Semester 1, 2010 - 2011. MA6000M: Topics in Applied Mathematics

Bifurcation Theory and Applications

Sketch of the course

This course will introduce ideas and methods from nonlinear dynamics which are widely and routinely used to understand models of a wide range of physical systems, for example fluid flows, population dynamics, chemical reactions and coupled oscillators. The `dynamical systems viewpoint' is to concentrate on features of the dynamics that are independent of the coordinate system, for example the long-term behaviour that the system `settles down to'.

The first half of the course will be concerned with the qualitative behaviour of solutions to nonlinear ordinary differential equations, with an emphasis on structural changes in response to variations in parameters (bifurcation theory). There will be a brief discussion of the generation of complicated dynamics.

The second half of the course will extend these ideas to the case of symmetric systems. Symmetry naturally arises either from physical constraints, or from modelling assumptions. The existence of symmetry in the differential equations is, however, not `generic'. This change in genericity comes with additional structure that provides ways of understanding the notion of a `typical' bifurcation in the presence of symmetry, and typical dynamical behaviours. Basic ideas from group theory and representation theory will be developed as needed to describe the action of the group on the phase space of the dynamical system. Local bifurcations will be discussed in some detail, after which a variety of directions are possible including applications to pattern formation in two and three dimensions, heteroclinic cycling and coupled cell systems.

Any time remaining will be used to discuss problems of current interest in the field, for example (and these are given only as examples) the extension of ODE methods to pattern-forming PDEs such as the Swift--Hohenberg equation, the generation of localised states near subcritical Turing instabilities, and the development of bifurcation theory for locally-symmetric systems.

There will be a number of problem sheets and problem classes.

Detailed course outline

Part I

Introduction (through which we will go quite fast): phase space and the qualitative description of solutions to ODEs. Topological equivalence, hyperbolicity and structural stability of flows. Stable and unstable manifolds. Codimension--one local bifurcations in flows and maps. Centre manifolds. Reduction to normal forms; normal form symmetries.

Global bifurcations: Lorenz and Shil'nikov mechanisms. Codimension-two bifurcations: degenerate Hopf, Takens--Bogdanov.

Part II

Groups and their irreducible representations. Fixed point subspaces and computation of their dimension. Isotropy subgroups. Steady-state bifurcations: the Equivariant Branching Lemma.

Stability. Isotypic decomposition. Computing invariants and equivariants.

Spatio-temporal symmetry. The Equivariant Hopf Theorem.

Steady-state planar pattern formation. Square and hexagonal lattices. Scalar and pseudoscalar cases.

Coupled cell systems. Robust heteroclinic cycles.

Further topics (time/interest/audience permitting):

Pattern-forming PDEs. Multiple-scales reduction to the Ginzburg--Landau equation. Eckhaus instability.

Localised states and the Hamiltonian-Hopf bifurcation.

Synchronisation.

Groupoid symmetry. Takens--Bogdanov as a codimension-one bifurcation.

Applications to fluid mechanics: Rayleigh--B\'enard convection, Taylor--Couette flow.

The above outline of the course is available as a PDF here. This PDF also contains details of potentially useful books, although we won't be following one in particular for the whole course. Much of the introductory material can be found in every serious nonlinear dynamics textbook.