Semester 1, 2011 - 2012. 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 qualitative study of models for structures and instabilities in spatially-extended continuum systems. These model equations are typically collections of nonlinear parabolic partial differential equations. Throughout the course we will concentrate on understanding generic behaviours, and those parts of the dynamics that are both of physical interest and typical of a wide class of problems. Physical symmetries and asymptotic scalings play crucial roles. Examples motivated by fluid mechanics (in which context many of these ideas were first developed) will be discussed in some detail.

There will be a number of problem sheets and problem classes. The style of the course will be to develop intuition and link theory with applications. Theorems will in many cases be motivated, stated and discussed but proofs will not be given.

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

Part II

Low-dimensional behaviour and bifurcations in Rayleigh--B\'enard convection.

Pattern-forming instabilities. The Ginzburg--Landau and Newell--Whitehead--Segel equations. \mbox{Secondary} instabilities, e.g. Eckhaus.

Oscillatory instabilities: the complex Ginzburg--Landau equation. Benjamin--Feir instability. The Kuramoto--Sivashinsky equation.

Further topics (time/interest/audience permitting):

Phase instabilities of fully nonlinear patterns. The Cross--Newell equation.

Localised states and the Hamiltonian-Hopf bifurcation.

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.