Math 110.421:  Dynamical Systems

Spring 2018 Course Page

Instructor:  Professor Richard Brown

Lecture:   TTh 3:00pm - 4:15pm  Room:  Krieger 309

Teaching Assistant/Grader:  Thomas Brazelton

Text: A Modern Introduction to Dynamical Systems, 0th Edition

Brown, Richard,

Oxford University Press, July 2018: PDF-version

 

Course Syllabus and Homework Assignment Schedule

 

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 does not yet exist in publication.  I am under contract with Oxford University Press to write this text, and we will use the unedited final manuscript, in PDF form, as the text.  The text at the bookstore, A First Course in Dynamics, by Hasselblatt and Katok, is an excellent second text for this course.  Please feel free to refer to it at any time.   

 

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.

 

Lectures:

  1. Week 1:  01/30,2018, 02/01/2018

  2. Week 2:  02/06/2018, 02/08/2018

  3. Week 3:  02/13/2018, 02/15/2018

  4. Week 4:  02/20/2018, 02/22/2018

  5. Week 5:  02/27/2018, 03/01/2018, http://bit.ly/2CSoJxv

  6. Week 6:  03/06/2018 (http://bit.ly/2I81P9g), 03/08/2018 (http://bit.ly/2Ic6pmY)

  7. Week 7:  03/13/2018 (http://bit.ly/2IpNRzS), 03/15/2018

  8. Week 8:  03/27/2018 (http://bit.ly/2GhNmGy), 03/29/2018 (http://bit.ly/2IaLoIv) 

  9. Week 9:  04/03/2018 (http://bit.ly/2GylamQ), 04/05/201

  10. Week 10:  04/10/2018, 04/12/2018

  11. Week 11:  04/17/2018,04/19/2018

  12. Week 12:  04/24/2018,04/26/2018

  13. Week 13:  05/01/2018, 05/03/2018

Problem Sets:

  1. Problem Set 1:  Due February 8 in class.  (Selected Solutions.)

  2. Problem Set 2:  Due February 15 in class. (Selected Solutions.)

  3. Problem Set 3:  Due February 22 in class. (Selected Solutions.)

  4. Problem Set 4:  Due March 1 in class. (Selected Solutions.)

  5. Problem Set 5:  Due March 8 in class. (Selected Solutions.)

  6. Problem Set 6:  Due March 15 in class.

  7. Problem Set 7:  Due March 29 in class.

  8. Problem Set 8:  Due April 5, in class.

  9. Problem Set 9:  Due April 12, in class.

  10. Problem Set 10:  Due April 19, in class.

  11. Problem Set 11:  Due April 26, in class.

  12. Problem Set 12:  Due May 3, in class.

Research Questions:  In upper level mathematics courses, there are always certain topics that are touched on in passing but never fully explored.  As we pass through some of thse topics, I will post here topics for further exploration.  Anyone interested in any of these (or any other) topics should let me know.  Perhaps we can talk through the boundaries of a deeper look into one or more of these.  Note that the topics here are not exercises, as in "I know the answer".  These are real undergraduate research projects. 

  1. Piecewise contractions:  A differentiable map on [0,1] whose derivative is strictly bounded in absolute value above by 1 is a contraction.  To what extent can we say the same for a map on [0,1] which is differentiable everywhere except for a single jump discontinuity?  It is easy to construct such a map that has no fixed point.  But must it have a single asymptotically stable periodic orbit.  Such a map we will call a piecewise contraction.

  2. Increasing maps on intervals and autonomous ODEs.  An increasing map on [0,1] can be studied by producing a "phase line" in a manner similar to that of a first order autonomous ODE.  In fact, it seems like we can always construct an increasing function whose dynamics are the same as that of a time-1 map of an autonomous ODE.  What is the relationship between increasing maps and autonomous ODEs?    

  3. The Matrix Exponential and its role in connecting linear maps on real space with linear systems of ODEs on real space. 

  4. Convex Billiards: Construct in Mathematica or MatLab a circular and elliptical billiard table. Then generalize to the strictly convex egg-shaped table given by $\left(x^2+y^2\right)^2 = ax^3+cxy^2$, $a\ge c\ge 0$. 

For those of you who would like help outside of that of the professor or TAs, there is a free service offered by the Mathematics Help Room.  Click for more details.

last updated 01/25/2018