{"id":126,"date":"2019-06-11T18:26:21","date_gmt":"2019-06-11T22:26:21","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=126"},"modified":"2019-06-28T09:51:05","modified_gmt":"2019-06-28T13:51:05","slug":"dynamical-systems-strogatz-chapter-3","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/11\/dynamical-systems-strogatz-chapter-3\/","title":{"rendered":"Dynamical Systems: Strogatz Chapter 3"},"content":{"rendered":"<p>&nbsp;<\/p>\n<ul>\n<li>Section 3.0: Introduction\n<ul>\n<li>I need to help students distinguish between parameters and variables.<\/li>\n<li>The beam bending example is ok, but the intuition isn&#8217;t so clear. \u00a0If I back up on the load does the beam straighten (is this a supercritical pitchfork)?<\/li>\n<li>It would be nice to introduce an intuitive example of a saddle-node bifurcation.<\/li>\n<\/ul>\n<\/li>\n<li>Section 3.1: Saddle-node bifurcation\n<ul>\n<li>Example 3.1.1 is algebraic, which is great. \u00a0Example 3.1.2 is geometric, which is also great. \u00a0For each of those examples, adding procedural instructions for creating an associated bifurcation diagram would be helpful.<\/li>\n<li>The explanation of normal forms, including the Taylor expansion and Figure 3.1.7 (where f(x) has a local minimum and is being shifted up and down) is clear in the text. \u00a0It comes across less clearly in Steve&#8217;s youtube videos.<\/li>\n<li>Figure 3.1.4 is the only example of a saddle-node bifurcation diagram. \u00a0Because it is associated with the normal form it might give the impression that the bifurcation happens at (0,0) and that saddle-node bifurcations always require a perfect parabola. \u00a0Adding more examples of bifurcation diagrams would be helpful.<\/li>\n<\/ul>\n<\/li>\n<li>Section 3.2: Transcritical bifurcation\n<ul>\n<li>Instructions \/ examples of making the bifurcation diagram are also missing here.<\/li>\n<li>A &#8220;bifurcation curve&#8221; in parameter space is an idea that is introduced here. \u00a0The idea of parameter space can be confusing (we have phase space, parameter space, and the mixed space that is used for bifurcation diagrams). \u00a0It is probably worth introducing this example and the idea of a stability diagram explicitly at this point.<\/li>\n<\/ul>\n<\/li>\n<li>Section 3.3: Laser threshold\n<ul>\n<li>I skip this section. \u00a0I should find a different application example of a transcritical bifurcation to replace it with.<\/li>\n<\/ul>\n<\/li>\n<li>Section 3.4: Pitchfork bifurcation\n<ul>\n<li>In the text, the idea of an equation being invariant under a change of coordinates is introduced. \u00a0We should do the coordinate replacement in class&#8230; (x to -x). \u00a0It would be worth building more intuition around the idea of &#8220;symmetry&#8221;<\/li>\n<li>I should assign 2.4.9 on critical slowing down. \u00a0I think I assign a modified version that isn&#8217;t very helpful. \u00a0The idea that solutions decay more slowly than exponential is hard to convey.<\/li>\n<li>The geometry associated with a pitchfork bifurcation doesn&#8217;t come across very well (that f(x) = g(x) &#8211; h(x) and the bifurcation happens when g(x) and h(x) become tangent).<\/li>\n<li>There are instructions for plotting a bifurcation diagram here. \u00a0The trick to find r in terms of x but then plot in rx-space is introduced.<\/li>\n<li>Steve&#8217;s youtube videos skip the subcritical pitchfork. \u00a0I have a video introducing it, but it could probably be better.<\/li>\n<li>The idea of slowly varying a parameter is introduced here. \u00a0When varying a parameter, the fixed point will change. \u00a0That the fixed point is chained to the parameter is a point of confusion for some students (they sometimes think of the parameter value responding to the fixed point value). \u00a0Explicitly defining hysteresis and talking about jumps would be a good thing to do here.<\/li>\n<\/ul>\n<\/li>\n<li>Section 3.5: Overdamped bead on a rotating hoop\n<ul>\n<li>I skip this section and replace it with an introduction to dimension and nondimensionalization. \u00a0Understanding this example requires being able to follow a nondimensionalization argument, as well the discussion about neglecting a term.<\/li>\n<\/ul>\n<\/li>\n<li>Section 3.6: Imperfect bifurcations and catastrophes\n<ul>\n<li>Some years I have explicitly introduced the imperfect bifurcation content. \u00a0Other times I have skipped it.<\/li>\n<li>Providing a reminder of the definition of a bifurcation curve and the distinction between parameter space and other coordinate planes that we use is important here.<\/li>\n<li>The definition of a cusp point and the term codimension-2 bifurcation both appear here.<\/li>\n<li>Stability diagrams, parameter space, etc are all introduced better in the text than they are in Steve&#8217;s youtube videos.<\/li>\n<li>The catastrophe example of a bead on a tilted wire seems fun to play with: could we work with it in simulation\/animation or do I need a physical version?<\/li>\n<\/ul>\n<\/li>\n<li>Section 3.7: Insect outbreak\n<ul>\n<li>Separation of time scales comes up here and is worth emphasizing.<\/li>\n<li>This is a nice modeling example.<\/li>\n<li>Making sure to present the terms cusp, bifurcation curve, stability diagram, and catastrophe in advance of this example might help make it easier to follow.<\/li>\n<li>Building more intuition for thinking about bifurcations via the intersection of two curves, f(x) = g(x) &#8211; h(x), could also help.<\/li>\n<li>The idea of bistability comes up in section 3.4 but is introduced here.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp; Section 3.0: Introduction I need to help students distinguish between parameters and variables. The beam bending example is ok, but the intuition isn&#8217;t so clear. \u00a0If I back up on the load does the beam straighten (is this a supercritical pitchfork)? It would be nice to introduce an intuitive example of a saddle-node bifurcation. [&hellip;]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[157888,1010],"tags":[],"class_list":["post-126","post","type-post","status-publish","format-standard","hentry","category-dynamical-systems","category-math"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-22","jetpack-related-posts":[{"id":132,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-5\/","url_meta":{"origin":126,"position":0},"title":"Dynamical Systems: Strogatz Chapter 5","author":"siams","date":"12 June 2019","format":false,"excerpt":"This chapter is mainly review of topics from prerequisite courses. \u00a0Steve does introduce the (Delta, tau)-plane for classifying fixed points of linear systems. \u00a0This chapter is a return to linear systems. There isn't a \"summary\" section in between Chapter 4 and Chapter 5. \u00a0That is probably a worthwhile spot to\u2026","rel":"","context":"In &quot;Dynamical Systems&quot;","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":130,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-4\/","url_meta":{"origin":126,"position":1},"title":"Dynamical Systems: Strogatz Chapter 4","author":"siams","date":"12 June 2019","format":false,"excerpt":"This chapter is not included in Steve's youtube videos. Section 4.0: Introduction The connection between putting the vector field on a circle and oscillation is not obvious. \u00a0Showing time series x(t) or y(t) for a uniform oscillator might help (the time series figures in the text have to do with\u2026","rel":"","context":"In &quot;Dynamical Systems&quot;","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":222,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2020\/05\/23\/python-in-my-dynamical-systems-class\/","url_meta":{"origin":126,"position":2},"title":"Python in my dynamical systems class","author":"siams","date":"23 May 2020","format":false,"excerpt":"I have been using Mathematica in my dynamical systems class for a few years. I don't have a systematic curriculum related to it, though, and need to develop clearer computational learning goals, as well as a pathway for students to develop computational skills. Ideally, by the end of the semester,\u2026","rel":"","context":"In &quot;Dynamical Systems&quot;","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":170,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/10\/meiss-differential-dynamical-systems-chaos\/","url_meta":{"origin":126,"position":3},"title":"Meiss: Differential Dynamical Systems (chaos)","author":"siams","date":"10 July 2019","format":false,"excerpt":"I am reading James Meiss' text Differential Dynamical Systems (SIAM). \u00a0I am specifically interested in how he tells the story of chaos. In the Preface, he mentions the following: That\u00a0\u00a0Chapter 5 focuses on invariant manifolds: stable and unstable sets heteroclinic orbits stable manifolds local stable manifold theorem global stable manifolds\u2026","rel":"","context":"In &quot;Dynamical Systems&quot;","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":118,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-strogatz-chapter-2\/","url_meta":{"origin":126,"position":4},"title":"Dynamical Systems: Strogatz Chapter 2","author":"siams","date":"10 June 2019","format":false,"excerpt":"Following this text, students study 1d, then 2d, then 3d flows. \u00a0In 1d, we find stability, construct phase portraits, and in chapter 3, make bifurcation diagrams. \u00a0We loop back to these topics with more complexity in 2d. \u00a0This creates natural \"spacing\". A few notes on spacing: Spacing improves induction\/generalization from\u2026","rel":"","context":"In &quot;Dynamical Systems&quot;","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":126,"position":5},"title":"Dynamical Systems: Math 21b differential equations background","author":"siams","date":"10 June 2019","format":false,"excerpt":"For students who have taken Math 1b, AM\/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in Math 1b + 9 classes in 21b). \u00a0See my prior post for the Math 1b diff eq content that is relevant to Dynamical Systems. Student diff eq background\u2026","rel":"","context":"In &quot;Dynamical Systems&quot;","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/126","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/users\/8032"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/comments?post=126"}],"version-history":[{"count":3,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/126\/revisions"}],"predecessor-version":[{"id":129,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/126\/revisions\/129"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}