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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.

Postgraduate Loans

If you study this module as part of an eligible qualification, you may be eligible for a Postgraduate Loan. For more information, see Fees and funding.

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
Find out more in Why the OU?
Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

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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

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance.

Contact us if you want to know more about study with The Open University before you register.

Assessment

The assessment details can be found in the facts box above.

You will be expected to submit your tutor-marked assignments (TMAs) online using eTMA system, unless there are some difficulties which prevent you from doing so. In these circumstances, you must negotiate with your tutor to get their agreement to submit your assignment on paper.

Course work includes

4 Tutor-marked assignments (TMAs)
Examination
No residential school

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Future availability

Nonlinear ordinary differential equations (M821) starts once a year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2024.

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 Register
    05 Oct 2019 Jun 2020 Not yet available

    Registration closes 12/09/19 (places subject to availability)

    Register
    This module is expected to start for the last time in October 2024.

    Future availability

    Nonlinear ordinary differential equations (M821) starts once a year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2024.

    Additional costs

    Study costs

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

    Study weekend

    This module normally includes an optional study weekend. For each day you choose to attend, you must pay an additional charge of around £50 to cover tuition and refreshments during the day. You’ll pay this charge when you book, after you’ve registered on the module. You’ll also have to pay for your own travel to and from the venues and your own accommodation if you need it.

    Ways to pay

    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. To find out more see Fees and funding.

    Study materials

    What's included

    Module notes, other printed materials.

    You will need

    We recommend that you have access to the internet at least once a week during the module and would like to point out that vital material, such as your assignments, will be delivered online.

    Computing requirements

    A computing device with a browser and broadband internet access is required for this module.  Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

    Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

    A desktop or laptop computer with either:

    • Windows 7 or higher
    • macOS 10.7 or higher

    The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

    To participate in our online-discussion area you will need both a microphone and speakers/headphones. 

    Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students. 

    Materials to buy

    Set books

    • Jordan, D. & Smith, P. Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers (4th edn) Oxford University Press £36.99 - 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.