MTH 442 Introduction to Difference Equations – Spring 2009

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

Text: We will use an unpublished book, which I will make available in electronic form.

Grading: Two Exams (20% ea.), Homework (20%), Maple hw (20 %), Final 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 computer (and Maple) 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, stair-step diagrams, linear equations with constant coefficients, linear equations with variable coefficients, equilibrium points and local stability, non-hyperbolic equilibrium points, global attractors and stability, periodic points and local stability, introduction to chaos, introduction to bifurcation theory, applications of first order equations, basic theory of second order linear equations, homogeneous equations with constant coefficients, the method of undetermined coefficients, variation of parameters, asymptotic behavior of equations with constant coefficients, reduction of order, applications of second order equations, linear systems of DEs with constant coefficients: basic theory, stability, nonlinear systems, applications.

Difference Equations at URI: 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