(Fall, 2007)

**Course Objectives:**
The course covers the qualitative theory of ordinary differential equations
and dynamical systems. After an introduction of main elements of the ODE theory (existence,
uniqueness, and smooth dependence, linear systems), properties of dynamical systems
(stability theory, invariant manifolds, hyperbolic and center points) are studied. The course finishes
at the theory of periodic and homoclinic solutions (Floquet multipliers, Poincare indices, Melnikov functions).
Since this is the first graduate course in applied mathematics, the material does not include
too special and advanced topics, but focuses 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 with Lyapunov functions, stable and unstable manifolds,
hyperbolic theory, center manifolds and nonlinear stability, periodic orbits and limit cycles, Poincare-Bendixcon
theory and phase spaces.

**Instructor:**
Dr. Dmitry Pelinovsky, HH-422, ext.23424,
e-mail: dmpeli@math.mcmaster.ca

**Hours:**

*Lectures:* Tuesday, Thursday (9:30-11:00), HH-207

*Office hours:* Monday, Thursday (14:30-15:30)

**Textbooks:**

*"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

*"Ordinary differential equations"*
by R.K. Miller and A.N. Michel (Academic Press, 1982)

**Assignments:**
Six home assignments will be distributed during the classes with specific deadlines.
The texts for assignments and solutions will be posted on the course webpage.

**Presentations:** The course is completed by student presentations.

**Marking scheme:**

Presentation - 40% Assignments - 60% |