Module 11: Two Dimensional Bifurcations (cont.)

Folder description

Select the folder link above to access the video lectures and related content for this week. This week’s materials include the following:

  • 11A: Saddle Node Bifurcation of Cycles
  • 11B: Infinite Period Bifurcation
  • 11C: Homoclinic Bifurcation
  • 11D: Poincaré Maps
  • 11E: Poincaré Map: Stability Consideration
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Module at a Glance

Cell description

In this module we'll extend an understanding of bifurcations to two dimensional systems; observe changes in fixed points or limit cycles as parameters are varied.

Objectives 

By the end of this module, you will be able to:

  • Analyze global bifurcation beyond the Hopf bifurcation.
  • Utilize a Poincaré Map to convert dynamics around a closed orbit to dynamics about a fixed point.
  • Investigate the logistic equation with periodic harvesting/carrying capacity.

How to Complete this Module:

  1. Watch the video lectures 11A through 11D.
  2. Complete the reading assignments.
  3. Respond to the discussion board question.
  4. Complete the assignment.
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Readings

Cell description

This week's reading assignment includes the following:

  • Strogatz, Steven HNonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and EngineeringCRC Press, 2018. 
    • Chapter 8 (pp. 264-287)
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Module 11 Assignment

Cell description
  • Complete the following problems from chapter 8 of the Strogatz textbook: 8.4.2 (8 pts.); 8.4.3 (8 pts.); 8.7.1 (6 pts.); 8.7.2 (6 pts.)
  • Upload your work in PDF format by selecting the link above.

Due: Tuesday 11:59 pm

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Module 11 Discussion

Cell description

Watch this video about coupled oscillators; ignore this specific application – just focus on the oscillator and the coupling. Work through Section 8.6 in the text.

  • Discuss the video in the terms of our study.
  • Describe if this is periodic or quasiperiodic? What is the difference?
  • How does, or would, phase lock present itself to us given this physical manifestation of a coupled oscillator?
Name: Three-way Coupled Pendulum & Neutrinos What the Physics (05:05)
Duration: 05:05
Added: 21 Aug 2019 09:05 AM
Added By: DIANE BANNER
Description:  
Tags: stein, 625.718

Due: Sunday 11:59 pm

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        "html": "\n\n<img alt=\"Course Link\" src=\"/images/ci/sets/set01/discussion_on.gif\" class=\"item_icon\">\n\n<div class=\"item clearfix\" id=\"_7048815_1\">\n\n <h3>\n  <span class=\"reorder editmode hideme\"><span><img src=\"/images/ci/icons/generic_updown.gif\" alt=\"\"></span></span>\n\n  <img title=\"linked item\" alt=\"linked item\" src=\"/images/ci/icons/file_link_ti.png\" class=\"item_icon\">\n\n            <a href=\"/webapps/blackboard/content/launchLink.jsp?course_id=_194578_1&amp;content_id=_7048815_1&amp;mode=view\"><span style=\"color:#0067ac;\">Module 11 Discussion</span></a>\n      </h3><a href=\"#\" class=\"ally-alternative-formats ally-prominent-af-download-button\" data-ally-external-id=\"content:_7048815_1\" data-ally-upload-type=\"RichContent\" title=\"Alternative formats\" style=\"display: none\">\n    <span class=\"ally-sr-only\">\n  \n\n  \n\n            Module 11 Discussion\n      </span>\n</a>\n\n   \n </div>\n\n\n\n<div class=\"details\">\n    <div class=\"vtbegenerated\"> \n <p>Watch this video about coupled oscillators; ignore this specific application – just focus on the oscillator and the coupling. Work through Section 8.6 in the text.</p> \n <ul> \n  <li>Discuss the video in the terms of our study.</li> \n  <li>Describe if this is periodic or quasiperiodic? What is the difference?</li> \n  <li>How does, or would, phase lock present itself to us given this physical manifestation of a coupled oscillator?</li> \n </ul> \n <div style=\"width: 604px; overflow: hidden;\"> \n  <div style=\"float: left; clear: left;\"> \n   <div style=\"float: left; clear: both;\"> \n    <iframe width=\"320\" height=\"240\" style=\"width: 604px; height: 386px;\" frameborder=\"0\" scrolling=\"no\" allowfullscreen=\"true\" allow=\"autoplay *; fullscreen *; encrypted-media *; camera *; microphone *;\" src=\"/webapps/osv-kaltura-bb_bb60/content/iframe.jsp?url=..%2FLtiMashupPlayIframeWrapperResponsive%3FplayUrl%3D%2Fbrowseandembed%2Findex%2Fmedia%2Fentryid%2F1_t8t5u84g%2FshowDescription%2Ffalse%2FshowTitle%2Ffalse%2FshowTags%2Ffalse%2FshowDuration%2Ffalse%2FshowOwner%2Ffalse%2FshowUploadDate%2Ffalse%2FplayerSize%2F608x402%2FplayerSkin%2F30749451%2F&amp;thumbnail=https://cdnsecakmi.kaltura.com/p/1458241/sp/145824100/thumbnail/entry_id/1_t8t5u84g/version/100001&amp;title=Three-way+Coupled+Pendulum+%26+Neutrinos++What+the+Physics+%2805%3A05%29&amp;course_id=_194578_1&amp;content_id=_7048815_1\"></iframe> \n   </div> \n  </div> \n  <div id=\"container_0.21136320961032184\" class=\"hide-icon hidden\" style=\"margin-top: 30px; float: left; clear: left;\"> \n   <div style=\"float: left; clear: left; margin-right: 8px;\"> \n    <div style=\"float: left; clear: both;\"> \n     <a class=\"lb\" style=\"text-decoration: none; color: #333333;\" href=\"../LtiMashupPlayIframeWrapperResponsive?playUrl=/browseandembed/index/media/entryid/1_t8t5u84g/showDescription/false/showTitle/false/showTags/false/showDuration/false/showOwner/false/showUploadDate/false/playerSize/608x402/playerSkin/30749451/&amp;course_id=_194578_1\" target=\"_blank\" title=\"Three-way Coupled Pendulum &amp; Neutrinos  What the Physics (05:05)\"> <img border=\"0\" src=\"https://cdnsecakmi.kaltura.com/p/1458241/sp/145824100/thumbnail/entry_id/1_t8t5u84g/version/100001\"> </a> \n    </div> \n    <div style=\"float: left; clear: both; width: 100%; background: url('/images/ci/ng/rumble_button_back.gif') 50% 50%; padding-top: 2px; padding-bottom: 2px; text-decoration: none; color: #333333; text-align: center;\"> \n     <a class=\"lb\" style=\"text-decoration: none; color: #333333;\" href=\"../LtiMashupPlayIframeWrapperResponsive?playUrl=/browseandembed/index/media/entryid/1_t8t5u84g/showDescription/false/showTitle/false/showTags/false/showDuration/false/showOwner/false/showUploadDate/false/playerSize/608x402/playerSkin/30749451/&amp;course_id=_194578_1\" target=\"_blank\" title=\"Three-way Coupled Pendulum &amp; Neutrinos  What the Physics (05:05)\"> Watch Media </a> \n    </div> \n   </div> \n   <div style=\"float: left; clear: right;\"> \n    <table style=\"margin: 0px !important; border-collapse: collapse;\"> \n     <tbody> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\"><b>Name:</b></td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\"><b>Three-way Coupled Pendulum &amp; Neutrinos What the Physics (05:05)</b></td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Duration:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">05:05</td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Added:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">21 Aug 2019 09:05 AM</td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Added By:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">DIANE BANNER</td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Description:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">&nbsp;</td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Tags:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">stein, 625.718</td> \n      </tr> \n     </tbody> \n    </table> \n   </div> \n  </div> \n </div> \n <p><br></p> \n <div style=\"font-family: 'Open Sans', sans-serif; text-shadow: none !important; letter-spacing: normal !important;\"> \n  <strong style=\"font-family: inherit; text-shadow: none !important; letter-spacing: normal !important; margin: 0px; padding: 0px; border: 0px; outline: 0px; font-weight: bold; font-style: inherit; font-size: 13px;\"><span data-contrast=\"auto\" xml:lang=\"EN-US\" style=\"font-family: inherit; text-shadow: none !important; letter-spacing: normal !important; margin: 0px; padding: 0px; border: 0px; outline: 0px; font-weight: inherit; font-style: inherit; font-size: 13px;\"></span></strong> \n </div> \n <p><strong style=\"font-family: 'Open Sans', sans-serif; text-shadow: none !important; letter-spacing: normal; margin: 0px; padding: 0px; border: 0px; outline: 0px; font-weight: bold; font-style: normal; font-size: 13px; color: #000000; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-style: initial; text-decoration-color: initial;\">Due: Sunday 11:59 pm</strong></p> \n <p style=\"font-family: inherit; text-shadow: none !important; letter-spacing: normal !important; margin: 0px 0px 1em; padding: 0px; border: 0px; outline: 0px; font-weight: inherit; font-style: inherit; font-size: 13px;\"></p> \n <p><br></p> \n</div>\n  </div>\n\n <div id=\"_7048815_1previewModule\" class=\"moduleSample\">\n </div>\n",
        "url": "https://blackboard.jhu.edu/webapps/blackboard/content/launchLink.jsp?course_id=_194578_1&content_id=_7048815_1&mode=view",
        "title": "Module 11 Discussion"
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    "html": "\n\n<img alt=\"Course Link\" src=\"/images/ci/sets/set01/discussion_on.gif\" class=\"item_icon\">\n\n<div class=\"item clearfix\" id=\"_7048815_1\">\n\n <h3>\n  <span class=\"reorder editmode hideme\"><span><img src=\"/images/ci/icons/generic_updown.gif\" alt=\"\"></span></span>\n\n  <img title=\"linked item\" alt=\"linked item\" src=\"/images/ci/icons/file_link_ti.png\" class=\"item_icon\">\n\n            <a href=\"/webapps/blackboard/content/launchLink.jsp?course_id=_194578_1&amp;content_id=_7048815_1&amp;mode=view\"><span style=\"color:#0067ac;\">Module 11 Discussion</span></a>\n      </h3><a href=\"#\" class=\"ally-alternative-formats ally-prominent-af-download-button\" data-ally-external-id=\"content:_7048815_1\" data-ally-upload-type=\"RichContent\" title=\"Alternative formats\" style=\"display: none\">\n    <span class=\"ally-sr-only\">\n  \n\n  \n\n            Module 11 Discussion\n      </span>\n</a>\n\n   \n </div>\n\n\n\n<div class=\"details\">\n    <div class=\"vtbegenerated\"> \n <p>Watch this video about coupled oscillators; ignore this specific application – just focus on the oscillator and the coupling. Work through Section 8.6 in the text.</p> \n <ul> \n  <li>Discuss the video in the terms of our study.</li> \n  <li>Describe if this is periodic or quasiperiodic? What is the difference?</li> \n  <li>How does, or would, phase lock present itself to us given this physical manifestation of a coupled oscillator?</li> \n </ul> \n <div style=\"width: 604px; overflow: hidden;\"> \n  <div style=\"float: left; clear: left;\"> \n   <div style=\"float: left; clear: both;\"> \n    <iframe width=\"320\" height=\"240\" style=\"width: 604px; height: 386px;\" frameborder=\"0\" scrolling=\"no\" allowfullscreen=\"true\" allow=\"autoplay *; fullscreen *; encrypted-media *; camera *; microphone *;\" src=\"/webapps/osv-kaltura-bb_bb60/content/iframe.jsp?url=..%2FLtiMashupPlayIframeWrapperResponsive%3FplayUrl%3D%2Fbrowseandembed%2Findex%2Fmedia%2Fentryid%2F1_t8t5u84g%2FshowDescription%2Ffalse%2FshowTitle%2Ffalse%2FshowTags%2Ffalse%2FshowDuration%2Ffalse%2FshowOwner%2Ffalse%2FshowUploadDate%2Ffalse%2FplayerSize%2F608x402%2FplayerSkin%2F30749451%2F&amp;thumbnail=https://cdnsecakmi.kaltura.com/p/1458241/sp/145824100/thumbnail/entry_id/1_t8t5u84g/version/100001&amp;title=Three-way+Coupled+Pendulum+%26+Neutrinos++What+the+Physics+%2805%3A05%29&amp;course_id=_194578_1&amp;content_id=_7048815_1\"></iframe> \n   </div> \n  </div> \n  <div id=\"container_0.21136320961032184\" class=\"hide-icon hidden\" style=\"margin-top: 30px; float: left; clear: left;\"> \n   <div style=\"float: left; clear: left; margin-right: 8px;\"> \n    <div style=\"float: left; clear: both;\"> \n     <a class=\"lb\" style=\"text-decoration: none; color: #333333;\" href=\"../LtiMashupPlayIframeWrapperResponsive?playUrl=/browseandembed/index/media/entryid/1_t8t5u84g/showDescription/false/showTitle/false/showTags/false/showDuration/false/showOwner/false/showUploadDate/false/playerSize/608x402/playerSkin/30749451/&amp;course_id=_194578_1\" target=\"_blank\" title=\"Three-way Coupled Pendulum &amp; Neutrinos  What the Physics (05:05)\"> <img border=\"0\" src=\"https://cdnsecakmi.kaltura.com/p/1458241/sp/145824100/thumbnail/entry_id/1_t8t5u84g/version/100001\"> </a> \n    </div> \n    <div style=\"float: left; clear: both; width: 100%; background: url('/images/ci/ng/rumble_button_back.gif') 50% 50%; padding-top: 2px; padding-bottom: 2px; text-decoration: none; color: #333333; text-align: center;\"> \n     <a class=\"lb\" style=\"text-decoration: none; color: #333333;\" href=\"../LtiMashupPlayIframeWrapperResponsive?playUrl=/browseandembed/index/media/entryid/1_t8t5u84g/showDescription/false/showTitle/false/showTags/false/showDuration/false/showOwner/false/showUploadDate/false/playerSize/608x402/playerSkin/30749451/&amp;course_id=_194578_1\" target=\"_blank\" title=\"Three-way Coupled Pendulum &amp; Neutrinos  What the Physics (05:05)\"> Watch Media </a> \n    </div> \n   </div> \n   <div style=\"float: left; clear: right;\"> \n    <table style=\"margin: 0px !important; border-collapse: collapse;\"> \n     <tbody> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\"><b>Name:</b></td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\"><b>Three-way Coupled Pendulum &amp; Neutrinos What the Physics (05:05)</b></td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Duration:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">05:05</td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Added:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">21 Aug 2019 09:05 AM</td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Added By:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">DIANE BANNER</td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Description:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">&nbsp;</td> \n      </tr> \n      <tr style=\"vertical-align: top;\"> \n       <td style=\"padding: 0px !important;\">Tags:</td> \n       <td style=\"padding: 0px 0px 0px 20px !important;\">stein, 625.718</td> \n      </tr> \n     </tbody> \n    </table> \n   </div> \n  </div> \n </div> \n <p><br></p> \n <div style=\"font-family: 'Open Sans', sans-serif; text-shadow: none !important; letter-spacing: normal !important;\"> \n  <strong style=\"font-family: inherit; text-shadow: none !important; letter-spacing: normal !important; margin: 0px; padding: 0px; border: 0px; outline: 0px; font-weight: bold; font-style: inherit; font-size: 13px;\"><span data-contrast=\"auto\" xml:lang=\"EN-US\" style=\"font-family: inherit; text-shadow: none !important; letter-spacing: normal !important; margin: 0px; padding: 0px; border: 0px; outline: 0px; font-weight: inherit; font-style: inherit; font-size: 13px;\"></span></strong> \n </div> \n <p><strong style=\"font-family: 'Open Sans', sans-serif; text-shadow: none !important; letter-spacing: normal; margin: 0px; padding: 0px; border: 0px; outline: 0px; font-weight: bold; font-style: normal; font-size: 13px; color: #000000; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-style: initial; text-decoration-color: initial;\">Due: Sunday 11:59 pm</strong></p> \n <p style=\"font-family: inherit; text-shadow: none !important; letter-spacing: normal !important; margin: 0px 0px 1em; padding: 0px; border: 0px; outline: 0px; font-weight: inherit; font-style: inherit; font-size: 13px;\"></p> \n <p><br></p> \n</div>\n  </div>\n\n <div id=\"_7048815_1previewModule\" class=\"moduleSample\">\n </div>\n",
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        "title": "Module 11 Discussion",
        "body": "\n    <div> \n <p>Watch this video about coupled oscillators; ignore this specific application – just focus on the oscillator and the coupling. Work through Section 8.6 in the text.</p> \n <ul> \n  <li>Discuss the video in the terms of our study.</li> \n  <li>Describe if this is periodic or quasiperiodic? What is the difference?</li> \n  <li>How does, or would, phase lock present itself to us given this physical manifestation of a coupled oscillator?</li> \n </ul> \n <div> \n  <div> \n   <div> \n     \n   </div> \n  </div> \n  <div id=\"container_0.21136320961032184\"> \n   <div> \n    <div> \n     <a href=\"../LtiMashupPlayIframeWrapperResponsive?playUrl=/browseandembed/index/media/entryid/1_t8t5u84g/showDescription/false/showTitle/false/showTags/false/showDuration/false/showOwner/false/showUploadDate/false/playerSize/608x402/playerSkin/30749451/&amp;course_id=_194578_1\" target=\"_blank\" title=\"Three-way Coupled Pendulum &amp; Neutrinos  What the Physics (05:05)\">  </a> \n    </div> \n    <div> \n     <a href=\"../LtiMashupPlayIframeWrapperResponsive?playUrl=/browseandembed/index/media/entryid/1_t8t5u84g/showDescription/false/showTitle/false/showTags/false/showDuration/false/showOwner/false/showUploadDate/false/playerSize/608x402/playerSkin/30749451/&amp;course_id=_194578_1\" target=\"_blank\" title=\"Three-way Coupled Pendulum &amp; Neutrinos  What the Physics (05:05)\"> Watch Media </a> \n    </div> \n   </div> \n   <div> \n    <table> \n     <tbody> \n      <tr> \n       <td><b>Name:</b></td> \n       <td><b>Three-way Coupled Pendulum &amp; Neutrinos What the Physics (05:05)</b></td> \n      </tr> \n      <tr> \n       <td>Duration:</td> \n       <td>05:05</td> \n      </tr> \n      <tr> \n       <td>Added:</td> \n       <td>21 Aug 2019 09:05 AM</td> \n      </tr> \n      <tr> \n       <td>Added By:</td> \n       <td>DIANE BANNER</td> \n      </tr> \n      <tr> \n       <td>Description:</td> \n       <td>&nbsp;</td> \n      </tr> \n      <tr> \n       <td>Tags:</td> \n       <td>stein, 625.718</td> \n      </tr> \n     </tbody> \n    </table> \n   </div> \n  </div> \n </div> \n  \n <div> \n   \n </div> \n <p><strong>Due: Sunday 11:59 pm</strong></p> \n  \n  \n</div>\n  "
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