WISM538 : Quasi-Periodic Bifurcation Theory

Heinz Hanßmann, Yuri A. Kuznetsov

fall	time	place
lectures	monday 13:15 - 15:00	HFG 610

ECTS : 7.5 credit points


Additional lectures on tuesday 6 december (15:15-17:00) and on wednesday 11 january (13:15-15:00, last lecture); both also in HFG 610.

Description

In bifurcation theory one examines how the behaviour of a dynamical system can change under variation of external parameters. This concerns in particular equilibria and other invariant sets: these can cease to exist or change their stability behaviour. An important example is the so-called Hopf bifurcation, where an equilibrium loses stability and a (stable) periodic orbite branches off.

Quasi-periodic orbits lie dense on invariant tori. During the course the possible bifurcations of these objects are treated. An important example are repeated (quasi)-periodic Hopf bifurcations, where a periodic orbit leads to 2-tori, 3-tori, ... Next to this Hopf-Landau-Lifschitz scenario for the development of turbulence one has the Ruelle-Takens scenario in the resonances between the occurring frequencies.

Learning Goals

During the course the student will achieve knowledge of both KAM theory and bifurcation theory to the extent that these are applicable to bifurcations of invariant tori.

Assumed knowledge

Ordinary Differential Equations,
Analysis,
Introductory Dynamical Systems.

Examination

Presentation and home work excercises.

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 (ed. V.I. Arnold)

Springer (1988)

H.W. Broer, H. Hanßmann and F.O.O. Wagener

Quasi-Periodic Bifurcation Theory: the geometry of KAM

(in preparation)

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)

Yu. Kuznetsov

Elements of applied bifurcation theory

Applied Mathematical Sciences 112, Springer (1995)

Contents

Monday 12 September. Dissipative systems. Exercise 1 (pdf, ps).

Monday 19 September. Conservative systems. Exercise 2 (pdf, ps).

Monday 26 September. Circle mappings. Exercise 3 (pdf, ps).

Monday 3 October. Circle Mapping Theorem. Exercise 4 (pdf, ps).

Monday 10 October. Linear problems. Exercise 5 (pdf, ps).

Monday 17 October. Persistence. Exercise 6 (pdf, ps).

Monday 24 October. Persistence of invariant tori. Exercise 7 (pdf, ps).

Monday 31 October. On matrices depending on parameters. Exercise 8 (pdf, ps).

Monday 7 November. Bifurcations of equilibria and periodic orbits. Exercise 9 (pdf, ps).

Monday 14 November. Symplectic structures in Hamiltonian systems. Exercise 10 (pdf, ps).

Monday 21 November. Persistence in Hamiltonian systems. Exercise 11 (pdf, ps).

Monday 28 November. Bifurcations in Hamiltonian systems. Exercise 12 (pdf, ps).

Monday 5 December. Normal forms. Exercise 13 (pdf, ps).

Tuesday 6 December. Families of quasi-periodic attractors and their bifurcations.

Monday 12 December. Action angle variables. Parametrised KAM theory. Exercise 14 (pdf, ps).

Monday 9 January. Denjoy's theorem. Preservation of structure

Wednesday 11 January. Reversibility.