Module 2: Bifurcations of the Dynamics of Vector Fields on a Line

Lectures and Related Content

Folder description

Please click the folder link above to access the Module 2 instructional content. Module 2 contains the following:

Lecture 2A: Saddle Node Bifurcations
Lecture 2B: Saddle Node Bifurcations, An Example
Lecture 2C: Transcritical Bifurcations
Lecture 2D: Transcritical Bifurcations, An Example
Lecture 2E: Supercritical Pitchfork Bifurcations
Lecture 2F: Subcritical Pitchfork Bifurcations (cont.)

Technical

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Module at a Glance ObjectivesHow to complete this module:

Cell description

Understand how the flow of a one-dimensional dynamical system is dependent on the parameters. Observe dynamics as parameters change.

Objectives

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

Identify a saddle point bifurcation and understand the dynamics associated with this type of fixed point.
Identify a  transcritical  bifurcation and understand the dynamics associated with this type of fixed point.
Identify pitchfork bifurcations and understand the dynamics associated with this type of fixed point.
Examine  th e real-world application of damped beads on a hoop.

How to complete this module:

Watch the video lectures 2A through 2F.
Complete the reading assignment.
Participate in the discussion assignment.
Complete the problem assignment.

Technical

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Readings

Cell description

This week's reading assignment includes the following:

Strogatz, Steven H. Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, 2018.
- Chapter 3 (pp. 45-61)

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Module 2 Discussion - Bifurcation

Cell description

Select the link above to access the discussion forum. After viewing the posted 15 minute video respond to the following prompt. In addition, comment on at least two of your classmates’ posts.

The video details a bead moving on a wire. Explain what bifurcation means in the context of this problem. Be as detailed or as conceptual as you feel necessary to communicate your answer. Can you think of any other examples of bifurcation?

Due: Sunday 11:59 pm

Technical

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Module 2 Assignment

Cell description

Complete the following problems from chapter 3 of the Strogatz textbook: 3.1.1 (5 pts.); 3.2.1 (5 pts.); 3.4.1 (5 pts.); 3.4.3 (5 pts.); 3.4.5 (5 pts.), 3.4.6 (5 pts.); 3.4.7 (5 pts.); 3.4.8 (5 pts.)
Upload your work in PDF format by selecting the link above.

Due: Tuesday 11:59 pm

Technical

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        "html": "\n\n<img alt=\"Assignment\" src=\"/images/ci/sets/set01/assignment_on.gif\" class=\"item_icon\">\n\n<div class=\"item clearfix\" id=\"_7048715_1\">\n\n <h3>\n  <span class=\"reorder editmode hideme\"><span><img src=\"/images/ci/icons/generic_updown.gif\" alt=\"\"></span></span>\n\n\n            <a href=\"/webapps/assignment/uploadAssignment?content_id=_7048715_1&amp;course_id=_194578_1&amp;group_id=&amp;mode=view\"><span style=\"color:#0067ac;\">Module 2 Assignment</span></a>\n      </h3><a href=\"#\" class=\"ally-alternative-formats ally-prominent-af-download-button\" data-ally-external-id=\"content:_7048715_1\" data-ally-upload-type=\"RichContent\" title=\"Alternative formats\" style=\"display: none\">\n    <span class=\"ally-sr-only\">\n  \n\n\n            Module 2 Assignment\n      </span>\n</a>\n\n   \n </div>\n\n\n\n<div class=\"details\">\n    <div class=\"vtbegenerated\"> \n <ul> \n  <li>Complete the following problems from chapter 3 of the Strogatz textbook:&nbsp;<strong>3.1.1</strong> (5 pts.);&nbsp;<strong>3.2.1</strong> (5 pts.);&nbsp;<strong>3.4.1</strong> (5 pts.); <strong>3.4.3</strong> (5 pts.);&nbsp;<strong>3.4.5</strong> (5 pts.), <strong>3.4.6</strong> (5 pts.);&nbsp;<strong>3.4.7</strong> (5 pts.); <strong>3.4.8</strong> (5 pts.)</li> \n  <li>Upload your work in PDF format by selecting the link above.</li> \n </ul> \n <p style=\"margin-left: 30px;\"></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: Tuesday 11:59&nbsp;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 style=\"margin-left: 30px;\"></p> \n</div>\n  </div>\n\n <div id=\"_7048715_1previewModule\" class=\"moduleSample\">\n </div>\n",
        "url": "https://blackboard.jhu.edu/webapps/assignment/uploadAssignment?content_id=_7048715_1&course_id=_194578_1&group_id=&mode=view",
        "title": "Module 2 Assignment"
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    "html": "\n\n<img alt=\"Assignment\" src=\"/images/ci/sets/set01/assignment_on.gif\" class=\"item_icon\">\n\n<div class=\"item clearfix\" id=\"_7048715_1\">\n\n <h3>\n  <span class=\"reorder editmode hideme\"><span><img src=\"/images/ci/icons/generic_updown.gif\" alt=\"\"></span></span>\n\n\n            <a href=\"/webapps/assignment/uploadAssignment?content_id=_7048715_1&amp;course_id=_194578_1&amp;group_id=&amp;mode=view\"><span style=\"color:#0067ac;\">Module 2 Assignment</span></a>\n      </h3><a href=\"#\" class=\"ally-alternative-formats ally-prominent-af-download-button\" data-ally-external-id=\"content:_7048715_1\" data-ally-upload-type=\"RichContent\" title=\"Alternative formats\" style=\"display: none\">\n    <span class=\"ally-sr-only\">\n  \n\n\n            Module 2 Assignment\n      </span>\n</a>\n\n   \n </div>\n\n\n\n<div class=\"details\">\n    <div class=\"vtbegenerated\"> \n <ul> \n  <li>Complete the following problems from chapter 3 of the Strogatz textbook:&nbsp;<strong>3.1.1</strong> (5 pts.);&nbsp;<strong>3.2.1</strong> (5 pts.);&nbsp;<strong>3.4.1</strong> (5 pts.); <strong>3.4.3</strong> (5 pts.);&nbsp;<strong>3.4.5</strong> (5 pts.), <strong>3.4.6</strong> (5 pts.);&nbsp;<strong>3.4.7</strong> (5 pts.); <strong>3.4.8</strong> (5 pts.)</li> \n  <li>Upload your work in PDF format by selecting the link above.</li> \n </ul> \n <p style=\"margin-left: 30px;\"></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: Tuesday 11:59&nbsp;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 style=\"margin-left: 30px;\"></p> \n</div>\n  </div>\n\n <div id=\"_7048715_1previewModule\" class=\"moduleSample\">\n </div>\n",
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        "title": "Module 2 Assignment",
        "body": "\n    <div> \n <ul> \n  <li>Complete the following problems from chapter 3 of the Strogatz textbook:&nbsp;<strong>3.1.1</strong> (5 pts.);&nbsp;<strong>3.2.1</strong> (5 pts.);&nbsp;<strong>3.4.1</strong> (5 pts.); <strong>3.4.3</strong> (5 pts.);&nbsp;<strong>3.4.5</strong> (5 pts.), <strong>3.4.6</strong> (5 pts.);&nbsp;<strong>3.4.7</strong> (5 pts.); <strong>3.4.8</strong> (5 pts.)</li> \n  <li>Upload your work in PDF format by selecting the link above.</li> \n </ul> \n  \n <div> \n   \n </div> \n <p><strong>Due: Tuesday 11:59&nbsp;pm</strong></p> \n  \n  \n</div>\n  "
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