WISM538 : Seminar

Symmetry and Bifurcations

Heinz Hanßmann




spring time place
lectures tuesday 15:15-17:00 MIN 011

ECTS : 7.5 credit points





An equilibrium of a dynamical system can bifurcate if, under parameter variation, the linearisation encounters purely imaginary eigenvalues. For a single pair ±iµ  the equilibrium persists throughout the resulting Hopf bifurcation, splitting off a periodic orbit. Zero eigenvalues let the equilibrium disappear during the bifurcation.

Symmetry already under a finite group enforces an equilibrium with eigenvalue µ = 0 to persist throughout the bifurcation as well. Lie group symmetries have an even greater influence on the dynamics, e.g. rendering a Hamiltonian system integrable (and also to be Hamiltonian in the first place can be phrased as a symmetry of the system).

We study the impact of symmetry on bifurcations of dynamical systems. For this we use tools from algebra like representation theory. Symmetries in bifurcations form a vast research area and the choices we have to make will be taylored to the interests of the participants of the seminar.


Each week one lecture is given on a particular topic. The lecturer also constructs an exercise for all other students, which is not too difficult (at least, not more than one or two hours work). Students have to hand in these exercises one week later, and who constructed the exercise grades the solutions handed in on a scale from 1 to 10.

Assumed knowledge

A good basic knowledge of differential equations.

Examination

The presentations (80%) and the home work excercises (20%).


Subjects for presentation

  1. Unfolding Theory (Myrthe van Leeuwen)
  2. Bifurcations with Z2-Symmetry (Aurora Faure Ragani)
  3. Lyapunov-Schmidt Reduction (Rita Mak)
  4. The Hopf Bifurcation (Gert Jan Gelderman)
  5. Monodromy (Erwin Dijkstra)
  6. Continuation methods (Aurora Faure Ragani)
  7. Symmetry-Breaking in Steady-State Bifurcation (Rita Mak)
  8. The Hamiltonian Hopf Bifurcation (Erwin Dijkstra)



Literature

V.I. Arnold
Geometrical Methods in the Theory of Ordinary Differential Equations
Springer (1983)

V.I. Arnold
Mathematical Methods of Classical Mechanics (2nd ed.)
GTM 60, Springer (1989)

K. Efstathiou
Metamorphoses of Hamiltonian systems with symmetries
LNM 1864, Springer (2005)

J. Guckenheimer and P. Holmes
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (2nd ed.)
Springer (1986)

M. Golubitsky and D.G. Schaeffer
Singularities and Groups in Bifurcation Theory I
Applied Mathematical Sciences 51, Springer (1985)

M. Golubitsky, I. Stewart and D.G. Schaeffer
Singularities and Groups in Bifurcation Theory II
Applied Mathematical Sciences 69, Springer (1988)

H. Hanßmann
Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems
LNM 1893, Springer (2007)

Y.A. Kuznetsov
Elements of Applied Bifurcation Theory
Applied Mathematical Sciences 112, Springer (2023)

J. Montaldi and T. Ratiu
Geometric Mechanics and Symmetry: the Peyresq Lectures
LMS Lecture Notes Series 306, Cambridge University Press (2005)

D.H. Sattinger
Group Theoretic Methods in Bifurcation Theory
LNM 762, Springer (1979)




Contents

6. February. Introduction

13. February. Distribution of (remaining) talks

27. February. Unfolding Theory

5. March. Bifurcations with Z2-Symmetry

12. March. Monodromy

19. March. Lyapunov-Schmidt Reduction

26. March. Continuation methods

9. April. The Hopf Bifurcation

16. April. Symmetry-Breaking in Steady-State Bifurcation

23. April. The Hamiltonian Hopf Bifurcation