Semester 1, 2011 - 2012. MA6000M: Topics in Applied Mathematics

Bifurcation Theory and Applications

This is a roughly graduate-level lecture course in bifurcation theory. The level and size of the audience will to a certain extent determine the speed and direction that the course actually ends up taking. Below is an outline of one possible collection of topics.

Essential Information

Location: Wolfson Lecture Theatre, 4 West 1.7 (i.e. Level 1).
Lecture times: Monday 10.15 - 12.05 AND Tuesday 15.15 - 16.05.
On Monday 10 October and Monday 14 November the lectures will be in 1WN 3.23 and only for 1 hour: 10.15-11.05.


Sketch of the course

... is available here. Note that lectures may continue after Xmas.

Related courses

You may also like to go to Robert MacKay's course given via the Taught Course Centre. Details can be found here. General details of Taught Course Centre courses can be found here. Lectures are on Thursdays 10am - 12noon in 3 West room 4.13 (Dept of Physics).

HEALTH WARNING Any notes and problem sheets below may well contain errors and omissions, even if these were pointed out and corrected in lectures or problem classes.

So far the contents of the course have been roughly as follows. Section numbers are given in bold.

Lectures 1-2 (Mon 3 Oct): Philosophy. The `dynamical systems viewpoint'. Brief description of thermal convection as a motivating example. Pictures and movies of pattern formation and dynamics. Definitions: equilibrium point, fixed point, periodic orbit, homoclinic/heteroclinic orbit, invariant set.

Lecture 3 (Tues 4 Oct): Stability: Lyapunov, quasi-asymptotic, asymptotic. Attracting sets, topological transitivity, attractors. Basin of attraction. Jacobian matrix. Poincare sections and return maps. Stable and unstable linear subspaces. Generalised eigenvectors (recall).

Lecture 4 (Mon 10 Oct): Local stable and unstable manifolds. Stable Manifold Theorem. Hartman-Grobman theorem. Phase plane sketching. Structural stability.

Lecture 5 (Tues 11 Oct): Definition of local and global bifns. Local bifns in 1D: saddle-node, transcritical and pitchfork. Codimension. Unfoldings.

Lecture 6-7 (Mon 17 Oct): Oscillatory (Hopf) bifurcation. Poincare-Bendixson Theorem. Dulac's criterion. Centre manifold theorem. Computation of extended centre manifold. Normal form.

Lecture 8 (Tues 18 Oct): Normal form symmetry. Local bifurcations in maps: the Floquest multiplier +1 and -1 cases (sn,t/c,pf; period-doubling).

Lecture 9-10 (Mon 31 Oct): Neimark-Sacker bifurcation. Dynamics in the non-resonant and weakly resonant cases. Circle maps. Frequency locking. Arnol'd tongues.

Lecture 11-12 (Mon 7 Nov): Global bifurcations: 2.1 Introduction. 2.2 The planar case. Chaos: 1.8.1 The gap-sawtooth map. Definitions of SDIC and TT. Proof that the gap-sawtooth map is SDIC and TT.

Lecture 13 (Tues 8 Nov): Symbolic dynamics. Semiconjugacy. Subshifts of finite type (SSFT). Definitions of horseshoe and chaos for 1D noninvertible maps.

Lecture 14 (Mon 14 Nov): 1.8.3 Dimensions: topological and box-counting. 1.8.4 Smale horseshoe construction.

Lecture 15 (Tues 15 Nov): 1.8.4 continued. 1.8.5. Homoclinic points in planar maps.

Lecture 16-17 (Mon 21 Nov): The Lambda Lemma. Chaotic dynamics in homoclinic tangles. 2.3 The Lorenz scenario: without, and then with, symmetry. Generation of complex dynamics: the `homoclinic explosion'.

Lecture 18-19 (Mon 28 Nov): 2.4 The Shilnikov scenario. Chapter 3: codimension-two local bifurcations. 3.1 Degenerate Hopf bifurcation. 3.2 Takens-Bogdanov bifurcation (without symmetry).

Lecture 20-21 (Mon 5 Dec): 3.3 Takens-Bogdanov bifuration with symmetry (2 cases). Gluing bifurcation. Chapter 4: Rayleigh-Benard and thermosolutal convection.

Lecture 22 (Tues 6 Dec): 4.1 (continued). Boussinesq approximation. Dimensionless numbers (Prandtl, Lewis, Rayleigh, solutal Rayleigh), streamfunction-vorticity formulation, boundary conditions. 4.2 Linear Theory.

Lecture 23-24 (Mon 12 Dec): 4.3 Modified perturbation theory for Rs=0. 4.4 Truncation of the PDEs: the Lorenz eqns. 4.5 Thermosolutal convection. 4.6 Truncation of the PDEs for Rs<>0.

Problem Sheet 1. Linear systems, phase portraits, local bifurcations in 1D.

Problem Sheet 2. Bifurcations in flows. (Extended) centre manifolds. Normal forms.

Problem Sheet 3. Global bifurcations. Takens-Bogdanov with Z2 symmetry.

Problem class 1 (sheet 1): Tuesday 1 November

Problem class 2 (sheet 2): Tuesday 22 November

Problem class 3 (sheet 2, continued): Tuesday 29 November