2008/09 Undergraduate Module Catalogue
MATH2391 Nonlinear Differential Equations
10 creditsClass Size: 200
Module manager: Professor A.V. Mikhailov
Email: sashamik@maths.leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2008/09
Pre-requisite qualifications
MATH1970 or MATH1932 or MATH1960 or MATH1400 or MATH2450, or equivalent. MATH2365 and MATH2420 are recommended but not essential.Co-requisites
MATH2015 | Analysis 2 |
This module is approved as an Elective
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:a) sketch phase plane portraits of second-order linear and nonlinear ODEs
b) sketch bifurcation diagrams and identify bifurcation points
c) determine the stability of equilibrium points using a variety of methods
d) determine the exsistence or otherwise of periodic orbits in second order autonomous nonlinear ODEs using Dulac's criterion, Lyapunov functions and the Poincare-Bendixson Theorem.
Syllabus
1. Existence and uniqueness of ordinary differential equations. Examples of finite time blow-up abd 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. Discussion of structural stability.
4. Second order nonlinear ODEs. Phase portraits. Construction of the exponential matrix, including Jordan canonical form for 2 x 2 matrices.
5. Second order nonlinear ODEs. Equilibrium solutions and linear stability theory. Using MAPLE to assist drawing phase portraits.
6. Elementary theory of periodic orbits. Dulac's criterion, Lyapunov functions, Poincare index theory, Poincare-Bendixson Theorem.
7. Bifurcations in second order nonlinear ODEs: the Hopf bifurcation., Only treated as a statement. without proof or extended study.
Teaching methods
Delivery type | Number | Length hours | Student hours |
Example Class | 9 | 1.00 | 9.00 |
Lecture | 22 | 1.00 | 22.00 |
Private study hours | 69.00 | ||
Total Contact hours | 31.00 | ||
Total hours (100hr per 10 credits) | 100.00 |
Private study
Regular examples sheetsMethods of assessment
Coursework
Assessment type | Notes | % of formal assessment |
In-course Assessment | . | 10.00 |
In-course MCQ | . | 10.00 |
Total percentage (Assessment Coursework) | 20.00 |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
Exams
Exam type | Exam duration | % of formal assessment |
Standard exam (closed essays, MCQs etc) | 2 hr | 80.00 |
Total percentage (Assessment Exams) | 80.00 |
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 websiteLast updated: 18/05/2009
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