{"id":118,"date":"2019-06-10T17:19:09","date_gmt":"2019-06-10T21:19:09","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=118"},"modified":"2019-06-11T17:48:43","modified_gmt":"2019-06-11T21:48:43","slug":"dynamical-systems-strogatz-chapter-2","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-strogatz-chapter-2\/","title":{"rendered":"Dynamical Systems: Strogatz Chapter 2"},"content":{"rendered":"<p>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 &#8220;spacing&#8221;.<\/p>\n<p>A few notes on spacing:<\/p>\n<p>Spacing improves induction\/generalization from examples (but learners perceive otherwise):\u00a0<a href=\"https:\/\/journals.sagepub.com\/doi\/abs\/10.1111\/j.1467-9280.2008.02127.x\">https:\/\/journals.sagepub.com\/doi\/abs\/10.1111\/j.1467-9280.2008.02127.x<\/a><\/p>\n<p>The gap between perception by a learner of what is effective for them, and the actual performance of learners, is reasonably well documented for retrieval practice. \u00a0I wasn&#8217;t aware that it also occurs with spacing.<br \/>\n&#8220;Across experiments, spacing was more effective than massing for 90% of the participants, yet after the first study session, 72% of the participants believed that massing had been more effective than spacing&#8221;. \u00a0<a href=\"https:\/\/doi.org\/10.1002\/acp.1537\">https:\/\/doi.org\/10.1002\/acp.1537<\/a>\u00a0(From Kornell. &#8220;Optimising learning using flashcards: Spacing is more effective than cramming&#8221;, Applied Cognitive Psychology 2009)<\/p>\n<p>It is also worth noting that spacing isn&#8217;t a panacea if info needs to be retained for a long time. \u00a0In a study where it was a long time between learning and testing (about a year), subjects in one study retained about 20% (even with spacing in their learning):<br \/>\n<a href=\"https:\/\/www-jstor-org.ezp-prod1.hul.harvard.edu\/stable\/pdf\/40064895.pdf?refreqid=excelsior%3A3cf8dd6aa66f4bf0ab7b6ba6428a14e0\">https:\/\/www-jstor-org.ezp-prod1.hul.harvard.edu\/stable\/pdf\/40064895.pdf?refreqid=excelsior%3A3cf8dd6aa66f4bf0ab7b6ba6428a14e0<\/a><\/p>\n<p>Back to my notes on chapter 2:<\/p>\n<ul>\n<li>Section 2.0: Introduction\n<ul>\n<li>I need to introduce Newton&#8217;s notation for differentiation with respect to time:\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Notation_for_differentiation#Newton's_notation\">https:\/\/en.wikipedia.org\/wiki\/Notation_for_differentiation#Newton&#8217;s_notation<\/a><\/li>\n<\/ul>\n<\/li>\n<li>Section 2.1: A geometric way of thinking\n<ul>\n<li>The idea of interpreting the differential equation via a vector field does not come through sufficiently. \u00a0I think more care in distinguishing the vector field itself from the phase portrait (phase line) would help.<\/li>\n<li>Sketching the qualitative shape of solutions occurs here. \u00a0I&#8217;ll call those plots &#8220;time series&#8221; plots, so that we have a way to refer to them.<\/li>\n<\/ul>\n<\/li>\n<li>Section 2.2: Fixed points and stability\n<ul>\n<li>When &#8220;phase portrait&#8221; is defined, the f(x) plot is present (but it doesn&#8217;t need to be: the phase portrait is just what is happening along the x-axis).<\/li>\n<li>Writing f(x) = g(x) &#8211; h(x) and comparing g(x) and h(x) to construct the phase portrait is a method introduced with a single example. \u00a0Writing out the procedural steps for the general procedure could be helpful.<\/li>\n<\/ul>\n<\/li>\n<li>Section 2.3: Population growth\n<ul>\n<li>Figure 2.3.1 doesn&#8217;t make it into the lecture videos and is great for explaining the logistic model (that we&#8217;re just choosing a simple way to have a carrying capacity).<\/li>\n<li>It would be nice to find some of the data for these population models.<\/li>\n<li>The &#8220;per capita&#8221; growth rate is something students have found confusing some semesters.<\/li>\n<\/ul>\n<\/li>\n<li>Section 2.4: Linear stability analysis\n<ul>\n<li>This section is the heavy lift in this chapter&#8230; \u00a0Adding a visual of the small perturbation may help. \u00a0I&#8217;m not sure how intuitive the idea of a small perturbation is&#8230;<\/li>\n<li>The big-O notation needs to be introduced. \u00a0When we talk about &#8220;higher order terms&#8221;, students need to be reminded about the meaning of the word &#8220;order&#8221; in this context. \u00a0In addition, that O(eta^2) encompasses all the higher order terms should be made explicit (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Big_O_notation#Infinitesimal_asymptotics\">https:\/\/en.wikipedia.org\/wiki\/Big_O_notation#Infinitesimal_asymptotics<\/a>).\n<ul>\n<li>It would also be helpful to define &#8220;is asymptotic to&#8221; in the context of two functions (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Asymptotic_analysis\">https:\/\/en.wikipedia.org\/wiki\/Asymptotic_analysis<\/a>)<\/li>\n<\/ul>\n<\/li>\n<li>The connection to exponential growth and decay is relying on students having the solution to x&#8217; = a x very accessible in their memories. \u00a0Without this being second nature, a lot of the intuition in this section is lost.<\/li>\n<li>Needed background is the definition of a separable diff eq and then how to solve it (this could be introduced in the population growth section).<\/li>\n<li>The f &#8216; (x*) = 0 case always leads to a number of questions. \u00a0I think introducing the terms &#8220;hyperbolic&#8221; and &#8220;nonhyperbolic&#8221; would actually help.<\/li>\n<li>Most of my students mix up f &#8216; (x) and d^2 x\/ dt^2. \u00a0I need to think about how to avoid that confusion&#8230;<\/li>\n<li>I&#8217;d like to emphasize that we can solve the linearized system. \u00a0We can even use it to get the timescale of decay and to sketch solutions near the equilibrium solution. \u00a0Is it worth using this to piece together time-series plots of trajectories? \u00a0The distinction between linear and nonlinear systems doesn&#8217;t currently come through very well.<\/li>\n<li>I&#8217;d like to compare the solution to the linearized system for small perturbations to numerical approximations via RK4 \/ Mathematica.<\/li>\n<\/ul>\n<\/li>\n<li>Section 2.5: Existence and uniqueness\n<ul>\n<li>I should provide some intuition for what we mean by the word &#8220;smooth&#8221; and the term &#8220;smooth enough&#8221;.<\/li>\n<\/ul>\n<\/li>\n<li>Section 2.6: Impossibility of oscillations\n<ul>\n<li>Reintroducing the definition of &#8220;monotonic&#8221; could be helpful here.<\/li>\n<li>The analogy to the &#8220;over-damped&#8221; limit is probably not illuminating for students without physics\/engineering interests\/background.<\/li>\n<\/ul>\n<\/li>\n<li>Section 2.7: Potentials\n<ul>\n<li>This is not introduced in Steve&#8217;s youtube videos. \u00a0It is an opportunity to review the definition of &#8220;gradient&#8221; (and to reinforce the vector field interpretation of the 1d system).<\/li>\n<li>Perhaps show figure 2 from\u00a0<a href=\"https:\/\/science-sciencemag-org.ezp-prod1.hul.harvard.edu\/content\/361\/6406\/eaat6412\/tab-figures-data\">https:\/\/science-sciencemag-org.ezp-prod1.hul.harvard.edu\/content\/361\/6406\/eaat6412\/tab-figures-data<\/a>\u00a0for an ecology example of thinking in terms of a potential function.<\/li>\n<\/ul>\n<\/li>\n<li>Section 2.8: Solving equations on the computer\n<ul>\n<li>Also not in Steve&#8217;s youtube videos. \u00a0I haven&#8217;t been introducing it explicitly, but probably should. \u00a0It converts a flow to a map&#8230;<\/li>\n<li>They should know about slope fields from Math 1b, so I could remind them and anchor on that.<\/li>\n<li>This is a good place to introduce computers.\n<ul>\n<li>Find fixed points symbolically (Solve) and with root finding (FindRoot).<\/li>\n<li>Take the derivative symbolically (and perhaps numerically?).<\/li>\n<li>Create time series plots.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>2.0 &#8211; 2.6 is the prep for a single class meeting. \u00a0In the following class, the goals would be to gain procedural fluency with<\/p>\n<ul>\n<li>finding fixed points (either given a diff eq or a plot of f(x))<\/li>\n<li>determining their stability from a graph of f(x)<\/li>\n<li>determining their stability by finding f &#8216; (x*) and interpreting it<\/li>\n<li>sketching approximate time series of trajectories<\/li>\n<li>making phase portraits on the x-axis<\/li>\n<\/ul>\n<p>Some extra stuff that would be good too:<\/p>\n<ul>\n<li>recognizing and setting up an integral for the solution of separable diff eqs<\/li>\n<li>distinguishing between linear and nonlinear equations<\/li>\n<li>identifying phenomena that occur in linear vs nonlinear autonomous diff eqs<\/li>\n<li>doing the procedural stuff above in Mathematica<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 &#8220;spacing&#8221;. A few notes on spacing: Spacing improves induction\/generalization from examples (but learners perceive otherwise):\u00a0https:\/\/journals.sagepub.com\/doi\/abs\/10.1111\/j.1467-9280.2008.02127.x [&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-118","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-1U","jetpack-related-posts":[{"id":132,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-5\/","url_meta":{"origin":118,"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":170,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/10\/meiss-differential-dynamical-systems-chaos\/","url_meta":{"origin":118,"position":1},"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":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":118,"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":222,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2020\/05\/23\/python-in-my-dynamical-systems-class\/","url_meta":{"origin":118,"position":3},"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":130,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-4\/","url_meta":{"origin":118,"position":4},"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":108,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-1b-differential-equations-background\/","url_meta":{"origin":118,"position":5},"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":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/118","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=118"}],"version-history":[{"count":7,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/118\/revisions"}],"predecessor-version":[{"id":125,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/118\/revisions\/125"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}