Mathematics 746: Stability and Bifurcations
(Winter, 2005)

Course Information

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:

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

"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%