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.
04 Oct 2014
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This module is expected to start for the last time in October 2016.
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 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.
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.
This is one of the intermediate modules in the MSc in Mathematics (F04). Normally, you should have completed at least one intermediate module before studying Advanced mathematical methods (M833).
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 at least one of the entry modules for the MSc in Mathematics (F04), Calculus of variations and advanced calculus (M820) or Analytic number theory I (M823).
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 contact our Student Registration & Enquiry Service.
M821 is an optional module in our:
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.
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.
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. Find out more about our services for disabled students.
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.
You will need a computer with internet access to study this module as it includes online activities, for use with a web browser. There is also software to download and install on your computer.
If you have purchased a new desktop or laptop computer since 2008 you should have no problems completing the online activities.
If you’ve got a netbook, tablet or other mobile computing device you may have difficulties with some software, check our Technical requirements section.
If you use an Apple Mac you will need OS X 10.7 or later.
You can also visit the Technical requirements section for further computing information (including details of the support we provide).
Materials to buy
- Jordan, D and Smith, P Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers (4th edn) Oxford University Press £35.99 - ISBN 9780199208258
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.
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.
Students also studied
Students who studied this course also studied at some time:
The details given here are for the module that starts in October 2014. We expect it to be available once a year, in October.
How to register
To register a place on this course return to the top of the page and use the Click to register button.
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.