MATH 645: Differential Equations and Dynamical Systems
                                               
                                                                      Fall, 2009
                                                                                                                                                                                                                                                                                                                                   Prof. Robert Gardner
LGRT  1430 (you can try 1623c too, but office hours will be held in 1430)
Phone: 5-0029
e-mail:   gardner@math.umass.edu             
link to course web page:   www.math.umass.edu/~gardner

Office hours:  Monday 10:45-12, Tuesday 11 - 12, Thursday 8:30-9:30 and by appointment
Quick links: homework  supplement syllabus

Prerequisites:
math 523: Students should be familiar with "epsilon-delta" proofs and mathematically rigorous analysis.
Students may have acquired a similar level of analytical  maturity in other courses in math/stat or perhaps
in other programs. If you are unsure, you should discuss you background with me after class or during office hours.
multivariable calculus (math 425): (vector fields , differentials and Taylor expansions, the implicit function theorem. Integration theory- Green/Gauss/Stokes is not needed here.) Students with only math 233, which focuses on
vector fields of low dimension might be able to get by with some additional reading, but should talk to me about their background.
linear algebra (matrix algebra, eigenvalues and generalized eigenvalues, diagonalization,  and to a limited extent, some familiarity with Jordan Canonical forms.)  Math 545 would be best, but it would be possible to get by with m235 plus some additional reading.  Jordan forms are needed in m645, but do not play a major role in terms of calculations students are asked to do with the exception of one short segment of the course. You should be able to do a Jordan form calculation for a matrix of low dimension for this segment.

            
   
  References:  There is no official text for the course, but I  recommend "Differential Equations and  Dynamical Systems by Laurance Perko, (Springer, 2001) for students who would like to use a text along with the lectures. There will be some segments of the course that do not follow Perko closely, but for the most part, the order of presentation of topics in Perko is similar to the order I will follow in the lectures, and it reads more like a text with ample explanatory material, worked examples, and problems at the ends of sections.  A copy of Perko will be on reserve at the Dubois library, along with several other texts and monographs that range from elementary  undergraduate level DE's (e.g Boyce and DiPrima) through advanced topics beyond the scope of m645. Several of the references (Hale,  and Coddington and Levinson) are also suitable for texts as m645, but assume a somewhat greater level of mathematical maturity than Perko.

Graded coursework:  Graded coursework will consist  an (evening)  midterm (date and time TBA), a final exam, and periodically assigned homework sets.  The two exams and your set of homework scores will be
recorded as percentages, and your final numerical average will be determined according to the formula
                                    fna=  .3 (midterm) + .3 (final exam) + .4 (homework percentage)
In terms of letter grades,  the minimum score for a B is fna= 60.


Submission and grading of homework problems (Please read carefully)
Working on homework problems is the most important thing you can do to learn and master the subject. Problems sets will usually include some simple problems, and others that are more demanding.  Homework will be assigned here in sets consisting of several problems: I will announce each new set and any updates in class; pdf files should  be downloaded or viewed from the homework link. If you miss a class, it is your responsiblity to check the homework link for any new or updated assignments.
Late homework: homeowrk must be submitted by the announced due date. Late submissions will not be accepted without a valid excuse such as illness, etc. You may turn the assignment in during the lecture or leave it in a
homework envelope marked "m645 homework submissions" on the bulletin board opposite 1430 LGRT. Do not
leave homework under my office door or in my mailbox. The homework submission envelope will be removed within a day of the announced due date; after that, you will need a valid excuse for a late submission.
Grading and resubmission of homework: Each problem will be returned with one of four comments:
correct: your solution is correct in every detail. Each correct answer receives 1 homework point.
essentially correct: your solution has some minor errors  or unclear exposition, but demonstrates a correct understanding of the main points of the problem. Each essentially correct answer receives  3/4 of a homework point, but may be resubmitted once prior to the cutoff date. If your resubmitted solution is correct, the score will be raised to 1 homework point; if there are also errors in the resubmitted problem,  the original score of 3/4  homework point will not be reduced.
not correct:  You made a visible effort to tackle the problem, but your answer has  significant mathematical errors and/or does not demonstrate an adequate understanding of what the problem was asking you to do.  Each problem marked "not correct" receives a score of 0 homework points, but can be resubmitted once prior to the relevant cutoff date.
not accepted: You did not make an adequate visible effort on this problem, or omitted it completely. Problems marked not accepted receive 0 homework points and cannot be resubmitted.
cutoff dates: The homework sets covered on the midterm exam will  have a cutoff date 1 week prior to the date of the midterm exam.  The remaining homework sets will have a cutoff date one week prior to the date of the final exam.
Homework solutions: On the cutoff dates, I will publish full solutions to the problems on the  relvevant homework sets. The solutions will be available  on  the homework link. Exam solutions will also be published on the homework page link.
Resubmitting homework: Resubmission of homework is intended to encourage students to persist on problems that seemed out of reach at first, but it is important that this not be used as a means procrastinating with homework until just before an exam. The right to resubmit a problem has to be earned by making a visible effort to solve the problem on a paper turned in prior to the due date.  For example, recopying the statement of the question on your homework paper with a brief comment like "I couldn't do much with this..." will be marked "not accepted".
Getting help: If you are having difficulty getting started on a problem, I will be happy to meet with you during office hours to help you get started in the right direction. No appointment is needed for office hours. If you are not free during at those times, you can arrange to meet with me at other times by e-mailing me or seeing me after class to make an appointment. This should usually provide sufficient hints and suggestions to at a minimum, earn the right to resubmit a solution after the due date.
Unclaimed homework: I will return graded homework papers in class. If you are not present to claim your paper,
it will be left in an envelope on the bulletin board opposite LGRT 1430 marked "Graded math 645 homework."
Collaboration on homework: While it is desirable for students to discuss course material amonst themselves, you should work individually on particular homework problems,  and your homework submissions should be entirely your own work. An important objective of math 645 is to prepare Ph.D. students to take the advanced qualifying exam in differential equations, which is a manifestly non-collaborative activity. There is a time and place for collaborative work, but this needs to occur within a structure that promotes true collaboration, with substantive contributions from all participants.

Supplementary material I will make supplementary material available at this link, including lecture notes (frequently with additional worked examples there isn't time to cover in the lectures), numerical simulations (click me and me-- but first tell your browser to open with VLC)and some Mathematica notebooks used to produce the  simulations. While numerical experimentation will not be a part of the graded coursework, it is important to develop some facility with packages such as Mathematica and Matlab, and I would be happy to work with students without prior experience with Mathematica who would like  to perform their own numerical experiments.

Calculators: Students will not be permitted to use calculators or laptops during exams. You can of course use them while working on homework problems, but with the proviso that homework problems must be justified by including an explanation, proof, or calculation showing how the  answer was obtained. Generally and unless specifically directed to the contrary, I exect you to show the details of calculations associated to steps that require  significant calculus or algebra.

Syllabus:Topics to be covered are described in detail on the course syllabus. The syllabus includes most, but not all of the  topics on dynamical systems PhD students are responsible for on the advanced qualifying exam on differential equations. The course is not intended to be a complete presentation of all topics described in the "axioms" concerning ODEs and their applications; some of the suggested readings in the references are intended to highlight any omitted topics.