Module 13: Difference Equations

Lectures and Related Content

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:

  • 13A: Introduction to One-Dimensional Maps
  • 13B: Fixed Points
  • 13C: Example
  • 13D: Cobwebs
  • 13E: Logistic Map
  • 13F: Logistic Map Analysis
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Module at a Glance

Cell description

In this module we take our concepts for analysis of dynamical systems where time is continuous and apply them to a class of dynamical system where time is discrete.   

Objectives 

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

  • Develop tools to determine fixed points and evaluate stability.
  • Analyze the discrete-time logistic map using analogous concepts from the study of the continuous time logistic equation.
  • Motivate a more complete understanding of difference methods using Newton’s Method.

How to complete this module:

  1. Watch the video lectures 13A through 13F.
  2. Complete the reading assignments.
  3. Respond to the discussion board question.
  4. Complete the assignment.
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Readings

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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 10 (pp. 355-368)
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Module 13 Assignment

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  • Complete the following problems from chapter 10 of the Strogatz textbook: 10.1.6 (4 pts.); 10.1.11 (10 pts.); 10.2.4 (6 pts.), 10.2.6 (6 pts.); 10.2.8 (6 pts.); 10.3.1 (3 pts.); 10.3.3 (6 pts.)
  • Upload your work in PDF format by selecting the link above.

Due: Tuesday 11:59 pm

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Module 13 Discussion

Cell description

Read this book chapter and broaden your understanding of difference equations and where they can serve us.

  • Discuss other potential applications and the viability of difference equations.
  • How do the techniques we have learned this semester help with these? 

Due: Sunday 11:59 pm

Book chapter credit

Langtangen, H. P. (2016) Mathematical Models Based on Difference Equations. A Primer on Scientific Programming with Python, 5th edition: Springer. Retrieved from http://hplgit.github.io/primer.html/doc/pub/diffeq/._diffeq-readable001.html

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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=\"_7048835_1\">\n\n <h3 id=\"anonymous_element_12\">\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=_7048835_1&amp;mode=view\"><span style=\"color:#0067ac;\">Module 13 Discussion</span></a>\n      </h3><a href=\"#\" class=\"ally-alternative-formats ally-prominent-af-download-button\" data-ally-external-id=\"content:_7048835_1\" data-ally-upload-type=\"RichContent\" title=\"Alternative formats\" style=\"display: none\">\n    <span class=\"ally-sr-only\">\n  \n\n  \n\n            Module 13 Discussion\n      </span>\n</a>\n\n   \n </div>\n\n\n\n<div class=\"details\">\n    <div class=\"vtbegenerated\"> \n <p>Read this <a href=\"http://hplgit.github.io/primer.html/doc/pub/diffeq/._diffeq-readable001.html\" title=\"Book chapter: Mathematical Models Based on Difference Equations\" target=\"_blank\">book chapter</a> and broaden your understanding of difference equations and where they can serve us.</p> \n <ul> \n  <li>Discuss other potential applications and the viability of difference equations.</li> \n  <li>How do the techniques we have learned this semester help with these?&nbsp;</li> \n </ul> \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><strong>Book chapter credit</strong></p> \n <p><strong></strong>Langtangen, H. P. (2016) <em>Mathematical Models Based on Difference Equations.</em> A Primer on Scientific Programming with Python, 5th edition: Springer. Retrieved from http://hplgit.github.io/primer.html/doc/pub/diffeq/._diffeq-readable001.html</p> \n</div>\n  </div>\n\n <div id=\"_7048835_1previewModule\" class=\"moduleSample\">\n </div>\n",
        "url": "https://blackboard.jhu.edu/webapps/blackboard/content/launchLink.jsp?course_id=_194578_1&content_id=_7048835_1&mode=view",
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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=\"_7048835_1\">\n\n <h3 id=\"anonymous_element_12\">\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=_7048835_1&amp;mode=view\"><span style=\"color:#0067ac;\">Module 13 Discussion</span></a>\n      </h3><a href=\"#\" class=\"ally-alternative-formats ally-prominent-af-download-button\" data-ally-external-id=\"content:_7048835_1\" data-ally-upload-type=\"RichContent\" title=\"Alternative formats\" style=\"display: none\">\n    <span class=\"ally-sr-only\">\n  \n\n  \n\n            Module 13 Discussion\n      </span>\n</a>\n\n   \n </div>\n\n\n\n<div class=\"details\">\n    <div class=\"vtbegenerated\"> \n <p>Read this <a href=\"http://hplgit.github.io/primer.html/doc/pub/diffeq/._diffeq-readable001.html\" title=\"Book chapter: Mathematical Models Based on Difference Equations\" target=\"_blank\">book chapter</a> and broaden your understanding of difference equations and where they can serve us.</p> \n <ul> \n  <li>Discuss other potential applications and the viability of difference equations.</li> \n  <li>How do the techniques we have learned this semester help with these?&nbsp;</li> \n </ul> \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><strong>Book chapter credit</strong></p> \n <p><strong></strong>Langtangen, H. P. (2016) <em>Mathematical Models Based on Difference Equations.</em> A Primer on Scientific Programming with Python, 5th edition: Springer. Retrieved from http://hplgit.github.io/primer.html/doc/pub/diffeq/._diffeq-readable001.html</p> \n</div>\n  </div>\n\n <div id=\"_7048835_1previewModule\" class=\"moduleSample\">\n </div>\n",
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        "body": "\n    <div> \n <p>Read this <a href=\"http://hplgit.github.io/primer.html/doc/pub/diffeq/._diffeq-readable001.html\" title=\"Book chapter: Mathematical Models Based on Difference Equations\" target=\"_blank\">book chapter</a> and broaden your understanding of difference equations and where they can serve us.</p> \n <ul> \n  <li>Discuss other potential applications and the viability of difference equations.</li> \n  <li>How do the techniques we have learned this semester help with these?&nbsp;</li> \n </ul> \n  \n <div> \n   \n </div> \n <p><strong>Due: Sunday 11:59 pm</strong></p> \n  \n <p><strong>Book chapter credit</strong></p> \n <p>Langtangen, H. P. (2016) <em>Mathematical Models Based on Difference Equations.</em> A Primer on Scientific Programming with Python, 5th edition: Springer. Retrieved from http://hplgit.github.io/primer.html/doc/pub/diffeq/._diffeq-readable001.html</p> \n</div>\n  "
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