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.

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

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

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

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