Welcome to MATH 741 - APPLIED MATHEMATICS I

Fall 2010
Time: 10:30-12:00 on Wednesdays and Fridays in HH/207
Place: HH/207

Instructor: Dr. Bartosz Protas
Office: HH 326, Ext. 24116
Office hours: Wednesdays 1:00-3:00pm, or by appointment

Announcements:

• Some students have reported problems downloading Maple worksheets from the course website. I guess the problem may be with the configuration of the browser which may not recognize the *.mw files. What you can try to do is save these files without actually opening them. On the other hand, if your save the files as *.xm, and then manually change the extension to *.mw, it should work too.

• The final exam is available here. Is due electronically by midnight on Monday, December 6. Good Luck!

• Solutions to Assignments #3 and #4 are posted (see links on the left); solutions to the final exam are available here: Question #1, Question #4 and Questions #2, #3 and #4

Outline of the Course:

The main goal of this course is to offer an introduction to classical methods of applied mathematics. We will focus on the qualitative theory of systems of ordinary differential equations (ODEs). Following a review of standard results concerning existence and uniqueness of solutions and their continuous dependence on parameters, we will study linear system, stability theory, invariant manifolds, ending with a survey of periodic and homoclinic solutions. A second objective of this course is to introduce students to modern methods of symbolic and numerical computing useful in quantitative analysis. We will use the software environment MAPLE to illustrate a number of problems discussed in the course. In the optimistic variant, the specific topics to be discussed will include (number is parentheses correspond to sections in the textbook by L. Perko):

1) Elements of the ODE Theory
a) existence of solutions (2.1, 2.2),
b) uniqueness of solutions (2.2)
c) dependence on parameters (2.3)
d) flows defined by differential equations (2.5)
2) Linear Systems and Stability
a) properties of linear systems (1.3, 1.4)
b) solutions with homogeneous systems with constant coefficients (1.6, 1.7, 1.8)
c) critical points and linearized stability (1.9, 2.6)
d) Lyapunov functions and nonlinear stability (2.9)
3) Hyperbolic Theory
a) stable and unstable manifolds of dynamical systems (2.7, 2.10)
b) linearization of hyperbolic systems (2.8)
c) center manifold and nonlinear stability (2.11, 2.12)
d) normal forms (2.3)
4) Periodic and Homoclinic Orbits
a) Floquet theory and stability of periodic solutions (3.3)
b) Poincare maps (3.4, 3.5)
c) Poincare--Bendixon theory (3.6, 3.7, 3.8)
d) index theory and separatrix orbits (3.12)
e) structural stability (4.1)

Primary Reference:

a) L. Perko, Differential Equations and Dynamical Systems, Third Edition, Springer, (2000), ISBN 0-387-95116-4.

Supplemental References:

b) S. Lynch, Dynamical Systems with Applications Using MAPLE, Second Edition, Birkhauser, (2010). ISBN 978-0-8176-4389-8.
c) R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, (1982). ISBN 0-12-497280-2.
d) J. Cronin, Ordinary Differential Equations - Introduction and Qualitative Theory, Third Edition, CRC Press, (2008). ISBN 987-0-8247-2337-8.

In addition to the above references, example MAPLE codes will be made available to students on the course webpage.

Prerequisites:

real analysis and basic differential equations; no programming skills in MAPLE are required

Homework Assignments:

There will be four homework assignment which may involve some elements of MAPLE programming. Submission method (electronic vs. hard copy) will be announced when an assignment is posted. The tentative post and due dates are indicated in the Table below

 # Post Date Due Date HW 1 Wednesday, September 29 Wednesday, October 6 HW 2 Wednesday, October 20 Wednesday, October 27 HW 3 Wednesday, November 10 Wednesday, November 17 HW 4 Wednesday, November 24 Wednesday, December 1

The final grades will be based on
a) four homework assignments (4 x 15% = 60%),
b) take-home final exam (40%).

The instructor reserves the right to alter your final grade, in which case, however, the grade may only be increased.