ECTS : 7.5 credit points
Dynamical systems describe the evolution of the possible states
of the system (forming the phase space) as time varies.
In practical examples these systems depend on parameters:
for some coefficients the values are only approximately known and other
parameters enter from the outset as values to be controled and
adjusted.
Bifurcation theory studies how the behaviour of dynamical systems
changes under variation of parameters, especially where a
quantitatively small change of a parameter value leads to a
qualitative change in the dynamics.
In Hamiltonian systems some phase space variables can act as
parameters.
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
- Bifurcations of Equilibria: Vitorio Courtens
- Symmetric Bifurcations: Jan Pieter van der Plas
- Bifurcations of Periodic Orbits: Harm Verheggen
-
-
-
-
-
-
Literature
- R. Abraham and J.E. Marsden
- Foundations of Mechanics (2nd ed.)
- Benjamin (1978)
- 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)
- V.I. Arnol'd, V.V. Kozlov and A.I. Neishtadt
- Mathematical Aspects of Classical and Celestial Mechanics
- in Dynamical Systems III
- Springer (1988)
- H.W. Broer, I. Hoveijn, G. Lunter and G. Vegter
- Bifurcations in Hamiltonian systems:
Computing Singularities by Gröbner Bases
- LNM 1806, Springer (2003)
- R.H. Cushman and L.M. Bates
- Global Aspects of Classical Integrable Systems
- Birkhäuser (1997)
- 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 (4th ed.)
- Applied Mathematical Sciences 112, Springer (2023)
- J.E. Marsden
- Lectures on mechanics
- LMS Lecture Notes Series 174, Cambridge University Press (1992)
- J.E. Marsden and T.S. Ratiu
- Introduction to Mechanics and Symmetry
- Springer (1994)
- K.R. Meyer
- Generic bifurcation of periodic points
- Trans. AMS 149 (1970) 95-107
- K.R. Meyer and G.R. Hall
- Introduction to Hamiltonian Dynamical Systems and the
N-Body Problem
- Applied Mathematical Sciences 90, Springer (1992)
- 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
9. September.
Hamiltonian pitchfork bifurcation, symplectic form, Poisson bracket.
16. September.
Poisson space, symplectic leaves, Hamiltonian cusp bifurcation.
23. September.
No seminar.
30. September.
Bifurcations of Equilibria.
7. October.
Symmetric Bifurcations.
14. October.
Bifurcations of Periodic Orbits.
21. October.
28. October.
4. November.
11. November.
No seminar.
18. November.
25. November.
2. December.