{"id":130,"date":"2019-06-12T18:22:02","date_gmt":"2019-06-12T22:22:02","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=130"},"modified":"2019-06-28T09:51:19","modified_gmt":"2019-06-28T13:51:19","slug":"dynamical-systems-strogatz-chapter-4","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-4\/","title":{"rendered":"Dynamical Systems: Strogatz Chapter 4"},"content":{"rendered":"<p>This chapter is not included in Steve&#8217;s youtube videos.<\/p>\n<ul>\n<li>Section 4.0: Introduction\n<ul>\n<li>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 bottlenecks).<\/li>\n<\/ul>\n<\/li>\n<li>Section 4.1: Examples and definitions\n<ul>\n<li>The mechanics of checking whether a vector field is well-defined on the circle are fine. \u00a0I think problems arise when this is approached via intuition instead of calculation.<\/li>\n<\/ul>\n<\/li>\n<li>Section 4.2: Uniform oscillator\n<ul>\n<li>The idea of a phase angle, and what it represents isn&#8217;t obvious.<\/li>\n<li>We are sometimes making a model of a phase angle and sometimes making a model of a phase difference. \u00a0These two contexts lead to different interpretations of the phase portrait, so being explicit about which one we&#8217;re talking about is important.<\/li>\n<\/ul>\n<\/li>\n<li>Section 4.3: Nonuniform oscillator\n<ul>\n<li>That nonuniform oscillators have a number of applications is stated. \u00a0Intuition for the functional form isn&#8217;t provided though.<\/li>\n<li>The calculation to find the period of oscillation, and how it changes as the bifurcation is approached, isn&#8217;t so easy to follow. \u00a0This type of period calculation comes back in the van der Pol.<\/li>\n<\/ul>\n<\/li>\n<li>Section 4.4: Overdamped pendulum\n<ul>\n<li>An overdamped pendulum isn&#8217;t so intuitive (because we tend to picture the swinging case), so I&#8217;m not sure about this example.<\/li>\n<\/ul>\n<\/li>\n<li>Section 4.5: Fireflies\n<ul>\n<li>Firefly phase locking is a fun example! \u00a0In this model, the fireflies reach a fixed phase difference (they phase-lock), but they don&#8217;t flash in unison. \u00a0That is a bit unintuitive: since they are not in unison, students sometimes struggle with the idea that they can reach a non-unison steady state (the math works out cleanly, but translating it to the model can be hard).<\/li>\n<li>It may be worth looking up the Ermentrout 1991 paper about a species that shifts its frequency.<\/li>\n<\/ul>\n<\/li>\n<li>Section 4.6: Superconducting Josephson junctions\n<ul>\n<li>I skip this section. \u00a0Other possible applications: jet lag, rat brain grid cells, ??<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>This chapter is not included in Steve&#8217;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 bottlenecks). Section 4.1: Examples and [&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_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-130","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-26","jetpack-related-posts":[{"id":118,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-strogatz-chapter-2\/","url_meta":{"origin":130,"position":0},"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":132,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-5\/","url_meta":{"origin":130,"position":1},"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":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":130,"position":2},"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":[]},{"id":170,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/10\/meiss-differential-dynamical-systems-chaos\/","url_meta":{"origin":130,"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":108,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-1b-differential-equations-background\/","url_meta":{"origin":130,"position":4},"title":"Dynamical systems: Math 1b differential equations background.","author":"siams","date":"10 June 2019","format":false,"excerpt":"I have been using the Strogatz textbook for teaching dynamical systems. \u00a0The course has multivariable calculus and linear algebra prerequisites. \u00a0Students might take the prerequisite courses different places. \u00a0For students who have taken Math 1b, AM\/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in\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":126,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/11\/dynamical-systems-strogatz-chapter-3\/","url_meta":{"origin":130,"position":5},"title":"Dynamical Systems: Strogatz Chapter 3","author":"siams","date":"11 June 2019","format":false,"excerpt":"\u00a0 Section 3.0: Introduction I need to help students distinguish between parameters and variables. The beam bending example is ok, but the intuition isn'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\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\/130","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=130"}],"version-history":[{"count":1,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/130\/revisions"}],"predecessor-version":[{"id":131,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/130\/revisions\/131"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}