{"id":112,"date":"2019-06-10T16:39:03","date_gmt":"2019-06-10T20:39:03","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=112"},"modified":"2019-06-10T17:21:20","modified_gmt":"2019-06-10T21:21:20","slug":"dynamical-systems-math-21b-differential-equations-background","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","title":{"rendered":"Dynamical Systems: Math 21b differential equations background"},"content":{"rendered":"<p>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.<\/p>\n<p>Student diff eq background from Math 21b:<\/p>\n<ul>\n<li>Linear first order systems\n<ul>\n<li>Interpret a linear system written in vector\/matrix notation.<\/li>\n<li>Use an eigenbasis and eigenvalues for the matrix of a linear system to construct a general solution.<\/li>\n<li>Sketch a phase portrait for a 2d system.<\/li>\n<li>Match a 2d system to its phase portrait.<\/li>\n<li>Work with the matrix exponential.<\/li>\n<li>Define &#8220;asymptotically stable&#8221; and &#8220;equilibrium point&#8221;.<\/li>\n<li>Use eigenvalues of a linear system (with fixed point at the origin) to determine whether the origin is asymptotically stable.<\/li>\n<li>Relate the eigenvalues, determinant, and trace of the matrix to the phase portrait about the origin for a linear system (with fixed point at the origin).<\/li>\n<\/ul>\n<\/li>\n<li>Nonlinear first order systems\n<ul>\n<li>Sketch nullclines, identify equilbrium points, and add vectors of the vector field to regions of the phase plane.<\/li>\n<li>Use the Jacobian to identify the behavior of a system near an equilibrium point.<\/li>\n<\/ul>\n<\/li>\n<li>Higher order linear constant coefficient homogeneous differential equations\n<ul>\n<li>Convert from a second order equation to a first order system.<\/li>\n<li>Find the characteristic polynomial.<\/li>\n<li>Construct a general solution by converting to a matrix equation.<\/li>\n<li>Solve higher order homogenous constant coefficient linear differential equations.<\/li>\n<li>For a nonhomogeneous equation with a sinusoidal forcing (not at the natural frequency), find the general solution.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>Many of these topics overlap with Math 1b, but are presented in matrix form in 21b.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 from Math 21b: Linear first [&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-112","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-1O","jetpack-related-posts":[{"id":108,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-1b-differential-equations-background\/","url_meta":{"origin":112,"position":0},"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":132,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-5\/","url_meta":{"origin":112,"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":118,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-strogatz-chapter-2\/","url_meta":{"origin":112,"position":2},"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":141,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/19\/hughes-hallett-et-al-chapter-11-differential-equations\/","url_meta":{"origin":112,"position":3},"title":"Hughes-Hallett et al Chapter 11: Differential equations","author":"siams","date":"19 June 2019","format":false,"excerpt":"11.1: What is a differential equation? Starts with an example: what sets the rate at which a person learns a new task? \u00a0Defines a diff eq and a solution to a diff eq. Defines order of a diff eq. \u00a0Example 1 is showing a function is not a solution to\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":[]},{"id":170,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/10\/meiss-differential-dynamical-systems-chaos\/","url_meta":{"origin":112,"position":4},"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":150,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/varburg-and-purcell-7th-edition-differential-equations-mainly-chapter-18\/","url_meta":{"origin":112,"position":5},"title":"Varburg and Purcell 7th edition.  Differential equations (mainly chapter 18)","author":"siams","date":"25 June 2019","format":false,"excerpt":"Section 5.2: What is a diff eq?\u00a0 Provides an example and two solution methods before defining diff eq (and doesn't define a solution...).\u00a0 Then presents separation of variables via an example.\u00a0 Then a falling body example and an escape velocity example. Section 7.5: exponential growth and decay.\u00a0 They motivate y'\u2026","rel":"","context":"In &quot;Differential equations&quot;","block_context":{"text":"Differential equations","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/differential-equations\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/112","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=112"}],"version-history":[{"count":2,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/112\/revisions"}],"predecessor-version":[{"id":114,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/112\/revisions\/114"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=112"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=112"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}