# NONLINEAR PATTERNS - 2025/6

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 module provides a mathematical framework for understanding the formation and evolution of these patterns using ordinary and partial differential equations and group theory.

The module builds on group theory from MAT1031 Algebra and MAT2048 Groups & Rings. The module also builds on ordinary differential equations from MAT2007 Ordinary Differential Equation and partial differential equations from MAT2011 Linear PDEs.

### Module provider

Mathematics & Physics

BRIDGES Tom (Maths & Phys)

## Overall student workload

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

Semester 1

None.

## Module content

Indicative content includes:

• A review of flows and stationary points of ordinary differential equations.

• Centre manifolds and bifurcations. Classification of simple bifurcations.

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

• Spatially-modulated patterns in terms of envelope equations. The Ginzburg-Landau equation and the properties of stripes.

## Assessment pattern

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

N/A

## Assessment Strategy

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

• Understanding of subject knowledge, and recall of key definitions and results in the theory of nonlinear patterns.

• Experience of the methods used to analyse and solve real-world problems involving pattern formation and evolution.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 to 3.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 5.

Formative assessment

There are two formative unassessed courseworks over an eleven week period, designed to consolidate student learning.

Feedback

Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in office hours.

## Module aims

• This module aims to introduce students to a range of symmetry-based techniques for describing the formation and evolution of regular patterns which occur in nature or in laboratory experiments.

## Learning outcomes

 Attributes Developed 001 Students will be able to locate and classify codimension-one bifurcations of ordinary differential equations. KC 002 Students will understand the concept of a centre manifold. K 003 Students will be able to identify the symmetry group relevant to simple pattern formation problems and use the Equivariant Branching Lemma in simple cases. KC 004 Students will understand how to describe patterns using amplitude equations, and be able to find solutions of these equations in simple cases. KC 005 Students will 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 students with a detailed introduction to nonlinear patterns and their analysis.

• Provide students with experience of methods used to analyse and solve problems involving pattern formation and evolution in physical and biological systems.

The learning and teaching methods include:

• Three one-hour lectures for eleven weeks, in which students will be encouraged to take lecture notes to facilitate their learning and engagement with the module material. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these courseworks as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

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.

Upon accessing the reading list, please search for the module using the module code: MATM031

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM031 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: The module MATM031 equips students with skills which significantly enhance their employability. The mathematical proficiency gained will hone their critical thinking and problem-solving abilities. Students will learn to analyse real-world problems relating to pattern formation and evolution in physical and biological systems, and apply mathematical techniques and logical reasoning to arrive at solutions. Mathematical modelling and analysis of real-world problems are highly sought after skills in many professions.

Global and Cultural Capabilities: Student enrolled in MATM031 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MATM031 is a module which demands the ability to analyse real-world problems in physical and biological systems, and apply methods relating to both abstract algebra and differential equations to solve these problems and interpret the results. Students will gain skills in analysing unseen problems using lateral thinking, and will complete assessments which challenge them and build resilience.

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.