MATH 4193.001/5093.001 - Introductory Mathematical Modeling / Mathematical Models - Spring 2017
TR 12:00--1:15 p.m., 359 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Monday 2:30-3:30, Wednesday 10:30-11:30, or by appointment, in 802 PHSC.

Course catalog description:
For 4193: Prerequisite: 3113 or 3413, 3333, 4733 or 4753, or permission of instructor. Mathematics models are formulated for problems arising in various areas where mathematics is applied. Techniques are developed for analyzing the problem and testing validity of proposed model. (F)
For 5103: Prerequisite: permission of instructor or admission to the M.S. program. May be repeated with change of content; maximum credit six hours. Mathematical models are formulated for problems arising in various areas in which mathematics has been applied. In each case, techniques are developed for analyzing the resulting mathematical problem, and this analysis is used to test the validity of the model. (Sp)

Text: We will use parts of the books below (all of them are feely available in pdf format for OU students from the OU library):
G. S. Layek. An Introduction to Dynamical Systems and Chaos, Springer, 2015.
M. W. Hirsch, S. Smale, R. L. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd edition, Academic Press, 2012.
J. D. Murray. Mathematical Biology, Vols. 1 and 2, 3rd edition, Springer, 2002, 2003.
A. C. King, J. Billingham, S. R. Otto. Differential Equations: Linear, Nonlinear, Ordinary, Partial. Cambridge University Press, 2003.
R. Seydel. Practical Bifurcation and Stability Analysis. 3rd edition, Springer, 2010.

A brief (tentative) list of topics to be covered:

A brief description of the class: The class will study basic theory of nonlinear dynamics. Since a nonlinear system of ODEs does not satisfy the Superposition Principle, finding the general solution of such a system is very complicated and often impossible. Because of this, one can instead try to describe the qualitative behavior of the solutions of the system.
In this class we will derive nonlinear (systems of) ODEs occurring in some simple mechanical problems, and will analyze the behavior of their solutions. We will discuss briefly questions of existence and uniqueness of solutions, and will develop some methods for studying their bifurcations, i.e., situations in which the solutions of the system change their qualitative behavior dramatically for a small change of some parameter. We will study the phase portraits of autonomous linear systems, some bifurcations occurring in such systems (saddle-node, transcritical, pitchfork, Hopf), presence or absence of certain types of asymptotic behavior of the solutions, relaxation oscillations, limit cycles, hysteresis, Poincaŕe aps, etc. We will discuss the famous Lorenz system and similar systems exhibiting highly irregular behavior ("chaos"). We will also study some concepts related to highly iterated maps, i.e., the behavior of , where the function (”map”) is composed with itself times, for . We will study bifurcations in some maps, their periodic orbits, Lyapunov exponents, and the universal behavior of such maps. If time permits, we will touch on different concepts of dimension that are important in the context of nonlinear dynamics.

Homework:

Content of the lectures:

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also expected. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Homework: Homework will be assigned regularly and will be posted on the this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. The lowest homework grade will be dropped. Giving just an answer to a problem is not worthy any credit - you have to write a complete solution which gives your step-by-step reasoning and is written in grammatically correct English. Although good exposition takes time and effort, writing your thoughts carefully will greatly increase your understanding and retention of the material.
You are allowed to discuss the homework problems with the other students in the class. However, each of you will need to prepare individual solutions written in your own words - this is the only way to achieve real understanding!
Your homework should have your name clearly written on it, and should be stapled. Please write the problems in the same order in which they are given in the assignment. No late homework will be accepted!

Exams: There will be one in-class midterm and a (comprehensive) final.
A tentative date for the midterm is March 23 (Thursday).
The final exam is scheduled from 1:30 to 3:30 p.m. on May 8 (Monday).
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking any of the exams at the scheduled time.

Grading: Your grade will be determined by your performance on the following coursework:

Coursework Weight
Homework (lowest grade dropped) 45%
Midterm exam 20%
Final exam 35%

Students who are taking the class for a graduate credit (i.e, as MATH 5103) will have some additional homework problems and will have to complete a final project.

Technology: We will frequently engage in computations to illustrate the mathematical results that we discuss. You will have to do some elementary programming in Mathematica and MATLAB which are avaliable on the computers in the University's computer labs. The amount of programming you will need to do will be small, and previous programming experience is not assumed.

Policy on W/I Grades : You can withdraw from the course with an automatic "W" no later than March 31 (Friday) for undergraduate students and February 24 (Friday) for graduate students. Dropping after April 3 (Monday) requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates in the academic calendar!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Code.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

MATLAB tutorials:

Good to know: