Module manager: Dr Tyler Cassidy
Email: T.Cassidy1@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2024/25
MATH1005 or (MATH1010 and MATH1012) or (MATH1400 and MATH1060) or (MATH1400 and MATH1331) or (PHYS1300 and MATH1060), or equivalent.
This module is not approved as a discovery module
Nonlinear systems occur widely in the real world, and may produce oscillations or even wild chaotic fluctuations even when influenced by a constant external force. This course provides a first introduction to the mathematics behind such behaviour.
On completion of this module, students should be able to do the following (where appropriate) for first and second order linear and nonlinear ODEs:
a) sketch phase portraits;
b) determine the stability of equilibrium points via the eigenvalues of its Jacobian;
c) sketch bifurcation diagrams, identify bifurcation points and classify fold (saddle-node), transcritical and pitchfork bifurcations;
1. Existence and uniqueness of ordinary differential equations. Examples of finite time blow-up and non-uniqueness of solutions.
2. First order nonlinear ODEs. Stability of equilibrium solutions. Interpretation of the nonlinear ODE as a vector field.
3. Bifurcation theory for first order nonlinear ODEs:
the saddle-node, transcritical and pitchfork bifurcations.
4. Second order linear ODEs. Phase portraits. Construction of the exponential matrix, including Jordan canonical form for 2 x 2 matrices.
5. Second order nonlinear ODEs. Equilibrium solutions, linear stability theory and drawing phase portraits.
6. Additional topics (at the module leader's discretion): first integrals, theory of periodic orbits; perturbation approaches, computational methods.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Workshop | 10 | 1 | 10 |
| Lecture | 22 | 1 | 22 |
| Private study hours | 68 | ||
| Total Contact hours | 32 | ||
| Total hours (100hr per 10 credits) | 100 | ||
Regular examples sheets
Regular problem solving assignments
| Assessment type | Notes | % of formal assessment |
|---|---|---|
| In-course Assessment | . | 15 |
| Total percentage (Assessment Coursework) | 15 | |
There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.
| Exam type | Exam duration | % of formal assessment |
|---|---|---|
| Standard exam (closed essays, MCQs etc) | 2.0 Hrs 0 Mins | 85 |
| Total percentage (Assessment Exams) | 85 | |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
The reading list is available from the Library website
Last updated: 4/29/2024
Errors, omissions, failed links etc should be notified to the Catalogue Team