(Winter, 2005)

**Course Objectives:**
The course explains methods of analysis and computations of normal forms in
problems of stability and bifurcations. These problems occur in many dynamical
systems, derived in physical and engineering sciences. We shall consider both
finite-dimensional (ODEs) and infinite-dimensional (DDEs, PDEs) dynamical systems,
which may possess additional symmetries (e.g. reversibility, translations, and
gauge invariance). Reviewing theory of invariant manifolds, center manifolds
and normal forms, we shall classify typical bifurcations of equilibrium points,
periodic solutions and homoclinic orbits in dynamical systems.
Stability analysis of equilibrium points, periodic solutions and homoclinic orbits
is also recovered within the normal form
computations. The course is finished with the review of recent results on stability
of solitary waves in Hamiltonian dynamical systems.

**Topics:**
equilibrium points of dynamical systems, stable, unstable and center manifolds theorem, normal form
theorem, symmetries, saddle-node and pitchfork bifurcations,
Hopf bifurcations, fold and flip bifurcations, Melnikov integrals,
Poincare maps, invariant tori, stability of solitary waves.

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

**Hours:**

*Lectures:* Monday (2:30-4:00pm), HH/207; Thursday (2:30-4:00pm), HH/312

**Textbooks:**

*"Elements of applied bifurcation theory"*
by Yu.A. Kuznetsov (Springer-Verlag, 1998), ISBN 0-387-98382-1

*"Introduction to applied nonlinear dynamical systems and chaos"*
by S. Wiggins (Springer-Verlag, 2003), ISBN 0-387-00177-8

*"Topics in bifurcation theory and applications"*
by G.Iooss and M. Adelmeyer (World Scientific, 1998), ISBN 981-02-3728-6

*"Stability, instability and chaos"*
by P.Glendinning (Cambridge University Press, 1994), ISBN 0-521-41553-5

**Assignments:**
Five home assignments will be handed out in class on Mondays every second
week, starting the week of January 10. The assignments are due
on Thursday in the following week. The texts for assignments
and solutions will be posted on the course webpage.

**Presentations:** The course is completed by a series of student presentations.

**Marking scheme:**

Presentation - 40% Assignments - 60% |