MATHEMATICS 454Ordinary Differential Equations and StabilityTheoryMWF 2:00-2:50, PAS 414“A very small cause which escapes our notice determines a considerableeffect that we cannot fail to see, and then we say that the effect is due tochance. If we knew exactly the laws of nature and the situation of the uni-verse at the initial moment, we could predict exactly the situation of thesame universe at a succeeding moment. But even if it were the case thatthe natural laws had no longer any secret for us, we could still know thesituation approximately. If that enabled us to predict the succeeding situ-ation with the same approximation, that is all we require, and we shouldsay that the phenomenon had been predicted, that it is governed by thelaws. But is not always so; it may happen that small differences in theinitial conditions produce very great ones in the final phenomena. A smallerror in the former will produce an enormous error in the latter. Predictionbecomes impossible.”–Henri Poincar´eText:• Strogatz: Nonlinear Dynamics and ChaosInstructor:• Dr. Karl Glasner• Office: Math 618• e-mail: [email protected] (This is the best way to reach me)• Office Hours: MWF 10:00-11:00, and by appointment• Web: http://www.math.arizona.edu/∼ kglasner/math454/math454.htmlCourse Description:This course is an outgrowth of the study of ordinary differential equations, andwould more typically be called “Dynamical Systems”. Rather than focusing onthe question “what IS the solution” (which is, in most problems, impossibleto fully address), we ask “what properties do solutions possess”? The study ofdynamical systems is therefore meant to address both qualitative and quantita-tive features rather than solution techniques (although some will be needed tomake things concrete). The end of this course is an introduction to some of themost significant (and well publicized!) developments in quantitative sciencethat have occurred over the last half century. We will encounter and make pre-cise familiar buzzwords such as “chaos” and “fractal”. Along the way, someadvanced ideas in applied mathematics will be carefully introduced.Grading Policy:Grades are based upon quizzes (derived from suggested homework, see theweb site), and projects that may involve computer work. The lowest quizgrade will be dropped. If a quiz is missed, this will automatically count asthe dropped quiz. The course grade will be determined by combining perfor-mance on each as follows:50% quizzes50% projectsGrades will be assigned roughly as 90-100=A, 80-89=B, etc., with some possi-ble adjustment downward being made as the semester progresses. The finalgrading scale will not reflect a “curve” that is sensitive to the class average, butis rather based upon some fixed benchmark of knowledge acquisition.Quizzes:Every 3-4 lecture periods, a quiz will be given concerning material coveredsince the previous quiz. The quiz questions will be heavily based on suggestedhomework problems listed on the web site, so doing these ahead of time willsignificantly enhance your performance.Projects:There will be several computer related projects assigned and posted on theweb-site. These are meant to give you more of a hands-on experience with thesubjects. Some of the projects require the use of MATLAB, a high-level mathe-matical environment and language. If you are unfamiliar with this program, atutorial is available on the web-site. Additionally, the first project gets you usedto this program. You will generally be given templates for MATLAB projects,so very little coding on your part will be needed. Projects MUST follow therules of writing and presentation detailed below.Attendance:Students should be familiar with the university attendance policy. Attendancein EVERY class is mandatory. As few as three (3) unexcused absences willresult in either an administrative withdrawal, OR automatic failure of the class,at the discretion of the instructor.Calendar:Week of Jan 10: Introduction, Chapter 2Week of Jan 17: Chapter 2 (No class Monday)Week of Jan 24: Chapter 3Week of Jan 31: Chapter 3,4Week of Feb 7: Chapter 4Week of Feb 14: Chapter 5,6Week of Feb 21: Chapter 6Week of Feb 28: Chapter 6,7Week of March 7: Chapter 7Week of March 21: Chapter 8Week of March 28: Chapter 8, 10Week of April 4: Chapter 10Week of April 11: Chapter 9, 11Week of April 18: Chapter 11Week of April 25: Chapter 11,12Week of May 2: Chapter 12Written project requirementsNOTE: It will be a general policy that work which does not follow theseguidelines will be returned without a grade.Presenting mathematical ideas should be no different than any other subject.The usual rules of good writing should apply, especially neatness, clarity andorganization. Good examples of mathematical writing can be found in many(but by no means all!) textbooks. This is the style you should emulate forwritten assignments in any advanced course.Some specific considerations are:(1) The most important rule: brevity. Answers should be as long as necessaryto convey all significant details, and NO LONGER!(2) Complete, grammatically correct sentences (yes, just like in other classes)are essential. Note that equations are parts of sentences and should be accom-panied by text and proper punctuation.(3) Organizing ideas is the most important part of writing. Large problemsor projects may require organization into paragraphs and even entire labelledsections. Sections in the project write-up must correspond to the sections ofthe project description.(4) Projects in 454 must be typeset or word processed. Equations are ideallytypeset (using LaTeX, equation editor, etc.) or can be neatly handwritten intothe narrative. Note that not every detail of a mathematical calculation is nec-essary - selecting which equations to display is a bit of an art.(5) Graphs should be annotated (by hand if necessary), and a textual explana-tion should accompany. In the case where computer code is used, a copy of thecode and graphical/ numerical results should be included at the end. If manyvariations of the same code is used, only a basic template in