WISM532 : Perturbation Theory

Heinz Hanßmann

fall	time	place
lectures	wednesday 11:00 - 13:00	room 610

ECTS : 7.5 credit points

On wednesday 24 october the course takes place in MIN 012; no course on wednesday 21 november.

Perturbation theory treats complicated systems as small perturbations of simpler systems and aims to use these for a description of the dynamics. In many models the simplifying assumptions (e.g. on the symmetry of the problem) already lean towards this direction. A mathematical theory that constructs to a given system a better amenable approximation is the theory of normal forms of dynamical systems. After an introduction to normal forms I will in particular treat quasi-periodic motions.

Literature

V.I. Arnold

Geometrical Methods in the Theory of Ordinary Differential Equations

Springer (1983)

V.I. Arnold, V.V. Kozlov and A.I. Neishtadt

Mathematical Aspects of Classical and Celestial Mechanics

in Dynamical Systems III

Springer (1988)

H.W. Broer

Normal forms in perturbation theory

p. 6310-6329 in Encyclopedia of Complexity and System Science (ed. R.A. Meyers)

Springer (2009)

H.W. Broer, F. Dumortier, S.J. van Strien and F. Takens

Structures in dynamics

Finite-dimensional deterministic studies

North-Holland (1991)

H.W. Broer and H. Hanßmann

Perturbation theory (dynamical systems)

Scholarpedia (2008)

M.C. Ciocci, A. Litvak-Hinenzon and H.W. Broer

Survey on dissipative KAM theory including quasi-periodic bifurcation theory

Chapter 5 of Geometric Mechanics and Symmetry: the Peyresq Lectures (eds. J. Montaldi and T.S. Ratiu)

LMS Lecture Notes Series 306, Cambridge University Press (2005)

J. Guckenheimer and P. Holmes

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (2nd ed.)

Springer (1986)

M.W. Hirsch, C.C. Pugh and N. Shub

Invariant manifolds

Lecture Notes in Mathematics 583, Springer (1977)

Yu. Kuznetsov

Elements of applied bifurcation theory

Applied Mathematical Sciences 112, Springer (1995)

J. Moser

Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics

Annales of Mathematics Studies 77, Princeton University Press (1973)

M.B. Sevryuk

Reversible systems

Lecture Notes in Mathematics 1211, Springer (1986)

S. Wiggins

Introduction to applied nonlinear dynamical systems and chaos

Texts in Applied Mathematics 2, Springer (1990)

Contents

12. September. Introduction.

19. September. Simplifying co-ordinates. The last exercise (pdf, ps) is homework.

26. September. Hyperbolic equilibria, Lie brackets.

3. October. Normal form theory. The last exercise (pdf, ps) is homework.

10. October. Reversible systems, resonant normal form.

17. October. Hopf bifurcation. The last exercise (pdf, ps) is homework.

24. October. Hopf bifurcation, normally hyperbolic invariant manifolds.

31. October. Floquet theory, periodic normalization. The last exercise (pdf, ps) is homework.

7. November. Normal forms for mappings, periodic Hopf bifurcation.

14. November. Linearization theorem of Siegel, Diophantine conditions. The last exercise (pdf, ps) is homework.

28. November. Quasi-periodic motions and attractors, theorem of Denjoy.

5. December. Families of conditionally periodic tori, theorem of Moser on persistence of Diophantine tori in reversible systems. The last exercise (pdf, ps) is homework.

12. December. Persistence of lower-dimensional tori in reversible systems.

19. December. Bifurcations of lower-dimensional tori in reversible systems, quasi-periodic Hopf bifurcation. The exam (pdf, ps) is due wednesday 16 january.