19215701

SoSe 21: Dynamical Systems 3: Bifurcation Theory

Bernold Fiedler

Kommentar

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 "Dynamical Systems 3 - Paralipomena".

Prerequisites are Dynamical Systems 1 and/or 2.

Literaturhinweise

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.

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