Postgraduate

Nonlinear ordinary differential equations

On this page

• What you will study
• Entry
• Qualifications
• If you have a disability or additional requirement
• Study materials
• Teaching and assessment
• Students also studied
• Future availability
• How to register

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.

Course facts

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This course is available for study in the countries shown.

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.

The course 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.

You will learn

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

Entry

You must be registered for the MSc in Mathematics or for another qualification towards which the course can count. You must normally pass at least one of the entry modules Calculus of variations and advanced calculus (M820) or Analytic number theory I (M823) before studying this module. For this module you should have a sound knowledge of the theory of calculus and ordinary linear differential equations with constant coefficients. An adequate preparation would be our undergraduate-level courses Mathematical methods and models (MST209) (or the discontinued MST207) and Pure mathematics (M208) (or the discontinued M201, M203, M231 and MST204). 

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 course, please contact our Student Registration & Enquiry Service.

Qualifications

M821 is an optional module in our:

• Postgraduate Certificate in Mathematics (C90)
• Postgraduate Diploma in Mathematics (E23)
• MSc in Mathematics (F04)

Some postgraduate qualifications allow study to be chosen from other subject areas. We advise you to refer to the relevant qualification descriptions for information on the circumstances in which this module can count towards these qualifications because from time to time the structure and requirements may change.

If you have a disability or additional requirement

The material contains small print and diagrams, which may cause problems if you find reading text difficult.

If you are a new student, or new to study using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our publication Meeting Your Needs.

You can also find information about accessible study materials, financial support and the Disabled Students' Allowance, equipment and other services, on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.

Study materials

What's included

Course notes, other printed materials.

You will need

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

Computing requirements

This course includes online computer activities – you can access these using a web browser that can play Flash and Shockwave.

You will need internet access and a computer. If you have purchased a new computer since 2005 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual.

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 our Student Registration & Enquiry Service 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 a special maths eTMA processor, which is used in place of the main 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.

You will, however, be granted the option of submitting on paper if typesetting electronically or merging scanned images of your answers to produce an electronic TMA would take you an unacceptably long time.

Assessment is an essential part of the teaching, so you are expected to complete it all. But if you unavoidably miss or do badly in an assignment, some courses allow you a ‘substitution score’, calculated as a weighted average of all your scores for the course. In M821 this rule can apply to one assignment only. You will be given more detailed information when you begin the course.

Students also studied

Students who studied this course also studied at some time:

• Calculus of variations and advanced calculus (M820)
• Applied complex variables (M828)
• Analytic number theory I (M823)
• Advanced mathematical methods (M833)
• Dissertation in mathematics (M840)

Future availability

The details given here are for the course that starts in February 2012. We expect it to then be available in February 2013, October 2014 and thereafter in October each year.

How to register

To register a place on this course return to the top of the page and use the Click to register button. For more information and advice about registration see OU Study Explained.

Study explained

Course

- a module of study that can count towards a qualification.

Credits

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Financial support

- find out if you qualify for support with your fees with our eligibility checker.

Study explained

- all you need to know about distance learning with the OU.

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