Math 134: Dynamical Systems
Fall 2005

(12/13) I just posted some review material for the final exam (see above). Btw, there will be six problems and you are allowed to bring a 3 X 5" cheat sheet.

(12/8) Here's a list of Main Questions we discussed in Math 134, to help you review for the final. Also, I will have no office hours on Friday, December 9.

(12/7) My office hours next week (12/12-12/16) will be:

Monday & Wednesday, 10-12 and 2:30-4
Thursday, 2-3

(12/5) Homework 7 solutions have been posted. Also, if you're planning to take Math 131B next semester with me, or know somebody who does, please register as soon as you can or urge that person to register. So far nobody has signed up and the class may be cancelled. Btw, the time for Math 131B will be changed from 12:30 to 11:30 to avoid conflict with Math 129B. Also:

The final exam will be on December 19, from 9:45 to 12, in room MH 222.

(12/4) You can choose to do your presentation on Monday, December 12 or Wednesday, December 14, between 10 and 12, or 2:30 and 3:30. Let me know when you'd like to to it. Also, our class photo-op will be tomorrow, 12/5 (not on 12/7).

(12/1) Hint for exercise 6 from Chapter 9 (Homework 8): do not try to find a global Lyapunov function (to see why, compute all equilibria of the system). Don't give up if you get mixed terms such as x^2 y and x y^2 in the flow derivative of L; simply use the fact that they are smaller than quadratic terms if x, y are close to zero.

(11/30) It seems that there is still some confusion about the definition and meaning of the flow of an ODE, so here's a handout called "The flow of a differential equation". Also note:

The final exam will be on December 19, from 9:45 to 12, room TBA.

(11/29) Homework 8 (the last one!) has been posted and is due on December 7. Also, if you chose to do a presentation, please come see me as soon as possible.

(11/28) Solutions to Homework 6 have been posted. Also: we'll have a class photo taken on December 7.

(11/23) I will bring the SOTEs (Student Evaluations of Teaching Effectiveness) on Monday, December 5.

Happy Thanksgiving!

(11/20) Homework 7 has been posted and is due November 30.

(11/11) Solutions to Homework 5 have been posted.

(11/8) One last chance to let me know if you disagree with moving the final exam to December 19! Please let me know by tomorrow.

(11/6) You can choose to substitute one of your midterms by a 15 minute presentation in December (after the end of classes, date yet to be determined). Here's a list of sections to choose from: 9.2, 9.3, 11.1, 11.2, 17.3, 17.4. Let me know if you plan to do this (and which section you want to present) as soon as possible. Also, Homework 6 has been posted and is due November 16.

(11/2) Midterm 2 solutions have been posted. (Note that you can still turn in the extra credit problem.) Also, I would like to change the date and time of the final exam to December 19 at 9:45. Please let me know if that works for you or not.

(10/30) I posted some practice questions for midterm 2 as well as a summary of the material. You don't need to turn in your solutions to the practice questions, but we'll discuss these in class tomorrow.

(10/29) Recall that Midterm 2 will cover sections 4.1, 4.2, 5.6, 6.4, and 6.5. To prepare, I suggest that you carefully go over the theory and homework assignments 4 and 5. I'll post a few more practice problems soon. You are allowed to bring a 3X5" index card with your notes to the exam.

(10/26) If you are interested in teaching math, then you may be interested in "So you want to teach math?", an event organized by the Math Club and the Math Department. It takes place in MH 523 on October 28 at 3:30.

(10/25) You can turn in Homework 4 on Friday, October 28. Also, I made a correction to the topology handout posted yesterday (thanks to Matt Low for pointing it out; there's also a correction to the last exercise).

(10/24) Solutions to Homework 4 have been posted. I also posted some notes on topology.

(10/16) I will have no office hours on Monday and Wednesday afternoon this week and no office hours on Friday. Sorry about that. I'll try to make up this lack of office hours in the coming weeks.

(10/13) I posted Homework 5 (due October 26), which includes a reading assignment. Since we'll have no class on Friday, October 21 (I'll be away at a conference), you can do the reading that day. Note that problems from Chapter 5 are based on the reading assignment. (Btw, I'll post solutions to the last homework as soon as everybody turns it in. If you already did, I can email you the solutions - just let me know.)

(10/11) Here's a reading guide for the near future. We'll skip chapter 5 and sections 6.1-6.3, but I strongly suggest that you read it all for your own mathematical education (also, in higher dimensions things are pretty similar to the 2D case). We will cover sections 6.4 and 6.5 and then move on to nonlinear systems and do sections 7.1-7.3.

(10/10) I posted some notes on the Implicit Function Theorem.

(10/9) It seems that Homework 4 is harder than I thought it would be, so you can turn in your solutions on Wednesday, October 12. We'll discuss it in class tomorrow, but here are some hints. Problems 1-4 shouldn't be too hard. For #5, first reduce both systems to canonical form, then look for a conjugacy between the canonical forms (as in the proof of the Theorem on p.66). For #6, one way of doing it would be to find a conjugacy between the associated equations in the complex plane, as I once explained in class. You could also use polar coordinates, after reducing to canonincal form. For #7 and #8, first reduce to canonical form. Solution to #9 is contained in solutions to #6-8.

(10/2) Homework 4 has been posted and is due on Monday, October 10.

(9/30) My favorite program for drawing phase portraits of planar (linear and non-linear) systems is pplane6, written by John Polking at Rice University. The only drawback is that it is based on Matlab. To download pplane6, follow this link. Btw, I forgot to post solutions to Homework 3. Here they are.

(9/28) Midterm 1 solutions have been posted.

(9/24) I posted some Midterm 1 review problems. You should consider them as a homework assignment, but you don't have to turn your work in.

Recall that the main concepts we covered (and that you should review) are: for 1-dimensional equations - the phase line, types of equilibria, periodic solutions, Poincare map, bifurcations; for planar linear systems - the linearity principle, eigenevalues and eigenvectors, finding the general solution, types of equilibria, changes of coordinates.

(9/21) I posted a little handout on linear transformations vs. matrices.

(9/19) Solution to Quiz 1 has been posted.

(9/18) Homework 3 has (finally!) been posted. It's due on September 26.

(9/17) Solutions to Homework 2 have been posted.

(9/16) I got stuck in class today trying to prove that in the case of complex eigenvalues, the general solution is a linear combination of X_Re and X_Im. Here's a correction to that proof. Sorry for the confusion.

(9/12) Quiz 1 will be on Monday, September 19 and will cover homework assignments 1 and 2.

(9/9) You can turn in Homework 2 on Wednesday, September 14.

(9/8) Solutions to some Homework 1 problems have been posted.

(9/6) I will not have an office tomorrow (9/7) between 2:30 and 3:30, but I will have an extra one on Friday (9/9) at the same time.

(9/5) Homework 2 has been posted and is due on Monday, September 12.

(8/25) Homework 1 has been posted and is due on Friday, September 2. Problem #15 is optional but is worth extra credit.

(8/19) The first class meeting is on Wednesday, August 24.

Lecture time and location

MWF 9:30-10:20 in MH 234

Prerequisite

Math 133A (with a grade of "C-" or better) or instructor consent. Some knowledge of linear algebra would be helpful, but not essential.

Office hours

Monday and Wednesday, 10:30-11:30 and 2:30-3:30, Friday 10:30-11:30 (only), and by appointment

Textbook

M. W. Hirsch, S. Smale, and R. L. Devaney: Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier/Academic Press, second edition, 2003.
ERRATA

Syllabus

Green sheet

First-order equations refresher (Chapter 1).

Planar linear systems (Chapter 2): second-order ODEs and planar systems, preliminaries from algebra, planar linear systems, eigenvalues and eigenvectors, solving linear systems, linearity principle.

Phase portraits depending on the eigenvalues (Chapter 3).

Classification of planar systems (Chapter 4).

Nonlinear systems (Chapter 7): dynamical systems, existence and uniqueness theorem, continuous dependence of solutions.

Equilibria in nonlinear systems (Chapter 8): examples, sinks, sources, saddles, stability, bifurcations.

Closed orbits and limit sets (parts of Chapter 10).

Applications: RLC circuit (parts of Chapter 12), Newtonian mechanics (parts of Chapter 13), predator/prey systems (parts of Chapter 11, time permitting).

Basics of discrete dynamical systems (parts of Chapter 15, time permitting).

Exams

There will be occasional short quizzes, two midterms, and a final exam. The exam schedule:

Midterm 1: September 28

Midterm 2: November 2

Final exam: December 19, 9:45-12, MH 222

There will be no make-up exams.

Midterm 1 solutions

Midterm 2 solutions

How to...

How to study for this and other math classes

How to take the exam

Grading policy

Homework 10%, Quizzes 10%, Midterms 40%, Final 40%

Handouts

The flow of a differential equation

A few words about topology

Some notes on the Implicit Function Theorem.

A write-up explaining the difference between matrices and linear transformations.

Homework

Late homework policy. One day late: 50% penalty; two days late: no credit.

# Due date Assignment Solutions

1 9/2 Chapter 1 exercises #1, 2abc, 3a, 5, 8, 9, 11, 12, 14.
#15 extra credit

2 9/14 Chapter 2, all exercises (on p.36-38) except 5, 8, 10, 13

3 9/26 Chapter 3, ex. 1, 2(vi), 3, 5, 8, 9, 11, 12, 14, 15, 16

* 9/28 Midterm 1 review:
Ch. 1: #2, 4, 8, 10. Ch. 2: #1, 2, 9, 10.
Ch. 3: #2, 3, 6, 11.

4 10/12 Chapter 4, ex. 1-4, 5a, 6-9

5 10/26 Reading assignment: sec. 5.6.
Ch. 5, ex. 13, 14, 15. Ch. 6, ex. 12abcdh, 13, 14.

* 11/2 Practice questions for midterm 2

6 11/16 Chapter 7, ex. 1a, 2, 5, 6, 7 (extra credit), 8, 9.

7 11/30 Chapter 8, ex. 1, 2, 3, 11; extra credit: 6.

8 12/7 Chapter 9, ex. 1a, 4, 6.

#	Due date	Assignment
1	9/2	Chapter 1 exercises #1, 2abc, 3a, 5, 8, 9, 11, 12, 14. #15 extra credit
2	9/14	Chapter 2, all exercises (on p.36-38) except 5, 8, 10, 13
3	9/26	Chapter 3, ex. 1, 2(vi), 3, 5, 8, 9, 11, 12, 14, 15, 16
*	9/28	Midterm 1 review: Ch. 1: #2, 4, 8, 10. Ch. 2: #1, 2, 9, 10. Ch. 3: #2, 3, 6, 11.
4	10/12	Chapter 4, ex. 1-4, 5a, 6-9
5	10/26	Reading assignment: sec. 5.6. Ch. 5, ex. 13, 14, 15. Ch. 6, ex. 12abcdh, 13, 14.
*	11/2	Practice questions for midterm 2
6	11/16	Chapter 7, ex. 1a, 2, 5, 6, 7 (extra credit), 8, 9.
7	11/30	Chapter 8, ex. 1, 2, 3, 11; extra credit: 6.
8	12/7	Chapter 9, ex. 1a, 4, 6.

Quizzes

Quizzes will be based on the homework. The date of each quiz will be announced in class.

Date Quiz # Solutions

9/19 Quiz 1 Solutions

Date	Quiz #	Solutions
9/19	Quiz 1	Solutions

Policy on calculators

Feel free to use them as a study tool, but don't get too dependent on them.

Calculators will not be permitted on exams.

I will occasionally use MATLAB and other ODE software.

Academic integrity

By default, I regard my students as honest individuals and expect them to abide by the University policy on academic integrity.

Anonymous feedback

If you have any comments or suggestions, please fill out this anonymous feedback form.

Slobodan N. Simić

Last modified: Wed Dec 21 11:13:21 PST 2005

Math 134: Dynamical Systems Fall 2005

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