2023/24 Undergraduate Module Catalogue

MATH2391 Nonlinear Differential Equations

10 creditsClass Size: 170

Module manager: Dr Tyler Cassidy
Email: T.Cassidy1@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2023/24

Pre-requisite qualifications

MATH1005 or (MATH1010 and MATH1012) or (MATH1400 and MATH1060) or (MATH1400 and MATH1331) or (PHYS1300 and MATH1060), or equivalent.

This module is approved as a discovery module

Module summary

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.

Objectives

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;

Syllabus

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.

Teaching methods

Delivery type	Number	Length hours	Student hours
Workshop	10	1.00	10.00
Lecture	22	1.00	22.00
Private study hours			68.00
Total Contact hours			32.00
Total hours (100hr per 10 credits)			100.00

Private study

Regular examples sheets

Opportunities for Formative Feedback

Regular problem solving assignments

Methods of assessment

Coursework

Assessment type	Notes	% of formal assessment
In-course Assessment	.	15.00
Total percentage (Assessment Coursework)		15.00

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.

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr 00 mins	85.00
Total percentage (Assessment Exams)		85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 18/08/2023

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