Summer 2015

BMS-Course Perturbation Theory

Dr. Juliette Hell

Schedule, Summer 2015

Lecture:

Wednesday 12.15-14.00, seminar room 130, Arnimallee 3

Preceded by Dynamical Systems III Wednesday 10.15-12.00, seminar room 130, Arnimallee 3.

Written examination:

Wednesday, July 15, 12h00-14h00, seminar room 130, Arnimallee 3

Results to be seen at office door Dr. Juliette Hell only

Written examinations might be checked individually in the following time frame: August 17 - August 28. Please make an appointment with Dr. Juliette Hell first.

Correction session to prepare for the resit:

Thursday, October 1st, 10h00-14h00, seminar room 140, Arnimallee 7

Written examination (resit):

Friday, October 16, 10h00-12h00, seminar room 140, Arnimallee 7

Results (PDF) (upload October 22st, 2015)

Topics

The general idea of perturbation theory is to write a complicated system as the sum of a simpler dynamical system for which information about the dyamics is available, and a small perturbation. We will study methods adapted to various type of problems such as averaging, systems with different time scales, geometric singular perturbation theory, analysis of degenerate equilibria via blow-up,...

Bifurcation theory is the study of qualitative changes of the dynamics as a parameter of the system varies. We will focus on local bifurcations for vector fields. A typical situation is when the vector field admits an equilibrium where an eigenvalue of the linearization crosses the imaginary axis as the parameter varies. With the sign of the (real part of the) eigenvalue changes the stability of the equilibrium. But also other invariant sets and heteroclinic connections might pop up nearby. The nature of the dynamics bifurcating from the reference equilibrium depends on the nonlinearity and the dimension of the parameter. The appearance of an eigenvalue with zero real part at the critical parameter value suggests that center manifolds will play an important role. We will explore the bifurcation zoo and illustrate the theory by examples coming from physics, biology and other fields of applications. Depending on the interests of the audience and the time available, we might make excursions to some of the following topics: bifurcations in discrete dynamical systems, in PDE's, bifurcations and symmetries, global bifurcations, bifurcation without parameter.

Bifurcation and perturbation theory are deeply related and often combined. Therefore we strongly recommand to attend both lectures.

Prerequisites are Dynamical systems I and II.

References

• K.T. Alligood, T.D. Sauer and J.A. Yorke: Chaos, Springer, 1997.
• H. Amann: Ordinary Differential Equations, de Gruyter, 1990.
• V.I. Arnold: Ordinary Differential Equations, Springer, 2001.
• V.I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1988.
• W.E. Boyce and R.C. DiPrima: Elementary Differential Equations and Boundary Value Problems, Wiley, 5th edition, 1992.
• S.-N. Chow and J.K. Hale: Methods of Bifurcation Theory, Springer, 1982.
• E.A. Coddington and N. Levinson: Theory of ordinary differential equations, McGill-Hill, 1955.
• P. Collet and J.-P. Eckmann: Concepts and Results in Chaotic Dynamics. A Short Course, Springer, 2006.
• R. Devaney, M.W. Hirsch and S. Smale: Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2003.
  (This is the updated version of
  M.W. Hirsch and S. Smale: Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.)
• Dynamical Systems I, D.K. Anosov and V.I. Arnold (eds.), Encyclopaedia of Mathematical Sciences Vol 1, Springer, 1988.
• J. Hale: Ordinary Differential Equations, Wiley, 1969.
• B. Hasselblatt, A. Katok: A First Course in Dynamics, Cambridge 2003.
• P. Hartmann: Ordinary Differential Equations, Wiley, 1964.
• A. Katok, B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems, Cambridge 1997.
• F. Verhulst: Nonlinear Differential Equations and Dynamical Systems, Springer, 1996.

Homework assignments, Summer 2015

Archive Summer 2014, Winter 2014/2015