MTH 442 Introduction to Difference Equations – Spring 2013

Instructor: Orlando Merino, 101C Lippitt Hall, 8744442, merino@math.uri.edu

Time/place:   TuTh 8:00 a.m.   Lippitt Hall 205

Text: There is no official text. You will be given access (through SAKAI) to an electronic version of the necessary material and course notes. A good reference is: Discrete Dynamical Systems and Difference Equations with Mathematica, by M. Kulenovic and O. Merino, Chapman & Hall/CRC, 2002, ISBN 1-58488-287-5

Grading: Two Exams (20% ea.), Homework and CAS Projects (40 %), Final Presentation and Project (20%).

About the Course: This course is an introduction to the basic concepts and techniques of difference equations for advanced undergraduates and beginning graduate students. Difference equations appear in situations where the (n + 1)st generation (or state) of a system depends upon some previous generations (or states). Such equations also appear naturally as discrete analogues of differential equations, and as numerical solutions of differential equations that model various diverse phenomena in biology, ecology, physiology, physics, engineering, economics, and other areas. In addition to performing mathematical analysis of difference equations, with the aid of a Compute Algebra System (Mathematica) you will experiment with difference equations, and discover that such equations possess fascinating properties with a great amount of structure. Some of these computer observations may be cast as theorems that you discover and prove!

Topics: Introduction to Difference Equations, First order DEs, linear equations with constant coefficients, variable coefficients, stability of equilibria in both hyperbolic and nonhyperbolic cases, bifurcations, period doubling bifurcations, chaotic behavior (one dimensional case), two dimensional systems of difference equations, linear theory, nonlinear systems, equilibria, stability, periodic solutions, basins of attraction, stable and unstable manifolds, area preserving maps, systems with order higher than 2, numerical issues in difference equations.

M. Kulenovic's DE website: www.math.uri.edu/~kulenm/diffeqaturi/dehomepage.html

The Dynamical Systems and Techn. Project at Boston University: math.bu.edu/DYSYS/

Discrete Dynamical Syst. and Nonlinear Difference Eqs website: www.discretedynamics.net/

Dynamical Systems at Scholarpedia: www.scholarpedia.org/article/Dynamical_systems