Teaching

MA 4791/5791 Continuous Modeling (Fall 2010)

Mon. and Wed. 5:30-6:45pm; CU 656

Lynn S. Bennethum

Office: CU 638, Phone (303) 556-4810

Office hours: Mon. and Wed. 4:00-5:30pm in CU 638, or by appointment.

e-mail: Lynn.Bennethum@ucdenver.edu (best way to reach me).

home page: http://www-math.cudenver.edu/~bennethm

fax: (303)556-8550

Mathematical modeling is the process of taking a problem of practical interest, casting it in a mathematical form, and arriving at a meaningful or useful solution. There are many ways to mathematically model: statistically, discretely, continuously, etc. In this course we’ll consider problems whose mathematical formulation consists of ordinary differential or difference equations. The emphasis will be on analytical (pencil and paper) methods for solving and analyzing differential equations; however, numerical methods will be mentioned, can be used to check work, and may be used for projects. Given the developments of the past thirty years, it is impossible to study differential equations at this level without an encounter with dynamical systems and chaos. During this semester we will explore both the modeling process as it applies to problems in engineering, biology, ecology, physics and physiology as well as the exciting new developments in nonlinear dynamics.

This course is taught both as an upper level undergraduate course and as a first year graduate course. It is expected that students are comfortable with the material of Ordinary Differential Equations (MATH 3200) with some minimal review. Graduate students will be expected to model and analyze more sophisticated systems.

Textbooks: Nonlinear Dynamics and Chaos by Steven H. Strogatz, 2000, Perseus Books Publishing. ISBN: 0-7382-0453-6.

Prerequisite: Ordinary Differential Equations (MATH 3200) and Linear Algebra (MATH 3191). Some review is presented in the course.

Grading: Homework and Projects will be weighted as follows:

· Homework 70 per cent

· 2 Projects (15 per cent each) 30 per cent

Homework: Homework will be assigned roughly every 1.5 weeks. Graduate students will have extra problems on every problem set. Late homework will be accepted but with a penalty: 20% off for every class period late.

Projects: It is not too soon to begin thinking about topics for projects. Your projects must deal with a specific example of modeling that involves differential or difference equations. The model may be analytical or numerical.

The first project is due Monday Oct. 11. The second project is due no later than December 13.

Each project should be on the order of 8-10 pages for undergraduates and 12-15 pages for graduate students. It should be carefully written and well-organized, typed, preferably using an application program which is designed to incorporate equations such as LaTeX, and should contain figures, tables, and numerical output as needed. The paper must include references and show evidence of library/web research. The paper itself needn’t contain original work. You should begin talking with me as soon as you get ideas so we can refine and focus your topic. See the project guidelines for additional details.

General Flow of the Course: The course will follow the book fairly closely because it contains so many relevant and exciting topics. However, there will be a few instances in which we will stray from the book to stress the modeling aspects of the course. Here is a list of anticipated topics.

1. Read Chapter 1 of the text.

2. ODE review – Models that involve first-order ODE’s

3. Stirred tank problems – Reality to models

4. Population models.

5. One-dimensional flows – Chapter 2

6. Linear Systems in 2-D – Chapter 5

7. Phase plane analysis and geometry – Chapter 6

8. Bifurcations – Chapter 3

9. Flows on the circle; fireflies – Chapter 4

10. A quick look at limit cycles – Chapter 7

11. The Lorenz system – Chapter 9

12. One dimensional mappings – Chapter 10

This is an ambitious list of topics; at the same time it is a regrettably brief survey of modeling and dynamical systems. As you read, look for omitted topics in the book for possible project topics.

References: There is a wealth of reading to be done on nonlinear dynamics and ODE/modeling at all levels. Please explore the library and the web. Here are a few relevant sources (in no particular order).

1. Understanding Nonlinear Dynamics, D. Kaplan and L. Glass, Springer-Verlag, 1994.

2. Chaos: An Introduction to Dynamical Systems, K. Alligood, T. Sauer, J. Yorke, Springer-Verlag, 1996.

3. Differential Equations and Their Applications, M. Braun. Springer-Verlag, 1993.

4. Population Biology, A. Hastings, Springer-Verlag, 1997.

5. Mathematical Modeling for Industry and Engineering, T. Svobodny, Prentice-Hall, 1998.

6. Perspectives of Nonlinear Dynamics, Volumes 1 and 2, E. Jackson, Cambridge University Press, 1991.

7. Dynamics: The Geometry of Behavior, Parts 1-4. R. Abraham and C. Shaw, 1981-1988.

8. Chaotic Vibrations: An Introduction for Applied Scientists and Engineers, F. Moon, Wiley, 1992.

9. Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers, F. Moon, Wiley, 1992.

10. Differential Equations, Dynamical Systems and Linear Algebra, M. Hirsch and S. Smale, Academic Press, 1974.

11. An Introduction to Chaotic Dynamical Systems, R. Devaney, Benjamin, 1986.

12. From Clocks to Chaos, L. Glass and M. Mackey, Princeton University Press, 1988.

13. Fractals Everywhere, M. Barnsley, Academic Press, 1988.

14. Chance and Chaos, D. Ruelle, Princeton University Press, 1991.

Software Help:

Boulder has a nice set of applets for visualizing solutions to ODE's (one, or systems). See: http://amath.colorado.edu/java/

You first need to Download MVT, and then afterwards you can START MVT for simple point-and-click plots. For the phase plane portraits you need to put in initial conditions to get each sample trajectory.

Fall 2010 CLAS Academic Policies

The following policies pertain to all students and are strictly adhered to by the College of Liberal Arts and Sciences (CLAS).

Every student MUST check and verify their schedule prior to the published drop/add deadlines. Failure to verify a schedule is not sufficient reason to justify a late add or drop later in the semester.
CLAS students must use their email.cudenver.edu email address. Email is the official method of communication for all University of Colorado at Denver and Health Sciences Center business. All email correspondence will take place using your UCDHSC email address. Go to http://www.ucdenver.edu/student-services/resources/registrar/students/policies/Pages/EmailPolicy.aspx to activate your email address.
Students are NOT automatically added to a course off a wait list after wait lists are dropped. If a student is told by a faculty member that they will be added off the wait list, it is the responsibility of the student to complete the proper paperwork to add a course.
Students are not automatically notified if they are added to a class from a wait-list. Again, it is the responsibility of the student to verify their schedule prior to any official dates to drop or add courses.
Students must complete and submit a drop/add form to make any schedule changes. Students are not automatically dropped from a class if they never attended, stopped attending or do not make tuition payments.
Late adds will be approved only when circumstances surrounding the late add are beyond the student’s control and can be documented independently. This will require a petition and documentation from the student. Please note that the signature of a faculty member on an add form does not guarantee that a late add petition will be approved. Undergraduates should contact the Advising office and Graduate students should contact the Dean's office to petition for a late add.
Late drops will be approved only when circumstances surrounding the late drop have arisen after the published drop deadlines, are beyond the student’s control, and can be documented independently. This will require a petition and documentation from the student. Pre-existing circumstances (circumstances that existed prior to the published drop deadlines) regarding illness, family or other confounding issues will not be considered adequate reason to drop or withdraw from courses after the published University and/or College drop deadlines. Please note that the signature of a faculty member does not guarantee that a late drop petition will be approved. Undergraduates should contact the Advising office and Graduate students should contact the Dean's office to petition for a late drop.
Undergraduate students wishing to graduate in fall of 2010 must meet with their academic advisor and complete their graduation application and intent to graduate form by 5 PM on September 8, 2010. You can obtain an application ONLY after meeting with your academic advisor so make your appointment early. There are no exceptions to this policy or date.
Graduate students wishing to graduate in fall semester 2010 must complete their Intent to Graduate form and have a Request for Admissions to Candidacy on file with the CLAS Dean's office no later than 5 PM, September 8, 2010.

Students are responsible for completing financial arrangements with financial aid, family, scholarships, etc. to pay their tuition. Students will be responsible for all tuition and fees for courses they do not officially drop using proper drop/add procedures and forms.

Important Dates

August 23, 2010; First day of Class
August 29, 2010 ; Last day to add a class or be added to a wait list for a class using the UCDAccess portal. Please note that if your course does not appear as "enrolled" on your schedule you are not enrolled in the class.
August 30, 2010: LAST DAY TO DROP WITHOUT DROP CHARGE - THIS INCLUDES SECTION CHANGES.
August 30, 2010: Wait Lists are dropped. Any student who was not added to a course automatically from the wait list by this date and time MUST complete a schedule adjustment form to be added to the class. Students are NOT automatically added to the class from the wait list after this date and time. If your name is not on the official student roster, you are not registered for the course.
August 31, 2010: First day instructor may approve request to add a student to a full course with a Schedule Adjustment Form. (Late start classes may be added up until the day the class starts).
September 6, 2010; Labor Day. CAMPUS CLOSED (no classes).
September 8, 2010 at 5 PM; Last day to add structured courses using a Schedule Adjustment Form with instructor signature without a written petition for a late add. This is an absolute deadline and is treated as such. This deadline does not apply to independent study, internships, project hours, thesis hours, dissertation hours, and late-starting modular courses.
September 8, 2010 at 5 PM; Last day to drop a fall 2010 course or completely withdraw from all fall 2010 courses with a tuition adjustment minus the drop charge and no transcript notation - this includes section changes. Drops after this date will appear on your transcript. Drops will require instructor approval and withdraw from all classes requires a dean's signature. This is an absolute deadline and is treated as such.
September 8, 2010 at 5 PM; Last day to request pass/fail or no credit option for a course.
September 8, 2010 at 5 PM: Last day for a graduate student to register for a Candidate for Degree.
September 8, 2010 at 5 PM: Last day for a Ph.D. student to petition for a reduction in hours.
September 8, 2010 at 5 PM: Last day to apply for fall 2010 graduation. Undergraduates must make an appointment and see your academic advisor before this date to apply for graduation if you are an undergraduate; graduate students must complete the intent to graduate and candidate for degree form.
September 20-29, 2010: Faculty can use the early alert system.
November 1, 2010 at 5 PM; Last day for students to drop or withdraw from all classes without approval from the student’s academic Dean.
November 15, 2010 at 5 PM; Last day CLAS students can obtain dean's permission to drop or withdraw without a full petition. This is treated as an absolute deadline.
After November 15, 2010 all schedule changes require a full petition. Undergraduates should contact their CLAS advisor, graduate students should contact the dean's office.
November 22-24; Fall Break (No Classes; Campus Open)
November 25; Thanksgiving Day Holiday (Campus Closed)
December 13-17, 2010 Finals Week.

No schedule changes will be granted once finals week has started. There are NO exceptions to this policy.