WISM415 : Seminar

Bifurcations in Dynamical Systems

Heinz Hanßmann, Yuri Kuznetsov




fall time place
lectures monday 11:00-13:00 HFG 610

ECTS : 7.5 credit points


Our seminar is moved from BBG 017 to HFG 610 in the 6th floor of the Hans Freudenthalgebouw.




Dynamical systems describe the evolution of the possible states of the system (forming the state 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. This concerns both discrete and continuous dynamical systems.


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. Centre manifolds and the saddle-node bifurcation: Kieran Power
  2. Normal form of the Hopf singularity (pdf, ps): Ernst Röell
  3. Matrices depending on parameters (pdf, ps): Valesca Peereboom
  4. A Lie-theoretical approach to normal forms: Ernst Röell
  6. You may have your own suggestion: please discuss with us



Literature

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

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

Yu. Ilyashenko and Weigu Li
Nonlocal bifurcations
Mathematical Surveys and Monographs 66, American Mathematical Society (2004)

Y.A. Kuznetsov
Elements of Applied Bifurcation Theory
Applied Mathematical Sciences 112, Springer (2004)

B. Sandstede, T. Theerakarn
Regularity of Center Manifolds via the Graph Transform
J. Dyn. Diff. Equat. 27 (2015) 989-1006

L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev and L.O Chua
Methods of qualitative theory in nonlinear dynamics. Part I & II
World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises 4 and 5, World Scientific (1998) and (2001)




Contents

10. September. Introduction, distribution of (remaining) talks

17. September. Normal form of the Hopf singularity. Homework exercise (pdf, ps)

1. October. Centre manifolds and the saddle-node bifurcation. Homework exercise (pdf, ps)

8. October. Matrices depending on parameters. Homework exercise (pdf, ps)

15. October. A Lie-theoretical approach to normal forms. Homework exercise (pdf, ps)

22. October. Bifurcations in volume-preserving systems. Homework exercise (pdf, ps)