Nonlinear ordinary differential equations arise in various circumstances: a simple pendulum, oscillations in electrical circuits, oscillations of mechanical structures, molecular vibrations, the motion of particles in accelerators, planetary motion, and the effects of strong electromagnetic fields of atoms and molecules. In biology, they occur as models of evolving populations, the spreading of infectious diseases, and the modelling of neural systems.
The module is based on the book Nonlinear Ordinary Differential Equations by D. W. Jordan and P. Smith. It introduces some of the basic theory and simpler approximation schemes. It deals mainly with systems that have two degrees of freedom and can be divided into three parts.
First, the geometric aspects of the two-dimensional phase space are discussed; we show why the fixed points are important and how they can be classified, and the notion of a limit cycle is introduced.
Then, we develop schemes for approximated solutions of autonomous and non-autonomous equations and begin to understand how the solutions behave. This section emphasises periodically forced nonlinear oscillators and nonlinear oscillators with periodically time-varying parameters, leading to parametric resonances.
Finally, the stability of these solutions is discussed, various stability tests are obtained, and methods to establish the existence of periodic solutions are described.
Successful study of this module should enhance your skills in understanding complex mathematical texts, constructing solutions to problems logically and communicating mathematical ideas clearly.
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Course work includes:
You’ll be provided with module notes covering the module's content, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques contained in the set book. In addition, you will have a printed module handbook.
You’ll have access to a module website, which includes:
Set books:
You can study this module on its own or use the credits you gain towards an Open University qualification.
M821 is an option module in our:
Nonlinear ordinary differential equations (M821) starts every other year – in October.
It will next start in October 2026.
We expect it to start for the last time in October 2032.
As a student of The Open University, you should be aware of the content of the academic regulations, which are available on our Student Policies and Regulations website.
If you’re studying this module on its own or as part of a postgraduate qualification, you must have passed (or be studying in parallel) one of the following modules:
If you’re studying towards our undergraduate Master of Physics (M06), you must have passed modules SM380 and SM381 (minimum Grade 2 passes recommended).
You should be familiar with ordinary differential equations at an advanced undergraduate level.
The material contains small print and diagrams, which may cause problems if you find reading text difficult.
To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.
| Start | End | Register by | England fee |
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| 03 Oct 2026 | 30 Jun 2027 | 10 Sep 2026 | Not yet available* |
| *This start date is open for pre-booking, which means you can reserve your place ahead of the fees being confirmed. We’ll publish updated 2026/27 fees and funding information on the 25th of March. |
There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
If your income is not more than £25,000 or you receive a qualifying benefit, you might be eligible for help with some of these costs after your module has started.
There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
This module may have an optional study weekend. There will be a charge to attend in person or online. If attending in person, you’ll also need to pay for your travel, and accommodation if you need it. More details will be available after the module has started.
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If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to help with your tuition fees. To find out more, see Postgraduate tuition fee loans in Northern Ireland.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to help with your tuition fees. To find out more, see Postgraduate loans in Scotland.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to support your study costs. To find out more, see Postgraduate loans in Wales.
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