**Content**: How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.

However, this theory is not only of interest to the applied mathematician: indeed, it is an integral part of any rigorous mathematical training, and is developed here in a systematic way. Just as a `pure' subject like group theory can be part of the daily armoury of the `applied' mathematician , so ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology.

In this module we will cover relatively simple examples, first order equations

,

linear second order equations

and coupled first order linear systems with constant coefficients, for most of which we can find an explicit solution. However, even when we can write the solution down it is important to understand what the solution means, i.e. its `qualitative' properties. This approach is invaluable for equations for which we cannot find an explicit solution.

We also show how the techniques we learned for second order differential equations have natural analogues that can be used to solve difference equations.

The course looks at solutions to differential equations in the cases where we are concerned with one- and two-dimensional systems, where the increase in complexity will be followed during the lectures. At the end of the module, in preparation for more advanced modules in this subject, we will discuss why in three-dimensions we see new phenomena, and have a first glimpse of chaotic solutions.

**Aims**: To introduce simple differential and difference equations and methods for their solution, to illustrate the importance of a qualitative understanding of these solutions and to understand the techniques of phase-plane analysis.

**Objectives**: You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations) and to interpret their qualitative behaviour; and to do the same for simple difference equations.

**Books**:

The primary text will be:

J. C. Robinson *An Introduction to Ordinary Differential Equations*, Cambridge University Press 2003.

Additional references are:

W. Boyce and R. Di Prima, *Elementary Differential Equations and Boundary Value Problems*, Wiley 1997.

C. H. Edwards and D. E. Penney, *Differential Equations and Boundary Value Problems*, Prentice Hall 2000.

K. R. Nagle, E. Saff, and D. A. Snider, *Fundamentals of Differential Equations and Boundary Value Problems*, Addison Wesley 1999.

**Status for Mathematics students: **List B for third years. If numbers permit second and fourth years may take this module as an unusual option, but confirmation will only be given at the start of Term 2.

**Commitment: **10 two hour and10 one hour seminars (including some assessed problem solving)

**Assessment: **10% from weekley seminars, 40% from assignment, 50% two hour exam in June

**Prerequisites: **None

**Introduction**

This module gives you the opportunity to engage in mathematical problem solving and to develop problem solving skills through reflecting on a set of heuristics. You will work both individually and in groups on mathematical problems, drawing out the strategies you use and comparing them with other approaches.

**General aims**

This module will enable you to develop your problem solving skills; use explicit strategies for beginning, working on and reflecting on mathematical problems; draw together mathematical and reasoning techniques to explore open ended problems; use and develop schema of heuristics for problem solving.

This module provides an underpinning for subsequent mathematical modules. It should provide you with the confidence to tackle unfamiliar problems, think through solutions and present rigorous and convincing arguments for your conjectures. While only small amounts of mathematical content will be used in this course which will extend directly into other courses, the skills developed should have wide ranging applicability.

### Intended Outcomes

**Learning objectives**

The intended outcomes are that by the end of the module you should be able to:

- Use an explicit problem solving scheme to control your approach to mathematical problems
- Explain the role played by different phases of problem solving
- Critically evaluate your own problem solving practice

**Organisation**

The module runs in term 2, weeks 1-10

Thursday 14:00-15:00 OC0.04 (new teaching and learning building)

Friday 15:00-17:00 OC0.04

Most weeks the Thursday slot will be used for the weekly (assessed) problem session, but this will not be the case every week. You are expected to attend all three timetabled hours.

**Assessment Details**

- A flat 10% given for ‘serious attempts’ at problems during the course. Each week, you will be assigned a problem for the seminar. At then end of the seminar, you should present a ‘rubric’ of your work on that problem so far. If you submit at least 7 rubrics, deemed to be ‘serious attempts’, you will get 10%.
- One problem-solving assignment (40%) (deemed to be the equivalent of 2000 words) due by noon on Monday 20th March 2017 by electronic upload (pdf).
- A 2 hour examination in Summer Term 2017 (50%).

This module investigates how solutions to systems of ODEs (in particular) change as parameters are smoothly varied resulting in smooth changes to steady states (bifurcations), sudden changes (catastrophes) and how inherent symmetry in the system can also be exploited. The module will be application driven with suitable reference to the historical significance of the material in relation to the Mathematics Institute (chiefly through the work of Christopher Zeeman and later Ian Stewart). It will be most suitable for third year BSc. students with an interest in modelling and applications of mathematics to the real world relying only on core modules from previous years as prerequisites and concentrating more on the application of theories rather than rigorous proof.

Indicative content (precise details and order still being finalised):

1. Typical one-parameter bifurcations: transcritical, saddle-node, pitchfork bifurcations, Bogdanov-Takens, Hopf bifurcations leading to periodic solutions. Structural stability.

2. Motivating examples from catastrophe and equivariant bifurcation theories, for example Zeeman Catastrophe Machine, ship dynamics, deformations of an elastic cube, D_4-invariant functional.

3. Germs, equivalence of germs, unfoldings. The cusp catastrophe, examples including Spruce-Budworm, speciation, stock market, caustics. Thom’s 7 Elementary Catastrophes (largely through exposition rather than proof). Some discussion on the historical controversies.

4. Steady-State Bifurcations in symmetric systems, equivariance, Equivariant Branching Lemma, linear stability and applications including coupled cell networks and speciation.

5. Time Periocicity and Spatio-Temporal Symmetry: Animal gaits, characterization of possible spatio-temporal symmetries, rings of cells, coupled cell networks, H/K Theorem, Equivariant Hopf Theorem.

Further topics from (if time and interest):

Euclidean Equivariant systems (example of liquid crystals), bifurcation from group orbits (Taylor Couette), heteroclinic cycles, symmetric chaos, Reaction-Diffusion equations, networks of cells (groupoid formalism).