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Nonlinear ordinary differential equations

Relevant to scientists and engineers as well as mathematicians, this introduction to basic theory and simpler approximation schemes covers systems with two degrees of freedom. It introduces the geometric aspects of the two-dimensional phase space, the importance of fixed points and how they can be classified, and the notion of a limit cycle. You’ll develop schemes to approximate the solutions of autonomous and non-autonomous equations to understand how these solutions behave. Periodically forced nonlinear oscillators and nonlinear oscillators with periodically time-varying parameters leading to parametric resonances are discussed, along with the stability of these solutions and tests for obtaining stability.

Qualifications

M821 is an option module in our:

Module

Module code
M821
Credits

Credits

  • Credits measure the student workload required for the successful completion of a module or qualification.
  • One credit represents about 10 hours of study over the duration of the course.
  • You are awarded credits after you have successfully completed a module.
  • For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.
30
Study level
Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU postgraduate modules correspond to these frameworks.
OU Postgraduate
SCQF 11
FHEQ 7
Study method
Distance learning
Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

Module details Entry requirements Module registration Study materials

What you will study

Nonlinear ordinary differential equations arise in a wide variety of circumstances: a simple pendulum, oscillations in electrical circuits, oscillations of mechanical structures, molecular vibrations, the motion of particles in accelerators, planetary motion, the effects of strong electromagnetic fields of atoms and molecules. In biology, they occur as models of evolving populations and the spreading of infectious diseases and also in the modelling of neural systems.

The module is based on the set book Nonlinear Ordinary Differential Equations by D. W. Jordan and P. Smith. It is an introduction to some of the basic theory and to the simpler approximation schemes. It deals mainly with systems that have two degrees of freedom, and it 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 by which the solutions of autonomous and non-autonomous equations can be approximated, and so begin to understand how the solutions behave. In this section there is some emphasis on periodically forced nonlinear oscillators and on nonlinear oscillators with periodically time-varying parameters, leading to parametric resonances.

Finally, the stability of these solutions is discussed and various tests for stability are obtained, together with methods to establish the existence of periodic solutions.

You will learn

Successful study of this module should enhance your skills in understanding complex mathematical texts, constructing solutions to problems logically and communicating mathematical ideas clearly.

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.

The module has a dedicated and moderated forum where you can join in online discussions with your fellow students. There are also online module-wide tutorials. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.

Assessment

The assessment details can be found in the facts box.

Course work includes

4 Tutor-marked assignments (TMAs)
Examination

Future availability

Nonlinear ordinary differential equations (M821) starts every other year – in October.

This page describes the module that will start in October 2026.

We expect it to start for the last time in October 2032.

Regulations

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.

Entry requirements

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 all your Stage 3 modules (minimum Grade 2 passes recommended).

Preparatory work

You should be familiar with ordinary differential equations at an advanced undergraduate level.

All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.

Register

Start End Fee
- - -

No current presentation - see Future availability
This module is expected to start for the last time in October 2032.

Future availability

Nonlinear ordinary differential equations (M821) starts every other year – in October.

This page describes the module that will start in October 2026.

We expect it to start for the last time in October 2032.

Additional costs

Study costs

There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

Study weekend

This module may have an optional study weekend; there will be a charge to attend in person or online. If attending in person, you must also pay for travel and accommodation if you need it. More details will be available after the module has started.

Ways to pay for this module

We know there’s a lot to think about when choosing to study, not least how much it’s going to cost and how you can pay.

That’s why we keep our fees as low as possible and offer a range of flexible payment and funding options, including a postgraduate loan, if you study this module as part of an eligible qualification. To find out more, see Fees and funding.

Study materials

What's included

You’ll be provided with module notes covering the content of the module, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques that are contained in the set book. In addition, you will have a printed module handbook.

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

  • a week-by-week study planner
  • course-specific module materials
  • audio and video content
  • assessment details and submission section
  • online tutorial access
  • access to student forums.

Computing requirements

  • Primary device – A desktop or laptop computer. It’s possible to access some materials on a mobile phone, tablet or Chromebook; however, they will not be suitable as your primary device.
  • Peripheral device – Headphones/earphones with a built-in microphone for online tutorials.
  • Our OU Study app operates on supported versions of Android and iOS.
  • Operating systems – Windows 11 or latest supported macOS. Microsoft will no longer support Windows 10 as of 14 October 2025.
  • Internet access – Broadband or mobile connection.
  • Browser – Google Chrome and Microsoft Edge are recommended. Mozilla Firefox and Safari may be suitable.

Materials to buy

Set books

  • Jordan, D. & Smith, P. Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers (4th edn) Oxford University Press £47.49 - ISBN 9780199208258

If you have a disability

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.

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