Finite Dimensional Dynamical Systems : Perturbation Theory

Henk Broer, Heinz Hanßmann

fall	time	place
lectures	wednesday 10:00 - 12:45	BBL 107A

ECTS : 8 credit points

Course material includes: H.W. Broer, Notes on Perturbation Theory 1991, Erasmus ICP Mathematics and Fundamental Applications, Aristotle University Thessaloniki (1993), 44p. For main chapter Structure Preserving Normal Forms see here. H.W. Broer, M.C. Ciocci and A. Litvak-Hinenzon, Survey on dissipative KAM theory including quasi-periodic bifurcation theory based on lectures by Henk Broer, p. 303 -- 355 in Geometric Mechanics and Symmetry: the Peyresq Lectures (eds. J. Montaldi and T. Ratiu) LMS Lecture Notes Series 306, Cambridge University Press (2005)

Contents

Introduction, non-resonant normalization, Lie brackets. Homework: 2.3, 2.4.

Resonant and structure preserving normalization, Hopf bifurcation. Homework: 2.5, 2.8.

Normalization for the Hopf bifurcation, linear area-preserving and volume-preserving vector fields. Homework: 2.7, 1.1, 1.2.

Normalization of volume-preserving vector fields, structural stability. Homework: 1.3, 1.4, 2.6, 2.9.

Quasi-periodic motion, quasi-periodic attractors. Homework: 1.14, 2.10.

KAM theory of circle maps, small denominators. Homework: 1.18, 1.19 (still from `Notes..').

Toward a KAM theory of vector fields. Homework: 4, 13=15 (now from `Survey..').

Proof of the dissipative Main Theorem. Homework: 5, 6.

Floquet theory, the normal linear part of quasi-periodic tori. Homework: Logarithm of a matrix, 7=9.

The frequency-halving bifurcation. Homework.

The quasi-periodic Hopf bifurcation. Homework.

The quasi-periodic saddle-node bifurcation, application to the degenerate Hopf bifurcation for maps (Chenciner). Homework: 1.16, 1.17 (again from `Notes..').

Dynamics in the bubbles, the skew Hopf bifurcation.

The quasi-periodic centre-saddle bifurcation, concluding overview.