# Semester 1, 2010 - 2011. 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 - 11:05 AND Thursday 14:15 - 15:05.

We will use the slot Friday 15:15 - 16:05 occasionally for going through problem sheets.

**There will be no lectures or problem classes on**** Friday 15 Oct; Friday 22 Oct; Monday 25 Oct; Thursday 28 Oct; Friday 29 Oct; Thurs 16 Dec**
### Sketch of the course

... is available

here. Note that lectures will

**continue after Xmas**.

**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**.

Lecture 1 (Mon 4 Oct): Philosophy. The `dynamical systems viewpoint'. Brief description of thermal convection. Pictures and movies of pattern formation and dynamics.

Lecture 2 (Thurs 7 Oct): **1.1** Definitions: dynamical system, trajectory, eqm pt, fixed pt, periodic, homoclinic and heteroclinic orbits. Invariant sets. omega-limit set.

Lecture 3 (Fri 8 Oct): Lyapunov and quasi-asymptotic stability. Asymptotic stability. Attracting sets, attractors, basin of attraction. Linear stability of equilibria and fixed points. Floquet multipliers. Poincare sections. **1.2** Topological equivalence.

Lecture 4 (Mon 11 Oct): Linear stable and unstable subspaces. Local stable and unstable manifolds. Stable manifold theorem. Hartman-Grobman theorem. Linear systems in 2D. Predator-prey example. Structural stability.
Definitions of local and global bifurcation.

Lecture 5 (Thurs 14 Oct): **1.3** Local bifurcations in 1D: saddle-node, transcritical, pitchfork. Codimension. Unfoldings. Perturbed transcritical and pitchfork cases. Oscillatory bifurcation (in normal form).

Lecture 6 (Mon 18 Oct): Poincare-Bendixson Theorem. Dulac's criterion. **1.4** Centre manifold theorem. Problem sheets 1 and 2 handed out.

Lecture 7 (Thurs 21 Oct): **1.5**
Normal form transformations. Normal form symmetry.
Example: Hopf bifurcation.

Lecture 8 (Mon 1 Nov):
**1.6**
Local bifurcations in maps: Floquet multipliers +1 and -1 (period-doubling). Complex FMs: normal form, definitions of strong, weak
and non-resonant cases. Discussion of the non-resonant case.

Lecture 9 (Thurs 4 Nov):
Dynamics in the weakly resonant case. Circle maps: rotation number, Arnol'd's *standard map*.
Arnol'd tongues and frequency locking.

Lecture 10 (Mon 8 Nov):
**2.** Global bifurcation theory. **2.1** Introduction. **2.2** The planar case (without symmetry).
Local and global parts of the return map. The saddle index.

Lecture 11 (Thurs 11 Nov):
**2.3** Real eigenvalues in 3D: without (simple or twisted cases), and with symmetry (Lorenz case).

Lecture 12 (Mon 15 Nov):
Chaos in the Lorenz case: the invariant Cantor set.

Lecture 13 (Thurs 18 Nov, in 4 West 4.8):
Brief interlude (note new section numbering scheme) on chaos. **C.1** 'Gap-Sawtooth' map.
**C.2** Symbolic dynamics. Sequence spaces, shift operator. Properties. (Semi)conjugacies.
Subshifts of finite type (SSFT).

Lecture 14 (Mon 22 Nov):
Horseshoes and chaos for 1D noninvertible maps. Period 3 implies orbits of all other periods.
**C.3** Dimensions. Topological dimension; box-counting dimension. Examples.

Lecture 15 (Thurs 25 Nov):
**C.4** The Smale Horseshoe and chaos in planar invertible maps.

Lecture 16 (Mon 29 Nov):
**C.5** Homoclinic points in planar maps. The 'lambda lemma' (statement only).

Lecture 17 (Thurs 2 Dec): The homoclinic tangle. **C.6** Horseshoes in the Lorenz case.

Lecture 18 (Mon 6 Dec): **2.4** Shilnikov scenario (complex pair of eigenvalues
in 3D). **2.5** Higher-dimensional global bifurcations.

Lecture 19 (Thurs 9 Dec, in 4 West 4.8): **3.1** Takens--Bogdanov
bifurcation without symmetry.

Lecture 20 (Mon 13 Dec): **3.2** Saddle-node-Hopf bifurcation. Breaking of the normal
form symmetry. Homoclinic orbits near the global bifurcation (heteroclinic connection).

**** Xmas break ****

Lecture 21 (Mon 10 January 2011): Bifurcations with symmetry.

Lecture 22 (Fri 14 Jan):

Lecture 23 (Mon 17 Jan):

Lecture 24 (Thurs 20 Jan):

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 Sheet 4. Bifurcations with symmetry.

**Problem class 1 (sheets 1 and 2): Friday 12 November**

**Problem class 2 (sheet 3, Qns 1-3): Friday 26 November**

**Problem class 3 (sheet 3, Qns 4 and 5): Friday 3 December**

**Problem class 4 (sheet 4): Friday 21 January 2011**