MA50215 Specialist Reading Course / MA50206 Advanced Mathematical Study 2

Topics in Applied Dynamical Systems

Semester 2, 2015 - 2016


This course will cover a selection of advanced topics in nonlinear dynamics and applications to classic problems in hydrodynamics. The course will be broadly in four parts. In each part there will be a small number of lectures to set the scene. Students will then choose one part to pursue in more detail and bring material to read and discuss during the second half of the course.


Part 1. Normal forms and global bifurcations.
Part 2. Local bifurcations: Codimension 2 cases and equivariant bifurcations.
Part 3. Rayleigh-Benard convection.
Part 4. Pattern-forming instabilities.

Problem sheets

There will be 3 or 4 problem sheets to assist with recall from previous study and to provide examples to complement the lecture material. They will be posted here as the semester progresses.
Problem sheet 1

Course notes and handouts

Additional material will be posted here as the course progresses.
Summary of background material.

Suggested reading (to be updated)

P. Glendinning, Stability, Instability and Chaos. CUP, 1994
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, 1985 - SECOND EDITION!
S.H. Strogatz, Nonlinear Dynamics and Chaos. Westview Press, 1994

Lecture schedule and contents

Lectures will be held at 10.15am in 4 West 4.8, unless specified otherwise below.

Week 1: 1 - 5 February 2016
FRIDAY. LECTURE 1. Review of background material (see handout).

Week 2: 8 - 12 February 2016
TUESDAY. Chapter 1. 1.1 Normal forms. Near-identity coordinate changes. Example in 2D.
THURSDAY. Normal form symmetry. Example: Hopf bifurcation. Problem sheet 1 handed out.

Week 3: 15 - 19 February 2016
THURSDAY. 1.2 Global bifurcation theory. Planar cases.
FRIDAY. Planar cases continued. Lorenz scenario.

Week 4: 22 - 26 February 2016
TUESDAY. Lorenz scenario: simple and twisted cases.
WEDNESDAY. 11.15am Problem Class on sheet 1.
FRIDAY. Sheet 2 handed out. Lorenz scenario with symmetry. Gluing bif'n.

Week 5: 29 February - 4 March 2016
MONDAY. 2.15pm 1.3 Shilnikov scenario.
TUESDAY. 1.3 (cont) Chaos in the Shilnikov bif'n. Double-pulse homoclinics.
THURSDAY. Chapter 2. Codimension 2 bifucations. 2.1 Takens-Bogdanov without symmetry.
FRIDAY. WOLFSON LECTURE THEATRE 4W1.7. 2.2 Takens-Bogdanov with Z2 symmetry. 2.2.1. Case P=+1, Q=-1.

Week 6: 7 - 11 March 2016
MONDAY. 2.15pm. 2.2.2 Case P=-1, Q=-1.
TUESDAY. 2.2.2 Case P=-1, Q=-1 (continued). Chapter 3. Thermosolutal convection. Introduction.
THURSDAY. 3.1 Basic equations. Boussinesq approximation. 3.1.2 Static solution ande dimensionless parameters.
FRIDAY. Boundary conditions. 3.2 Linear theory.

Week 7: 14 - 18 March 2016


Week 8: 11 - 15 April 2016
MONDAY. 2.15pm 3.3 Modified perturbation theory for R_s=0.
THURSDAY. 3.4 The Lorenz equations.

Week 9: 18 - 22 April 2016
MONDAY. 2.15pm 3.5 Thermosolutal convection for R_s <>0. Truncation approach to the PDEs. Takens-Bogdanov bifurcation.
THURSDAY. Chapter 4.4.1 Introduction, and the Swift--Hohenberg equation. 4.2 The Ginzburg--Landau equation. Variational structure.

Week 10: 25 - 29 April 2016
TUESDAY. 10.15am. 4.3 Steady spatially-varying solutions to the GL equation. Phase plane sketches. The Eckhaus instability.
THURSDAY. 10.15am. The Eckhaus instability as a phase diffusion instability. 4.4 Symmetries and the GL equation; equivariance and normal form symmetry.

Week 11: 2 - 6 May 2016
TUESDAY. 10.15am. Kate to lead discussion.
THURSDAY. 10.15am. Jack to lead discussion.

Week 12: 9 - 13 May 2016
TUESDAY. 10.15am. Anna to lead discussion.