Course Objectives: The course covers the qualitative theory of ordinary differential equations and the concepts of dynamical systems. The main focus is on analytical methods for differential equations in applications to physically relevant nonlinear systems. After an introduction of main elements of the ODE theory (existence, uniqueness, properties of the flow), we will study properties of dynamical systems (stability theory, invariant manifolds, center manifolds and normal forms, bifurcations of critical points) and finishing with periodic and homoclinic orbits (Floquet theory, Melnikov functions). Within the first graduate course in applied mathematics, we shall avoid too special and advanced topics, but focus on general methods and theorems.
Topics: existence and uniqueness of solutions, continuous dependence on initial data and parameters, general theory of linear ODE systems, stability theory, stable and unstable manifolds, periodic orbits and limit cycles, center manifolds and normal forms, homoclinic orbits, averaging, local and global bifurcations.
Instructor: Dr. Dmitry Pelinovsky, HH-422, ext.23424, e-mail: email@example.com
Lectures: Tuesday (10:30-12:00), Wednesday (12:00-13:30), HH-410
Office hours: Monday, Thursday (11:30-12:30)
"Differential equations and dynamical systems" by L. Perko (Springer-Verlag, 2001), ISBN 0-387-95116-4
"Ordinary differential equations with applications" by C. Chiconi (Springer-Verlag, 1999), ISBN 0-387-98535-2
Assignments: Six home assignments will be handed out, starting the week of September 12, 2005. The texts for assignments and solutions will be posted on the course webpage.
Presentations: The course is completed by a sequence of student presentations.
Presentation - 40%
Assignments - 60%