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Accessibility statement
Qualification dates
StartEnd
03 Oct 2026Jun 2027
This module covers basic theory and simpler approximation schemes with two degrees of freedom. It introduces geometric aspects of two-dimensional phase space, fixed points and their classifications, and limit cycles. You’ll approximate solutions of autonomous and non-autonomous equations to understand their behaviour, and examine periodically forced nonlinear oscillators and nonlinear oscillators with periodically time-varying parameters leading to parametric resonances. Finally, you’ll explore bifurcations in dynamical systems and gain a solid foundation in chaos theory. Relevant to scientists, engineers and mathematicians, the concepts have applications in many fields like climate modelling, electrical circuits, mechanical systems, and biological rhythms.
Nonlinear ordinary differential equations arise in various circumstances: a simple pendulum, oscillations in electrical circuits, oscillations of mechanical structures, molecular vibrations, the motion of particles in accelerators, planetary motion, and the effects of strong electromagnetic fields of atoms and molecules. In biology, they occur as models of evolving populations, the spreading of infectious diseases, and the modelling of neural systems.
The module is based on the book Nonlinear Ordinary Differential Equations by D. W. Jordan and P. Smith. It introduces some of the basic theory and simpler approximation schemes. It deals mainly with systems that have two degrees of freedom and 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 for approximated solutions of autonomous and non-autonomous equations and begin to understand how the solutions behave. This section emphasises periodically forced nonlinear oscillators and nonlinear oscillators with periodically time-varying parameters, leading to parametric resonances.
Finally, the stability of these solutions is discussed, various stability tests are obtained, and methods to establish the existence of periodic solutions are described.
If you’re studying this module on its own or as part of a postgraduate qualification, you must have passed (or be studying in parallel) one of the following modules:
If you’re studying towards our undergraduate Master of Physics (M06), you must have passed modules SM380 and SM381 (minimum Grade 2 passes recommended).
You should be familiar with ordinary differential equations at an advanced undergraduate level.
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Course work includes:
We regularly review the assessments in our modules, so we may update the examination method used for this module to an in-person exam or a remotely invigilated (proctored) exam. When we are making such a change, we will make it clear on this page. If we need to make a change after you have registered, we will notify you as soon as possible. If you have additional requirements, we will support you to complete your assessments.
You’ll be provided with module notes covering the module's content, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques contained in the set book. In addition, you will have a printed module handbook.
You’ll have access to a module website, which includes:
The material contains small print and diagrams, which may cause problems if you find reading text difficult.
To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.
Nonlinear ordinary differential equations (M821) starts every other year – in October.
It will next start in October 2026.
We expect it to start for the last time in October 2032.
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