WISM538 : Seminar

Bifurcation Theory

Heinz Hanßmann, Yuri Kuznetsov




fall time place
lectures monday 13:15-15:00 BBG 017

ECTS : 7.5 credit points


For monday 14 february the seminar was moved from BBG 020 to MIN 2.06. For the weeks after that, we again move to other rooms (BBG 106 for monday 21 february, MIN 0.09 for monday 7 march and BBG 017 from monday 14 march on). I'll also post the rooms at the specific dates below.




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. In Hamiltonian systems some state 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

  1. The Hamiltonian period-doubling bifurcation: Qi Li
  2. Cusp bifurcation of parabolic equilibria in Hamiltonian systems: Kostas Staikos
  3. Smale Horseshoe: Isis Marsman
  4. Saddle homoclinic bifurcation: Hannah van der Zande
  5. Saddle-focus homoclinic bifurcation: Hannah van der Zande
  6. Degenerate hyperbolic periodic orbits: Qi Li
  7. Saddle-saddle homoclinic bifurcation: Isis Marsman
  8. Umbilical bifurcation of equilibria with zero linear part: Kostas Staikos
  9. The quasi-periodic centre-saddle bifurcation: Machiel Kruger



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)

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

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

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)

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

S.G. Nikolov and V.M. Vassilev
Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation
Mathematics 9 352 (2021) 1-17

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

7. February (BBG 020). Introduction, local bifurcations of dissipative systems and of Hamiltonian systems, distribution of talks

14. February (MIN 2.06). Global bifurcations (pdf), distribution of (remaining) talks

21. February (BBG 106). The Hamiltonian period-doubling bifurcation (pdf)

7. March (MIN 0.09). Cusp bifurcation of parabolic equilibria in Hamiltonian systems

14. March (BBG 017). Smale Horseshoe

21. March (BBG 017). Saddle homoclinic bifurcation (pdf)

28. March (BBG 017). Degenerate hyperbolic periodic orbits

4. April (BBG 017). Umbilical bifurcation of equilibria with zero linear part (pdf)

2. May (BBG 017). Saddle-Saddle Homoclinic Bifurcation

9. May (BBG 017). Saddle-focus homoclinic bifurcation (pdf); the article of the homework is the above paper by Nikolov and Vassilev.

16. May (BBG 017). The quasi-periodic centre-saddle bifurcation