Math 421: Dynamical Systems

Spring 2017

Textbook

A First Course in Dynamics, Boris Hasselblatt and Anatole Katok,

Cambridge University Press (2003), ISBN 0 521 58750 6 (paperback).

 

Course Description

This course is an introduction to the theory of dynamical systems, with an emphasis on both theoretical aspects and applications. The material covered will include most sections from Part 1 of the book, Chapters 1-8.

 

Instructor

Hang Xu

Email: hxu@math.jhu.edu

Office Location: Krieger Hall Rm 220

Office Hours: Th 3:00-5:00 pm

 

Lecture

TTh 1:30-2:45 at Krieger 309

 

Grading

is based on 350 points distributed as follows:

12 homework problem sets, worth 10 points each, due in the lecture on the following Thursday. The lowest two score will be dropped.

Midterm on March 7 in lecture, worth 100 points.

Final Take home, worth 150 points.

 

Course Policies

No late homework is accepted. Staple your problem sets! Study groups are encouraged, but homework has to be written down independently.

No makeup exam in this course. If you have to miss an exam for a documented, legitimate reason, please inform me as early in the semester as possible.

You are responsible for lecture notes, any course material handed out, and attendance in class. The lectures will be conducted as if you have already read the material and attempted some homework problems. In this manner, you can focus mainly on those parts of the lectures that cover the areas of your reading you found difficulty to understand.

 

If you have any math questions, please feel free to ask me anytime. You can find me after lectures, in office hours, or you can reach me by email.

 

Help Room

Krieger Hall 213. 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. 

 

Students with disabilities

Students with documented disabilities or other special needs who require accommodation must obtain an accommodation letter from Student Disability Services, 385 Garland, (410)516-4720, studentdisabilityservices@jhu.edu. After that, remind the instructor of the specific needs at two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.

 

JHU ethics statement

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of student affairs and/or the chairman of the Ethics Board beforehand. See the guide on “Academic Ethics for Undergraduates” and the Ethics Board Web site (http://e-catalog.jhu.edu/undergrad-students/student-life-policies/#UAEB) for more information.

 

Tentative Schedule (will be updated as the course progresses)

Homework will be posted by every Tuesday and due in lecture on the next Thursday.

Week 1 (1/31 & 2/2)

Chapter 1, 2.1.1--2.1.3

Homework 1:  Lecture 1 Exercises 1, 3, 4, 8, 9.

                        Lecture 2 Exercises 13 (You only need to describe where orbits go), 14, 22.

                        Textbook: Exercises 2.6.1, 2.6.5 (You only need to show the interior is open).  

Week 2 (2/7 & 2/9)

2.2 Contractions in Euclidean Space

2.3 Interval Maps

Homework 1 due. 

Homework 2:  Lecture 2 Exercises 31 (This example is Lipschitz continuous but not differentiable), 32 (This example is continuous but not Lipschitz continuous), 39 (This example is differentiable but not continuously differentiable), 40 (This example shows the condition that interval I is closed is necessary for Prop. 2.32).

                       Lecture 3 Exercises 41, 44, 45.

                       Please read 2.2.3-2.2.5 about interesting applications of contraction principle in Lecture notes 3.

Week 3 (2/14 & 2/16)

2.3 Interval Maps

2.4 Bifurcations

Homework 2 due.

Homework 3: Lecture 3 Exercises 46.

                       Lecture 4 Exercises 65 (You only need to do parts a and b), 66, 67 (Read the paragraph above this problem), 68, 73, 74. Example 2.66.    

 

Week 4 (2/21 & 2/23)

2.4 Bifurcations of Interval Maps

2.5 First Return Maps

2.6 A Quadratic Interval Map: The Logistic Map

Homework 3 due.

Homework 4: Lecture 4 Exercise 76 (Read through Example 2.66.)

Lecture 5 Exercises 79, 83, 84, 85, 87 (Hint: local version of contraction principle).

Week 5 (2/28 & 3/2)

3.1 Topology on Sets

3.2-3.3 More on Lipschitz Continuity and Metrics

3.4 Some Non-Euclidean Metric Spaces

Homework 4 due.

Homework 5:  Lecture 6 Exercises 90, 91, 95, 97, 98, 100, 101, 106.

 

Week 6 (3/7 & 3/9)

Midterm

3.4 Some Non-Euclidean Metric Spaces

3.5 A Cantor Set

Homework 5 due.

Homework 6:  Lecture 6 Exercises 105, 110, 111, 112, 113, 115.

Week 7 (3/14 & 3/16)

No class on Tuesday due to the storm.

4.3 Linear Planar Map

Homework 6 due.

Homework 7: Lecture 7 Exercises 126 (Only the first part), 130 (Read Remark 4.20), 131, 132, 133, 139, 141, 142, 144 (See Example 4.22).   

 

Week 8 (3/21 & 3/23) Spring Break

Week 9 (3/28 & 3/30)

4.3 Applications of Linear Planar Map

4.4 The Relation between a Linear Flow and Linear Map

Homework 7 due.

Homework 8: Lecture 7 Exercises 146, 147. Read Propositions 4.33, 4.34, 4.35.

                       Lecture 8 Exercises 161, 168.

                       Textbook Exercise 4.1.1.

Week 10 (4/4 & 4/6) 

5.1 Rotations of the Circle

5.2 Equidistribution and Weyl’s Theorem 

Homework 8 due.

Homework 9: Lecture 8 Read Remark 5.4 and Definition 5.5. Then finish exercises 162 and 163.

   Exercises 169, 171(Parts a and b), 172.

Week 11 (4/11 & 4/13)

5.3 Linear Flows on the Torus

5.3.2 Application: A Polygonal Billiard

5.5 Invertible Circle Maps

Homework 9 due.

Homework 10: Lecture 8 Exercises 173(Part a), 177, 178 (You do NOT need to draw the graph), 179.

                         Textbook Exercise 4.3.1 on page 134.

Week 12 (4/18 & 4/20)

5.5 Invertible Circle Maps

Homework 10 due.

Homework 11: Lecture 8 Exercises 180, 181(Hint: Use Proposition 5.37.), 183.

Week 13 (4/25 & 4/27)

8.1 Compact space dimension

8.2.1-4 Topological Entropy

Homework 11 due.

Homework 12:

Week 14 (5/2 & 5/4)

8.2.1-4 Topological Entropy

11.1-3 Quadratic Maps and Chaos

Homework 11 due.

Homework 12:

Final Exam