A postgraduate course in Mathematics and Statistics.

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.

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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.

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 declare the MSc in Mathematics (or another qualification towards which the course can count) as your qualification intention. 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.

Regulations

As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are available on our Essential documents website.

If you have a disability

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

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

You will need a computer with internet access to study this course as it includes online activities, which you can access using a web browser.

• If you have purchased a new desktop or laptop computer since 2007 you should have no problems completing the online activities.
• If you’ve got a netbook, tablet or other mobile computing device check our Technical requirements section.
• If you use an Apple Mac you will need OS X 10.6 or later.

You can also visit the Technical requirements section for further computing information including the details of the support we provide.

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.

Future availability

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

How to register

We regret that we are currently unable to accept registrations for this course. Where the course is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.

Student Reviews

Distance learning

The Open University is the world’s leading provider of flexible, high quality distance learning. Unlike other universities we are not campus based. You will study in a flexible way that works for you whether you’re at home, at work or on the move. As an OU student you’ll be supported throughout your studies – your tutor or study adviser will guide and advise you, offer detailed feedback on your assignments, and help with any study issues. Tuition might be in face-to-face groups, via online tutorials, or by phone.

For more information about distance learning at the OU read Study explained.

Course satisfaction survey

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