Deterministic and stochastic dynamics
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&/courses/qualifications/details/ms327/39;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 .
Entry requirements
There is no formal pre-requisite study, but you must have the required mathematical skills.
You can .
if you’re not sure you’re ready.
Preparatory work
You should aim to be confident and fluent with the concepts covered in the , 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.