England.

# Deterministic and stochastic dynamics

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This module is an accessible introduction to the study of applied mathematics at OU level 3. It introduces core topics in applied mathematics at this level and is structured around three books: Fundamental concepts of dynamics, Deterministic dynamics and Stochastic processes and diffusion. The module uses the Maxima computer algebra system to illustrate how to explore the properties of dynamical systems with a computer.

## What you will study

This module is presented as three books.

Book 1: Fundamental concepts of dynamics
The first book considers ordinary differential equations, Newton's second law, conservation of energy, and the concepts of fixed point, limit cycles and constants of motion. It will also introduce a framework for discussing random processes, such as random walks.

Book 2: Deterministic dynamics
The second book will develop some more advanced concepts. In the case of conservative systems, it introduces the calculus of variations and develops Lagrangian dynamics from Hamilton’s principle. In the case of dissipative systems, it will consider the use of maps to model dynamical processes. ‘Chaos’ will be defined and explored using the strange attractor. The book will introduce the notions of Lyapunov exponents, fractal dimensions of attractors, and their connection via the ‘Lyapunov dimension’ formula.

Book 3: Stochastic processes and diffusion
Finally, the third book will investigate the random walk as the archetypical random dynamical process, and explain its connection to the diffusion equation. Fourier methods (both series and transforms) will be treated by illustrating their role in treatment of the diffusion equation and probability theory. The module will conclude with a look at some further applications of random dynamical systems, including the models used for option pricing in mathematical finance.

The module will use the Maxima computer algebra system to illustrate how computers are used to explore properties of dynamical systems. You will be required to use Maxima in some of the assignments, but it will be possible to complete the module without very extensive use of this package. However, there will be plenty of optional exercises which illustrate the power of computers for exploring the properties of dynamical systems.

You can find the full content list on the Open mathematics and statistics website.

## Entry requirements

There is no formal pre-requisite study, but you must have the required mathematical skills.

Talk to an advisor if you’re not sure you’re ready.

## Preparatory work

You should aim to be confident and fluent with the concepts covered in the Are you ready? quiz here, and follow the advice in the quiz.

The key topics to revise include:

• differential equations
• mechanics.

Mathematical methods, models and modelling (MST210) or Mathematical methods (MST224) is ideal preparation.

## What's included

You’ll have access to a module website, which includes:

• a week-by-week study planner
• course-specific module materials
• audio and video content
• Maxima algebra software and associated guidance
• assessment details, instructions and guidance
• online tutorial access
• access to student and tutor group forums.

You’ll be provided with printed books covering the content of the module, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques. You’ll also receive a printed module handbook.

## You will need

A scientific calculator.

### Computing requirements

You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.

Any additional software will be provided or is generally freely available.

To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).

Our module websites comply with web standards, and any modern browser is suitable for most activities.

Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.

It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.

## Teaching and assessment

### Support from your tutor

Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:

• Marking your assignments (TMAs) and providing detailed feedback for you to improve.
• Guiding you to additional learning resources.
• Providing individual guidance, whether that’s for general study skills or specific module content.
• Facilitating online discussions between your fellow students, in the dedicated module and tutor group forums.

Module tutors also run online tutorials throughout the module. Where possible, recordings of online tutorials will be made available to students. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part.

### Assessment

The assessment details for this module can be found in the facts box.

## If you have a disability

The OU strives to make all aspects of study accessible to everyone and this Accessibility Statement outlines what studying MS327 involves. You should use this information to inform your study preparations and any discussions with us about how we can meet your needs.

## Future availability

Deterministic and stochastic dynamics (MS327) starts once a year – in October.

This page describes the module that will start in October 2024.

We expect it to start for the last time in October 2027.

### Course work includes:

4 Tutor-marked assignments (TMAs)
Examination

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