Online course!

Tuesday 10:15-11:45

The course will take place via Zoom:

Meeting-ID: 815 1182 4130
Password: dyn2020

Klausur:

The exam will take place on Tuesday July 13, 10:00-12:00. Specific information can be found here EXAM INFO.

Nachklausur:

The exam will take place on Tuesday October 12, 10:00-12:00, in the usual online format, more information can be found here.

Tutorials:

Wednesday 16:15-17:45

Via Zoom:

Meeting ID: 361 072 2559
Passcode: dyn2020

Course description:

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 nonzero eigenvalues of the linearization cross the imaginary axis as the parameter varies. This leads to Hopf bifurcation: the appearance of periodic oscillations around the equilibrium. Other invariant sets and heteroclinic connections, however, might also arise nearby. We will explore the bifurcation zoo and illustrate the theory by examples coming from physics, biology and other fields of applications.

See also the lecture "Dynamical Systems 3 - Paralipomena".

Prerequisites are Dynamical Systems 1 and/or 2.

Homework assignments:

Please form teams of two and hand in your joint solutions.

1. Assignment PDF, due 29.04.21.
2. Assignment PDF, due 06.05.21.
3. Assignment PDF, due 13.05.21.
4. Assignment PDF, due 20.05.21.
5. Assignment PDF, due 27.05.21.
6. Assignment PDF, due 03.06.21.
7. Assignment PDF, due 10.06.21.
8. Assignment PDF, due 17.06.21.
9. Assignment PDF, due 24.06.21.
10. Assignment PDF, due 05.07.21.

Basic questions:

Basic questions Bifurcation Theory PDF

Basic questions Paralipomena (PDF)

References:

• V.S. Afraimovich and S.-B. Hsu: Lectures on Chaotic Dynamical Systems, AMS (2003).

• K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer (1996).

• V.I. Arnol’d: Catastrophe Theory, Springer (1984).

• V.I. Arnol’d Geometrical: Methods in the Theory of Ordinary Differential Equations, Springer (1988).

• V.I. Arnold: Ordinary Differential Equations, Springer (1992).

• H. Broer, F. Takens: Dynamical Systems and Chaos, Springer (2011).

• P. Chossat, R. Lauterbach: Methods in Equivariant Bifurcations and Dynamical Systems, World Scientific (2000).

• S.N. Chow and J.K. Hale: Methods of Bifurcation Theory, Springer (1982).

• G. Dangelmayr, K. Kirchgässner, B. Fiedler and A. Mielke: Dynamics of Nonlinear Waves in Dissipative Systems, Addison Wesley (1996).

• B. Fiedler: Global Bifrucation of Periodic Solutions with Symmetry, Springer (1988).

• B. Fiedler and J. Scheurle: Discretization of Homoclinic Orbits, Rapid Forcing and “Invisible” Chaos, Memoirs of the AMS (1996).

• M. Golubitsky and I. Stewart: The Symmetry Perspective, Springer, Birkhäuser (2002).

• M. Golubitsky, I. Stewart and D.G. Schaeffer: Singularities and Groups in Bifurcation Theory, Volumes 1 and 2, Springer (1985, 1988).

• J. Guckenheimer and P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer (1983).

• M.W. Hirsch, S. Smale and R.L. Devaney: Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier (2004).

• A. Katok and B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems, Cambridge 1997.

• H. Kielhöfer: Bifurcation Theory, an Introduction with Applications to PDEs, Springer (2004).

• Y. Kuznetsov: Elements of Applied Bifurcation Theory, Springer (1995).

• S. Liebscher: Bifurcation without Parameters, Springer (2014).

• J.E. Marsden and M. McCracken: The Hopf Bifurcation and Its Applications, Springer (1976).

• J. Palis and W. de Melo: Geometric Theory of Dynamical Systems, Springer (1982).

• L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev and L.O. Chua: Methods of Qualitative Theory in Nonlinear Dynamics, World Scientific (2001).

• C. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer (1982).

• A. Vanderbauwhede: Center Manifolds, Normal Forms and Elementary Bifurcations, in Dynamics Reported Volume 2, John Wiley & Sons (1989).

• A. Vanderbauwhede: Local bifurcation and symmetry, Pitman (1982).

• J. Wainwright and G.F.R. Ellis: Dynamical Systems in Cosmology, Cambridge University Press (1997).

• Handbook of Dynamical Systems, Volumes 1-3, Elsevier (2002-2010).

• Encyclopaedia of Mathematical Sciences: Dynamical Systems, Volumes 1-5, Springer (1994).

• Scholarpedia: Dynamical Systems, doi:10.4249/scholarpedia.1629