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# Mathematical Modelling: Nonlinearity, Uncertainty, & Computational Methods - MAT00110M

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• Department: Mathematics
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2023-24
• See module specification for other years: 2024-25

## Module summary

At the heart of Applied Mathematics is the pairing of rigorous analytic results with numerically derived insights in order to understand and solve real-world problems. This module explores this approach, which can be employed on a range of quantitative problems in science, industry, finance and management. Students will engage analytically with advanced dynamical systems theory on the one hand, whilst simultaneously learning how these ideas can be taken forward computationally with coding to vastly increase the scope of problems that can be tackled.

## Related modules

• None

### Prohibited combinations

Any MSc students taking this course should be familiar with 1D & 2D dynamical systems, classification of stable points, bifurcations and phase-plane analysis.

## Module will run

Occurrence Teaching period
A Semester 1 2023-24

## Module aims

At the heart of Applied Mathematics is the pairing of rigorous analytic results with numerically derived insights in order to understand and solve real-world problems. This module explores this approach, which can be employed on a range of quantitative problems in science, industry, finance and management. Students will engage analytically with advanced dynamical systems theory on the one hand, whilst simultaneously learning how these ideas can be taken forward computationally with coding to vastly increase the scope of problems that can be tackled.

## Module learning outcomes

By the end of this module students will be expected to be able to:

1. Analytically characterise the dynamics of low dimensional systems of ordinary differential equations in continuous time (e.g. periodic behaviour, Hopf bifurcations, excitability, tipping points, chaos)

2. Supplement and visualise these analytic insights with computational approaches introduced throughout the module

3. Push beyond the limits of analytical tractability to tackle a broader range of real-world modelling challenges (e.g. high dimensional systems of ordinary differential equations, discrete and stochastic simulations of reactions, stochastic differential equation)

4. Understand and explain the behaviour of these more complicated models using the analytic insight developed in simpler model analogues

5. Critically analyse relevant contemporary studies in context of module content.

6. Construct and analyse models of real-world systems using the approaches developed in the module.

## Module content

This module combines advanced dynamical systems theory with the practical computational methods required to understand such models in real-world scenarios. From a theoretical standpoint students will learn about limit cycles and periodic behaviour (which can explain the gene expression oscillations governing the circadian rhythm), excitability (used to model neuron firing), and tipping points (an important feature of climate and financial models alike). Deeper understanding of these models will be developed through a computational approach that will also broaden the scope of models that can be addressed. This will include dealing with complex networks (present in ecological, social and financial systems), stochastic differential equations (used to model ecological and financial time series), and stochastic simulation.

The computational element will be taught with reference to MATLAB (the language of choice in many fields of engineering and mathematical sciences), with explicit coding lessons introducing its syntax and visualisation functions. However the use of other languages is welcomed where appropriate.

Although a range of application areas will be covered, many will take inspiration from the biosciences, reflecting the research-led ethos of this module.

Additional M-level content will be signposted at appropriate points in the teaching schedule and will typically involve guided self-study of published research papers. This additional learning will be synthesised in week 9 “Developing and assessing useful models for complex problems”, and assessed by an individual report due in week 11.

## Indicative assessment

Essay/coursework 20
Essay/coursework 10
Essay/coursework 50
Essay/coursework 20

### Special assessment rules

None

If a student has a failing module mark, only failed components need be reassessed.

Coursework consists of 3 components, that build cumulatively over the duration of the module:

M-level: computational proficiency assignment (10%), group report (20%), final individual report (50%), additional assignment (20%)

### Indicative reassessment

Essay/coursework 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.