:1; mso-generic-font-family:roman; mso-font-format:other; mso-font-pitch:variable; mso-font-signature:0 0 0 0 0 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;} p {mso-style-noshow:yes; mso-style-priority:99; mso-margin-top-alt:auto; margin-right:0cm; mso-margin-bottom-alt:auto; margin-left:0cm; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt;} @page WordSection1 {size:595.3pt 841.9pt; margin:72.0pt 72.0pt 72.0pt 72.0pt; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} --> </style> <!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--><!-- Created by Maple 10, Windows XP --><!--[if gte mso 9]><xml> <o:shapedefaults v:ext="edit" spidmax="2050"/> </xml><![endif]--><!--[if gte mso 9]><xml> <o:shapelayout v:ext="edit"> <o:idmap v:ext="edit" data="1"/> </o:shapelayout></xml><![endif]--> </head> <a href="Chap8.html#MapleAutoBookmark1">Chap8.mw</a> <ul> </ul> <hr> <a name="MapleAutoBookmark1" /> <body bgcolor=white lang=EN-GB style='tab-interval:36.0pt'> <div class=WordSection1> <p><strong><span style='font-size:18.0pt;color:black'>Maple Text Commands for Chapter 8 - </span></strong>&nbsp;</p> <p><strong><span style='font-size:18.0pt;color:black'>POINCARE MAPS AND NONAUTONOMOUS SYSTEMS IN THE PLANE</span></strong>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>with(DEtools):with(plots):</span></strong><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p><span style='font-size:10.0pt;font-family:"Courier New";color:blue'>Warning, the name changecoords has been redefined</span>&nbsp;</p> <p><strong><span style='color:black'>Example 1: </span></strong><span style='color:black'>Poincare first returns for the ODE </span><span style='color:black;mso-no-proof:yes'><img width=132 height=48 id="_x0000_i1062" src="images/Chap8_1.gif" alt="dr/dt = -r^2, d*theta/dt = 1."></span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>sys:=diff(r(t),t)+r(t)^2: </span></strong><b><span style='font-family:"Courier New";mso-fareast-font-family:"Times New Roman"; color:red'><br> <strong><span style='font-family:"Courier New"'>returns:=dsolve({sys,r(0)=1},numeric,range=0..16*Pi): </span></strong><br> <strong><span style='font-family:"Courier New"'>evalf(seq(r[i]=returns(2*i*Pi),i=1..8),5);</span></strong></span></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=576 height=30 id="_x0000_i1061" src="images/Chap8_2.gif" alt="r[1] = [t = 6.2832, r(t) = .13730], r[2] = [t = 12.566, r(t) = 0.73712e-1], r[3] = [t = 18.850, r(t) = 0.50379e-1], r[4] = [t = 25.133, r(t) = 0.38266e-1], r[5] = [t = 31.416, r(t) = 0.30849e-1], r[6]..."></span><span style='color:blue'><br> </span><span style='color:blue;mso-no-proof:yes'><img width=576 height=30 id="_x0000_i1060" src="images/Chap8_3.gif" alt="r[1] = [t = 6.2832, r(t) = .13730], r[2] = [t = 12.566, r(t) = 0.73712e-1], r[3] = [t = 18.850, r(t) = 0.50379e-1], r[4] = [t = 25.133, r(t) = 0.38266e-1], r[5] = [t = 31.416, r(t) = 0.30849e-1], r[6]..."></span><span style='color:blue'><br> </span><span style='color:blue;mso-no-proof:yes'><img width=576 height=30 id="_x0000_i1059" src="images/Chap8_4.gif" alt="r[1] = [t = 6.2832, r(t) = .13730], r[2] = [t = 12.566, r(t) = 0.73712e-1], r[3] = [t = 18.850, r(t) = 0.50379e-1], r[4] = [t = 25.133, r(t) = 0.38266e-1], r[5] = [t = 31.416, r(t) = 0.30849e-1], r[6]..."></span><span style='color:blue'><br> </span><span style='color:blue;mso-no-proof:yes'><img width=576 height=30 id="_x0000_i1058" src="images/Chap8_5.gif" alt="r[1] = [t = 6.2832, r(t) = .13730], r[2] = [t = 12.566, r(t) = 0.73712e-1], r[3] = [t = 18.850, r(t) = 0.50379e-1], r[4] = [t = 25.133, r(t) = 0.38266e-1], r[5] = [t = 31.416, r(t) = 0.30849e-1], r[6]..."></span>&nbsp;</p> <p><strong><span style='color:black'>Example 5: </span></strong><span style='color:black'>Hamiltonian system with two-degrees of freedom </span><span style='color:black;mso-no-proof:yes'><img width=307 height=56 id="_x0000_i1057" src="images/Chap8_6.gif"></span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>omega1:=2:omega2:=2:H:=(omega1/2)*(p1^2+q1^2)+(omega2/2)*(p2^2+q2^2): </span></strong><b><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'><br> <strong><span style='font-family:"Courier New"'>hamilton_eqs(H); </span></strong></span></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=576 height=48 id="_x0000_i1056" src="images/Chap8_7.gif" alt="(Typesetting:-mprintslash)([[diff(p1(t), t) = -2*q1(t), diff(p2(t), t) = -2*q2(t), diff(q1(t), t) = 2*p1(t), diff(q2(t), t) = 2*p2(t)], [p1(t), p2(t), q1(t), q2(t)]], [[diff(p1(t), t) = -2*q1(t), diff..."></span><span style='color:blue'><br> </span><span style='color:blue;mso-no-proof:yes'><img width=576 height=48 id="_x0000_i1055" src="images/Chap8_8.gif" alt="(Typesetting:-mprintslash)([[diff(p1(t), t) = -2*q1(t), diff(p2(t), t) = -2*q2(t), diff(q1(t), t) = 2*p1(t), diff(q2(t), t) = 2*p2(t)], [p1(t), p2(t), q1(t), q2(t)]], [[diff(p1(t), t) = -2*q1(t), diff..."></span>&nbsp;</p> <p><strong><span style='color:black'>Figures 8.5(a) and (b): </span></strong><span style='color:black'>Projections of the Poincare surface-of-section.</span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>poincare(H,t=-100..100,{[0,0.5,1.5,0.5,0]},stepsize=0.1,iterations=4, </span></strong><b><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'><br> <strong><span style='font-family:"Courier New"'>scene=[p1=-1.5..1.5,q1=-1.5..1.5,q2=-1.5..1.5],3);</span></strong></span></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=432 height=30 id="_x0000_i1054" src="images/Chap8_9.gif" alt="_____________________________________________________"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=471 height=30 id="_x0000_i1053" src="images/Chap8_10.gif" alt="`H = 2.75`, ` Initial conditions:`, t = 0, p1 = .5, p2 = 1.5, q1 = .5, q2 = 0"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=286 height=30 id="_x0000_i1052" src="images/Chap8_11.gif" alt="`Maximum H deviation : .8677000000e-4 %`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=432 height=30 id="_x0000_i1051" src="images/Chap8_12.gif" alt="_____________________________________________________"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=179 height=30 id="_x0000_i1050" src="images/Chap8_13.gif" alt="`Time consumed: 3 seconds`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:black;mso-no-proof: yes'><img width=398 height=398 id="_x0000_i1049" src="images/Chap8_14.gif" alt=Plot></span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>poincare(H,t=0..30,{[0,0.5,1.5,0.5,0]},stepsize=0.005,iterations=3,scene=[p1,q1]); </span></strong><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=432 height=30 id="_x0000_i1048" src="images/Chap8_15.gif" alt="_____________________________________________________"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=471 height=30 id="_x0000_i1047" src="images/Chap8_16.gif" alt="`H = 2.75`, ` Initial conditions:`, t = 0, p1 = .5, p2 = 1.5, q1 = .5, q2 = 0"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=356 height=30 id="_x0000_i1046" src="images/Chap8_17.gif" alt="`Number of points found crossing the (p1,q1) plane: 19`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=194 height=30 id="_x0000_i1045" src="images/Chap8_18.gif" alt="`Maximum H deviation : 0. %`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=432 height=30 id="_x0000_i1044" src="images/Chap8_19.gif" alt="_____________________________________________________"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=179 height=30 id="_x0000_i1043" src="images/Chap8_20.gif" alt="`Time consumed: 2 seconds`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:black;mso-no-proof: yes'><img width=398 height=398 id="_x0000_i1042" src="images/Chap8_21.gif" alt=Plot></span>&nbsp;</p> <p><strong><span style='color:black'>Example 6: </span></strong><span style='color:black'>The Henon-Heiles Hamiltonian </span><span style='color: black;mso-no-proof:yes'><img width=355 height=62 id="_x0000_i1041" src="images/Chap8_22.gif"></span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>H:=(p1^2+q1^2+p2^2+q2^2)/2+q1^2*q2-q2^3/3:</span></strong><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p><strong><span style='color:black'>Figures 8.6(a) and (b): </span></strong><span style='color:black'>Three-dimensional and two-dimensional surface-of-section.<strong> </strong></span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>poincare(H,t=-100..100,{[0,0.06,0.1,-0.2,-0.2]},stepsize=0.1,iterations=4, </span></strong><b><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'><br> <strong><span style='font-family:"Courier New"'>scene=[p1=-0.3..0.3,q1=-0.3..0.3,q2=-0.3..0.3],3); </span></strong></span></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=432 height=30 id="_x0000_i1040" src="images/Chap8_23.gif" alt="_____________________________________________________"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=567 height=30 id="_x0000_i1039" src="images/Chap8_24.gif" alt="`H = .41466667e-1`, ` Initial conditions:`, t = 0, p1 = 0.6e-1, p2 = .1, q1 = -.2, q2 = -.2"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=286 height=30 id="_x0000_i1038" src="images/Chap8_25.gif" alt="`Maximum H deviation : .1690000000e-5 %`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=432 height=30 id="_x0000_i1037" src="images/Chap8_26.gif" alt="_____________________________________________________"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=179 height=30 id="_x0000_i1036" src="images/Chap8_27.gif" alt="`Time consumed: 3 seconds`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:black;mso-no-proof: yes'><img width=398 height=398 id="_x0000_i1035" src="images/Chap8_28.gif" alt=Plot></span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>poincare(H,t=-100..100,{[0,0.06,0.1,-0.2,-0.2]},stepsize=0.1,iterations=4,3); </span></strong><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=432 height=30 id="_x0000_i1034" src="images/Chap8_29.gif" alt="_____________________________________________________"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=567 height=30 id="_x0000_i1033" src="images/Chap8_30.gif" alt="`H = .41466667e-1`, ` Initial conditions:`, t = 0, p1 = 0.6e-1, p2 = .1, q1 = -.2, q2 = -.2"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=286 height=30 id="_x0000_i1032" src="images/Chap8_31.gif" alt="`Maximum H deviation : .1690000000e-5 %`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=432 height=30 id="_x0000_i1031" src="images/Chap8_32.gif" alt="_____________________________________________________"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:blue;mso-no-proof: yes'><img width=179 height=30 id="_x0000_i1030" src="images/Chap8_33.gif" alt="`Time consumed: 3 seconds`"></span>&nbsp;</p> <p align=center style='text-align:center'><span style='color:black;mso-no-proof: yes'><img width=398 height=398 id="_x0000_i1029" src="images/Chap8_34.gif" alt=Plot></span>&nbsp;</p> <p><strong><span style='color:black'>Figure 8.11(b): </span></strong><span style='color:black'>Phase portrait for the Duffing equation </span><span style='color:black;mso-no-proof:yes'><img width=256 height=54 id="_x0000_i1028" src="images/Chap8_35.gif"></span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>Gamma:=0.5:omega:=1.25:k:=0.3: </span></strong><b><span style='font-family:"Courier New";mso-fareast-font-family:"Times New Roman"; color:red'><br> <strong><span style='font-family:"Courier New"'>DEplot([diff(x(t),t)=y(t),diff(y(t),t)=x(t)-k*y(t)-(x(t))^3+Gamma*cos(omega*t)],[x(t),y(t)],t=0..300,[[x(0)=1,y(0)=0.5]],x=-2..2,y=-2..2,stepsize=0.1,linecolor=blue,thickness=1);</span></strong></span></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:black;mso-no-proof: yes'><img width=398 height=398 id="_x0000_i1027" src="images/Chap8_36.gif" alt=Plot></span>&nbsp;</p> <p><strong><span style='color:black'>Figure 8.11(b): </span></strong><span style='color:black'>Poincare first returns for the Duffing system.</span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>ff:=dsolve({diff(x(t),t)=y(t),diff(y(t),t)=x(t)-k*y(t)-(x(t))^3+Gamma* </span></strong><b><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'><br> <strong><span style='font-family:"Courier New"'>cos(omega*t),x(0)=1,y(0)=0.5},{x(t),y(t)}, </span></strong><br> <strong><span style='font-family:"Courier New"'>type=numeric,method=classical,output=procedurelist,maxfun=0): </span></strong><br> <strong><span style='font-family:"Courier New"'>pt:=array(0..10000):x1:=array(0..10000):y1:=array(0..10000): </span></strong><br> <strong><span style='font-family:"Courier New"'>imax:=4000: </span></strong><br> <strong><span style='font-family:"Courier New"'>for i from 0 to imax do </span></strong><br> <strong><span style='font-family:"Courier New"'>x1[i]:=eval(x(t),ff(i*2*Pi/omega)): </span></strong><br> <strong><span style='font-family:"Courier New"'>y1[i]:=eval(y(t),ff(i*2*Pi/omega)): </span></strong><br> <strong><span style='font-family:"Courier New"'>end do: </span></strong><br> <strong><span style='font-family:"Courier New"'>pts:=[[x1[n],y1[n]]$n=10..imax]: </span></strong><br> <strong><span style='font-family:"Courier New"'># Plot the points on the Poincare section # </span></strong><br> <strong><span style='font-family:"Courier New"'>pointplot(pts,style=point,symbol=circle,symbolsize=1,color=blue,axes=BOXED,scaling=CONSTRAINED,font=[TIMES,ROMAN,15]);</span></strong></span></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:black;mso-no-proof: yes'><img width=398 height=398 id="_x0000_i1026" src="images/Chap8_37.gif" alt=Plot></span>&nbsp;</p> <p><strong><span style='color:black'>Figure 8.14: </span></strong><span style='color:black'>Bifurcation diagram for the Duffing system.</span>&nbsp;</p> <table class=MsoNormalTable border=0 cellpadding=0 style='mso-cellspacing:1.5pt; mso-yfti-tbllook:1184'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>&gt; </span></strong><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> </td> <td valign=top style='padding:.75pt .75pt .75pt .75pt'> <p class=MsoNormal><strong><span style='font-family:"Courier New";mso-fareast-font-family: "Times New Roman";color:red'>G:=array(0..10000):Y:=array(0..10000): </span></strong><b><span style='font-family:"Courier New";mso-fareast-font-family:"Times New Roman"; color:red'><br> <strong><span style='font-family:"Courier New"'>x1:=array(0..10000):y1:=array(0..10000):r:=array(0..10000): </span></strong><br> <strong><span style='font-family:"Courier New"'># Increase Gamma from 0 to 0.45 # </span></strong><br> <strong><span style='font-family:"Courier New"'>jmax:=1000:k:=0.3:omega:=1.25:step:=0.00045:interval:=jmax*step: </span></strong><br> <strong><span style='font-family:"Courier New"'>x1[0]:=1:y1[0]:=0:r[0]:=1: </span></strong><br> <strong><span style='font-family:"Courier New"'>for j from 0 to jmax do </span></strong><br> <strong><span style='font-family:"Courier New"'>G[j]:=step*j: </span></strong><br> <strong><span style='font-family:"Courier New"'>ff := </span></strong><br> <strong><span style='font-family:"Courier New"'>dsolve({diff(x(t),t)=y(t),diff(y(t),t)=-k*y(t)+x(t)-(x(t))^3+G[j]*cos( </span></strong><br> <strong><span style='font-family:"Courier New"'>omega*t),x(0)=x1[j],y(0)=y1[j]},{x(t),y(t)}, type=numeric, </span></strong><br> <strong><span style='font-family:"Courier New"'>output=procedurelist); </span></strong><br> <strong><span style='font-family:"Courier New"'>x1[j+1]:=eval(x(t),ff(2*Pi/omega)): </span></strong><br> <strong><span style='font-family:"Courier New"'>y1[j+1]:=eval(y(t),ff(2*Pi/omega)): </span></strong><br> <strong><span style='font-family:"Courier New"'>r[j+1]:=sqrt((x1[j+1])^2+(y1[j+1])^2): </span></strong><br> <strong><span style='font-family:"Courier New"'>end do: </span></strong><br> <strong><span style='font-family:"Courier New"'>l :=[[G[n],r[n]] $n=0..jmax]: </span></strong><br> <strong><span style='font-family:"Courier New"'>P1:=plot(l, x=0..interval,y=0..2, </span></strong><br> <strong><span style='font-family:"Courier New"'>style=point,symbol=circle,symbolsize=1,color=blue): </span></strong><br> <strong><span style='font-family:"Courier New"'># Decrease Gamma from 0.45 to 0 # </span></strong><br> <strong><span style='font-family:"Courier New"'>Gb:=array(0..10000): </span></strong><br> <strong><span style='font-family:"Courier New"'>xb:=array(0..10000):yb:=array(0..10000):rb:=array(0..10000): </span></strong><br> <strong><span style='font-family:"Courier New"'>xb[0]:=x1[jmax+1]:yb[0]:=y1[jmax+1]:rb[0]:=sqrt((xb[0])^2+(yb[0])^2): </span></strong><br> <strong><span style='font-family:"Courier New"'>for j from 0 to jmax do </span></strong><br> <strong><span style='font-family:"Courier New"'>Gb[j]:=interval-step*j: </span></strong><br> <strong><span style='font-family:"Courier New"'>ff := </span></strong><br> <strong><span style='font-family:"Courier New"'>dsolve({diff(x(t),t)=-y(t),diff(y(t),t)=-(-k*y(t)+x(t)-(x(t))^3+Gb[j]* </span></strong><br> <strong><span style='font-family:"Courier New"'>cos(omega*t)),x(0)=xb[j],y(0)=yb[j]},{x(t),y(t)}, type=numeric, </span></strong><br> <strong><span style='font-family:"Courier New"'>output=procedurelist); </span></strong><br> <strong><span style='font-family:"Courier New"'>xb[j+1]:=eval(x(t),ff(-2*Pi/omega)): </span></strong><br> <strong><span style='font-family:"Courier New"'>yb[j+1]:=eval(y(t),ff(-2*Pi/omega)): </span></strong><br> <strong><span style='font-family:"Courier New"'>rb[j+1]:=sqrt((xb[j+1])^2+(yb[j+1])^2): </span></strong><br> <strong><span style='font-family:"Courier New"'>end do: </span></strong><br> <strong><span style='font-family:"Courier New"'>l := [[Gb[n], rb[n]] $n=0..jmax]: </span></strong><br> <strong><span style='font-family:"Courier New"'>P2:=plot(l,x=0..interval,y=0..2, </span></strong><br> <strong><span style='font-family:"Courier New"'>style=point,symbol=circle,symbolsize=1,color=blue): </span></strong><br> <strong><span style='font-family:"Courier New"'>display({P1,P2},labels=['Gamma','r']);</span></strong></span></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> </td> </tr> </table> <p>&nbsp;</p> <p align=center style='text-align:center'><span style='color:black;mso-no-proof: yes'><img width=398 height=398 id="_x0000_i1025" src="images/Chap8_38.gif" alt=Plot></span>&nbsp;</p> <p><span style='font-size:13.5pt;color:black'>End of Chapter 8 Commands</span>&nbsp;</p> </div> </body> </html>