I am reading James Meiss’ text Differential Dynamical Systems (SIAM). \u00a0I am specifically interested in how he tells the story of chaos.<\/p>\n
In the Preface, he mentions the following: That\u00a0\u00a0Chapter 5 focuses on invariant manifolds:<\/p>\n
That the “stable and unstable manifolds, proved to exist for a hyperbolic saddle, give rise to one prominent mechanism for chaos — heteroclinic intersection”.<\/p>\n
That Chapter 7 is background for understanding chaos (“Lyapunov exponents, transitivity, fractals, etc”):<\/p>\n
And that in Chapter 8 he’ll discuss Melnikov’s method (onset of chaos): sections 8.12 and 8.13.<\/p>\n
He notes that he doesn’t discuss discrete dynamics (maps).<\/p>\n
<\/p>\n
After the preface, doing a word search for “chaos” or “chaotic”:<\/p>\n
Chaos next comes up in the examples in section 1.4: Meiss introduces an example called the “ABC flow” from Arnold 1965. \u00a0He mentions this is a “prototype chaotic system” and introduces the idea that “nearby trajectories will often diverge exponentially quickly in time”. \u00a0Then he defines the Lyapunov exponent.<\/p>\n
Section 1.7 is about Quadratic ODEs: the simplest chaotic systems, after the Lorenz model is introduced in section 1.6. \u00a0The introduction of the Lorenz model includes an image of the setup, a mention of convective rolls, the idea of the Galerkin truncation, etc. \u00a0So he introduces this system by deriving the ODEs for it.<\/p>\n
In section 1.7 he says “informally, chaos corresponds to aperiodic motion that exhibits ‘sensitive dependence on initial conditions'”. \u00a0He’ll provide a formal definition in chapter 7. \u00a0He mentions that 3-dimensional systems “are the lowest dimensional autonomous ODEs that can exhibit chaos”. \u00a0There is a chart of Sprott’s quadratic chaotic differential equations (the simplest quadratic systems with chaos).<\/p>\n
In section 4.1 Definitions, “orbits can be quasiperiodic, aperiodic, or chaotic”. \u00a0When he introduces orbits, he introduces the idea of a periodic orbit as well as other options.<\/p>\n
Meiss returns to chaos in section 5.2 Heteroclinic orbits. \u00a0(See Diacu and Holmes 1996 for the story of Poincare, his mistake, and its correction). \u00a0He defines a heteroclinic orbit as an orbit that is backward asymptotic to one invariant set and forward asymptotic to a different one. \u00a0The homoclinic orbit (doubly asymptotic) is then a special case that is forward and backward asymptotic to the same invariant set.<\/p>\n
In a 2d system, if a branch of W^u intersects a branch of W^s then the branches coincide. \u00a0Orbits that separate phase space are called separatrices: “they separate phase space into regions that cannot communicate”. \u00a0In section 8.13, we’ll see that higher-dimensional systems are different from 2d ones, and that this doesn’t have to happen! \u00a0Meiss also defines “saddle connection” and mentions that Hamiltonian systems in the plane often have separatrices.<\/p>\n
Chaos comes up again in section 5.5 Global stable manifolds. \u00a0The global set comes from flowing the local set backward in time. \u00a0For finite time, it will be smooth. \u00a0To think about its structure in general, Meiss introduces the idea of an “embedding”. \u00a0He also defines an “immersion” and notes that “an immersion is locally a smooth surface”. \u00a0Note that immersions can cross themselves. \u00a0The topologist’s sine curve is an example that “has infinitely many oscillations and accumulates upon the interval [-1,1] on the y-axis”. \u00a0“We will see later that the global stable manifold can have this accumulation problem: indeed, this is one of the indications of chaos”.<\/p>\n
Next, in section 6.6, when the Poincar\u00e9-Bendixson theorem is introduced, it is introduced as a statement that “There is no chaos in two dimensions”. \u00a0So Meiss is building intuition for the idea of chaos from the very first day in the course, and is distinguishing between it and what happens in 2d, even as he introduces 2d.<\/p>\n
Chapter 7 focuses on chaotic dynamics, so is where the story will be more fully built out. \u00a0The chapter begins with quotes from Poincar\u00e9 and Lorenz and an informal definition. \u00a0To formalize it will require defining “unpredictable” and “sensitive dependence”. \u00a0This chapter occurs before the chapter on bifurcations.<\/p>\n
Two very simple linear examples with sensitive dependence are given to build some intuition for the stretching between nearby trajectories that happens for some initial conditions in a system with sensitive dependence, and to show that sensitive dependence alone is not “chaotic”.<\/p>\n
Aperiodicity or “wanders everywhere” on an invariant set is the next idea introduced, leading to a definition of “transitive”. \u00a0Then “a flow is chaotic on a compact invariant set X if the flow is transitive and exhibits sensitive dependence on X”. \u00a0This gives us an idea of mixing and unpredictability.<\/p>\n
For a lot of systems, their trajectories look chaotic “when solved numerically”. \u00a0Chaos was verified in the Lorenz system for r = 28 in Tucker 2002.<\/p>\n
Note: I also should read the text for references to the Lorenz system, because that system is used as an example.<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
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 center manifolds That the “stable […]<\/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-170","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-2K","jetpack-related-posts":[{"id":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":170,"position":0},"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 "Dynamical Systems"","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":170,"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 "Dynamical Systems"","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":170,"position":2},"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 "Dynamical Systems"","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":170,"position":3},"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 "Dynamical Systems"","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":170,"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 "Dynamical Systems"","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":170,"position":5},"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. 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