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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 optional module in our:

This module can also count towards M03, which is no longer available to new students.

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
No residential school

Future availability

Nonlinear ordinary differential equations (M821) starts every other 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 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

    To study this module you must declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention.

    Normally, you should have also completed Calculus of variations and advanced calculus (M820), one of the entry modules for the MSc in Mathematics (F04). This will prepare you for studying applied mathematics at this level. Some familiarity with ordinary differential equations at advanced undergraduate level would be an advantage.

    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.

    If you have any doubt about the suitability of the module, please speak to an adviser.

    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 2024.

    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 has an optional study weekend. You must pay £60 for tuition and refreshments. You must also pay for your travel to and from the venue, and accommodation if you need it.

    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 and tutor group forums.

    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 (11 'Big Sur' 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.

    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.

    If you have particular study requirements please tell us as soon as possible, as some of our support services may take several weeks to arrange. Visit our Disability support website to find more about what we offer.

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