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
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,
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.
Localised states and the Hamiltonian-Hopf bifurcation.
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.