{"id":132,"date":"2019-06-12T18:33:51","date_gmt":"2019-06-12T22:33:51","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=132"},"modified":"2019-06-28T09:51:29","modified_gmt":"2019-06-28T13:51:29","slug":"dynamical-systems-strogatz-chapter-5","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-5\/","title":{"rendered":"Dynamical Systems: Strogatz Chapter 5"},"content":{"rendered":"<p>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.<\/p>\n<p>There isn&#8217;t a &#8220;summary&#8221; section in between Chapter 4 and Chapter 5. \u00a0That is probably a worthwhile spot to review the differences between the behaviors we see in linear systems vs in nonlinear systems in 1d. \u00a0It isn&#8217;t noted in chapter 4, but I suppose the only possible linear system for flow on the circle is actually the constant velocity one (because of the periodicity requirement for the vector field). \u00a0What are the phenomena that we can encode via a nonlinear model that we can&#8217;t get to via a linear model?<\/p>\n<ul>\n<li>Section 5.0: Introduction\n<ul>\n<li>We are back to linear systems, now in 2d. \u00a0There is a bit more to this than there was in 1d. \u00a0We&#8217;ll use this info to classify fixed points in nonlinear systems. \u00a0Is it possible to build that out a bit more in 1d? \u00a0(To think of classification in 1d as seeing which type of linear fixed point our fixed point is similar to?) \u00a0I wonder if building out an analogy there would make the distinction between linear and nonlinear systems more intuitive.<\/li>\n<\/ul>\n<\/li>\n<li>Section 5.1: Definitions and examples\n<ul>\n<li>Students probably need to be reminded how to translate a higher order linear differential equation into a linear system (see Example 5.1.1).<\/li>\n<li>Distinguishing hyperbolic from non-hyperbolic fixed points could be helpful again here.<\/li>\n<li>Some students confuse &#8220;saddle point&#8221; and &#8220;saddle-node&#8221;.<\/li>\n<\/ul>\n<\/li>\n<li>Section 5.2: Classification of linear systems\n<ul>\n<li>The role of eigenvalues and eigenvectors in solutions is something some students remember and others find confusing. \u00a0Same with complex eigenvalues.<\/li>\n<li>Classifying fixed points both using the (Delta, tau)-plane, and providing the eigenvalue classification information in the (Real, Imaginary)-plane would be a good idea because one can memorize the Delta-tau plane without remembering where it came from.<\/li>\n<\/ul>\n<\/li>\n<li>Section 5.3: Love affairs\n<ul>\n<li>This example actually gets used a lot (so may have been seen in other courses). \u00a0For that reason, I skip it.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>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&#8217;t a &#8220;summary&#8221; section in between Chapter 4 and Chapter 5. \u00a0That is probably a worthwhile spot to review the differences between the [&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-132","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-28","jetpack-related-posts":[{"id":118,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-strogatz-chapter-2\/","url_meta":{"origin":132,"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":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":132,"position":1},"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":220,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/09\/18\/notes-on-ljung-system-identification\/","url_meta":{"origin":132,"position":2},"title":"Notes on Ljung: System Identification","author":"siams","date":"18 September 2019","format":false,"excerpt":"Reading Ljung. \u00a0System Identification: theory for the user. 1: Introduction. Goal: infer a model from observations. \u00a0\"Model\" refers to the set of relationships between variables in the system. \u00a0System identification involves analyzing input and output signals from the system. Example: assume a linear difference equation relates inputs to outputs. \u00a0Use\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"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":132,"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":132,"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":188,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-2\/","url_meta":{"origin":132,"position":5},"title":"Notes on &#8220;Calculus Blue&#8221; Volume 1, Chapter 2","author":"siams","date":"22 July 2019","format":false,"excerpt":"More notes on the Calculus Blue Multivariable Volume 1 videos on YouTube by \"Prof Ghrist Math\". Chapter 2 introduces curves in the plane and surfaces in 3-space with implicit and parametric definitions for curves in the plane and for surfaces in 3-space. \u00a0They also introduce the names and images for\u2026","rel":"","context":"In &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/132","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=132"}],"version-history":[{"count":1,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/132\/revisions"}],"predecessor-version":[{"id":133,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/132\/revisions\/133"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=132"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=132"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}