# NONLINEAR PATTERNS - 2023/4

Module code: MATM031

## Module Overview

Regular patterns arise naturally in many physical and biological systems, from hexagonal convection cells on the surface of the sun to stripes on a zebra's back. This course provides a basic framework for understanding the formation and evolution of these patterns using ordinary and partial differential equations and group theory.

### Module provider

Mathematics

BRIDGES Tom (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 79

Lecture Hours: 33

Seminar Hours: 5

Captured Content: 33

Semester 2

## Prerequisites / Co-requisites

MAT2048 Groups & Rings, MAT2011 Linear PDEs or PHY2065

## Module content

Indicative content includes:

• a review of flows and stationary points of ordinary differential equations, moving on to the concepts of centre manifolds and bifurcations. Simple bifurcations will be described and classified.

• group theoretic methods for analysing pattern-forming systems. Patterns on lattices and in boxes will be studied using symmetry groups. The Equivariant Branching Lemma and representations of groups will be covered.

• The module will conclude with the description of spatially-modulated patterns in terms of envelope equations. The Ginzburg-Landau equation will be derived and used to study the properties of stripes.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 mins) 20
Examination Exam (2 hrs) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate

• Subject knowledge through the recall of key definitions, theorems and methods.

• Analytical ability through the solution of unseen problems in the test and exam.

Thus, the summative assessment for this module consists of:

• One final examination worth 80% of the module mark.

• One in-semester test worth 20% of the module mark.

Formative assessment and feedback

Students receive written feedback via two marked coursework assignments over an 11 week period.  In addition, verbal feedback is provided by lecturer/class tutor at seminars and weekly tutorial lectures.

## Module aims

• To become familiar with a range of symmetry-based techniques for describing the behaviour of regular patterns that occur in nature or in laboratory experiments.

## Learning outcomes

 Attributes Developed 001 Be able to locate and classify codimension-one bifurcations of ordinary differential equations KC 002 Be familiar with the concept of a centre manifold K 003 Be able to identify the symmetry group relevant to simple pattern formation problems and use the Equivariant Branching Lemma in simple cases KC 004 Understand how to describe patterns using amplitude equations, and be able to find solutions of these equations in simple cases KC 005 Understand how to describe spatially modulated patterns using envelope equations, and be able to find solutions of these equations in simple cases KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

- A detailed introduction to nonlinear patterns and their analysis

- Experience (through demonstration) of the methods used to analyse, understand and solve problems involving nonlinear patterns.

The learning and teaching methods include:

• 3 x 1 hour lectures per week x 11 weeks, with projector-displayed written notes to supplement the module handbook and Q + A opportunities for students.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.