Summer 2021

BMS-Course Dynamical Systems III:
Bifurcation Theory

Prof. Dr. Bernold Fiedler

Recitation sessions: Alejandro López Nieto

Schedule, Sommersemester 2021

Lecture:

Tuesday, 10:15-11:45

Tutorials:

Alejandro López, Wednesday 16:15-17:45

ONLINE COURSE!

Check the KVV site of the course for information on how to join the lecture. Please, don't hesitate to contact Alejandro in case you want to join, but don't have a KVV account.

Nachklausur:

The Nachklausur (resit exam) will take place on October 12, 10:00-12:00 in the usual online format, you can find more information here (PDF)

Pass Criteria

Solve correctly at least 25% of the assignments. Hand in solution attempts for at least 50% of the assignments.
Present a correct solution to an assignment on the blackboard in the recitation session at least once.
Pass the written exam.

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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 equiibrium. 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 Paralipomena.

Prerequisites are Dynamical Systems 1 and/or 2.

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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

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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)