Spring 2020 Math 421 Course Homepage

This is a first course in the study of the mathematics underlying the idea of what it means to model something mathematically. When we attempt to study a phenomenon mathematically, we seek a functional relationship between an unknown, measurable quantity, and one we do know and can control. This functional relationship means that we can use the known quantity to calculate values of the unknown quantity, predict its values, and study the properties of the dependence. Obviously, this involves the calculus of functions of one or more variables. Since vector calculus is involved, it also involves linear algebra. But in a deeper sense, dynamical systems involves the use of tools and concepts from all branches of mathematics, from topology, geometry and analysis, to algebra. The basic models in this study are either discrete models, functions from a topological space to itself, or continuous ones, involving differential equations dictating flows on topological spaces. The spaces in general can be any set with a topology on it (a well-defined notion of open subset, or neighborhood), but will almost always be subsets of Euclidean Space in the course. We will use applications at times to study interesting behavior, but in general we will address a different question: Just what kind of dynamical information can be gleaned from a particular type of function or ODE on a particular type of space? We will start with very basic models exhibiting what we call simple dynamics, where almost all of the information can be described in very few words, and evolve toward very complicated systems, where fully describing the dynamical behavior is complex. Along the way we will develop a language to describe the dynamical systems and their behavior, classify systems according to their properties, broaden our perceptions of what it means to "move around" in a space, and gain insight as to just how important the subject can be to understanding both the properties of functions and spaces as well as applications to the natural sciences and engineering.

For full disclosure, the above text is mine and it will be required for the course. A great second text for this course is "A First Course in Dynamics", by Hasselblatt and Katok. For now, keep an active link to this page and site. I will update often with new information and any links I find relevant and/or useful. I will also place here announcements and such to keep you informed of anything I would like to know.

Welcome to the course! It is a great course to teach, and I hope you enjoy the time we have together.

The material for this course is scattered throughout this text. We cannot possibly cover everything, but we will tour through most all of the topics. Please see below in the weekly schedule for a list of topics and relevant text sections.

There will be weekly problem sets, a final examination, and maybe a project. The schedule of homework and the exams will be given below.

Homework, based on the current week's lectures, will be posted here, as well as in Blackboard, sometime on Thursday after lecture. These assignments will be due the following Thursday in lecture. Mathematics is best learned in an environment where active discussion is a fundamental part of the learning process. You are strongly encouraged to work together in the understanding phase of the homework preparation process. You are required, however, to work alone when writing up homework solutions for submission. Homework is the essential educational part of this course. You will be graded not only on your ability to work through problems completely and concisely, but also on the presentation of your solutions and the arguments you make. It is the process with which you develop your argument that must be clearly and comprehensively represented in your proofs and calculations. If the audience (the reader of your submissions) must read between the lines to understand your arguments, then your work is not complete.

This is a proof-based course. This means that many, if not most, of your homework problems will be to establish facts through argument rather than calculation. This means that you will be working with ideas more than numbers in many cases. I do understand that you may not be proficient at or have any experience in proof-writing. However, in time, you will adapt and learn.

You are responsible for lecture notes, any course material handed out, and attendance in class. There are no actual attendance requirements. However, I will easily get to know you and your rate of presence over time. And even though I will be using the text, the lectures will be a serious embellishment of the material. Good advice is the following: Treat the lectures as if you have already read the material and have already attempted some homework problems. In this manner, you can focus mainly on parts of the lectures not found in the text or that covering areas of your reading you found difficult to understand. My teaching style is that of interactive discussion and I will rely on your input in developing the material. Active participation in the classroom is a great way to generate the discussion necessary to fully grasp the material.

213 Kreiger Hall. The hours are 9am - 9pm on Monday through Thursday, and 9am - 5pm on Friday. This free service is a very valuable way to get one-on-one help on the current material of a class from other students outside the course. It is staffed by graduate students and advanced undergraduates. Outside of me and the Grader for the course, definitely take your questions to the Help Room. This course is simply an analysis course directed toward particular maps and differential equations. Most graduate students should be able to "see" through the many problems stated in this course. And your attempts to help guide them will be of huge benefit to you also.

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Cheating is wrong. Cheating hurts our community by undermining academic integrity, creating mistrust, and fostering unfair competition. The university will punish cheaters with failure on an assignment, failure in a course, permanent transcript notation, suspension, and/or expulsion. Offenses may be reported to medical, law, or other professional or graduate schools when a cheater applies. Violations can include cheating on exams, plagiarism, reuse of assignments without permission, improper use of the Internet and electronic devices unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition. Ignorance of these rules is not an excuse. In this course, as in many math courses, working in groups to study particular problems and discuss theory is strongly encouraged. Your ability to talk mathematics is of particular importance to your general understanding of mathematics.

You should collaborate with other students in this course on the general construction of homework assignment problems. However, you must write up the solutions to these homework problems individually and separately. If there is any question as to what this statement means, please see the professor or the grader.

For more information, see the guide on "Academic Ethics for Undergraduates" and the Ethics Board web site: http://ethics.jhu.edu.

Week	Topics	Homework	Due	Extra Material
January 28 - 30 01/28, 01/30	Course Orientation Chapter 1: What is a Dynamical System?	Problem Set 1	February 6	PS1-Solutions What is a model?
February 4 - 6 02/04, 02/06	2.1 Preliminaries	Problem Set 2	February 13	PS2-Solutions The DN: L4 What is an open set? What is Lipschitz continuity?
February 11 - 13 02/11, 02/13	2.2 The Contraction Principle 2.3 Interval Maps	Problem Set 3	February 20	PS3-Solutions What is a contraction? What is cobwebbing? Logistic Map Cobwebbing Coord. Transformation
February 18 - 20 02/18, 02/20	2.3 Interval maps 2.4 Bifurcations of interval maps 2.5 First return maps 2.6 Quadratic maps	Problem Set 4	February 27	PS4-Solutions What is a first return map? What is the logistic map? Logistic map video
February 25 - 27 02/25, 02/27	3.1 Topology on sets 3.2 More on metrics 3.3 Some non-Euclidean metric spaces 3.4 The Cantor set	Problem Set 5	March 5	PS5-Solutions What is a topology on a set? What is an equiv. of metrics? What is an equivalence class? What is a torus? What is a Cantor set?
March 3 - 5 03/03, 03/05	4.1 Linear, first-order ODEs in the plane 4.3 Linear Maps of the Plane	Problem Set 6	March 12	Solutions What is bufurcation theory? What is a matrix exponential?
March 10 - 12 03/10	5.1 Rotations of the circle 5.2 Equidistribution and Weyl's Theorem		March 26	Solutions
March 12	Class canceled due to university closure
March 17 - 19	Spring Break: No classes
March 24 - 26 03/24, 03/26	5.3 Linear flows on a torus 5.4 Toral translations	Problem Set 7	April 2	Solutions What is a linear toral flow? What is rational independence?
March 31 - April 2 03/31, 04/02	5.5 Circle homeomorphisms 6.1 Incompressibility 6.2 Newtonian Systems 6.3 Poincare Recurrence	Problem Set 8	April 9	Solutions
April 7 - 9 04/07, 04/09	6.4 Billiards	Problem Set 9	April 16	Solutions
April 14 - 16 04/14, 04/16	7.1 Counting Periodic Orbits 7.3 Chaos and Mixing 7.4 Sensitive dependence on initial conditions	Problem Set 10	April 23	Solutions
April 21 - 23 04/23	7.5 Quadratic maps 7.6 Complex dynamics	Problem Set 11	April 30	Solutions
April 28 - 30 04/28, 04/30	8.1 Topological conjugacy 8.2 Topological entropy	No assignment

May 9	Final Exam	9:00am - 12:00pm, Croft B32