Bifurcation Theory

Recommended Prerequisites

Basic knowledge of Linear Algebra, Calculus of several real variables, Group Theory, Rings and Modules, and Ordinary Differential Equations.

Teaching Methods

Lectures and tutorials: The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts. There are also tutorials, where exercises and related problems are solved.

Learning Outcomes

The students will be provided with the basic tools for studying bifurcations in differential equations and difference equations.

Work Placement(s)

No

Syllabus

Introduction to the study of qualitative changes in differential equations and difference equations with parameters. These changes include, for instance, the creation and destruction of equilibrium states, changes in periodic behaviour of solutions, changes in stability and transition to chaotic behaviour. In particular: fold or saddle-node points, pitchforks, higher codimension singularities of equilibria or fixed points; bifurcation at homoclinic cycles; Hopf bifurcation; period doubling; period doubling cascades.

Special classes of dynamical systems may also be treated, for instance differential equations with symmetry or coupled cell systems.

Assessment Methods

Assessment
Exam: 100.0%

Bibliography

M. Golubitsky, I.N. Stewart, and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. 2, Applied Mathematical Sciences 69, Springer-Verlag, New York, 1988.

M. Golubitsky and D.G. Schaeffer, Singularities and groups in bifurcation theory, vol. I, Applied Mathematical Sciences 51, Springer-Verlag, New York, 1985.

M. Golubitsky, I.N. Stewart, and D.G. Schaeffer, The Symmetry Perspective: from equilibrium to chaos in phase space and physical space, Birkhäuser, Basel, 2002.

J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York, 1983.

Y.A. Kuznetsov, Elements of applied bifurcation theory, Springer-Verlag, New York, 1995.