MA307-7-AU-CO:
Advanced Ordinary Differential Equations and Dynamical Systems
2022/23
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Postgraduate: Level 7
Current
Thursday 06 October 2022
Friday 16 December 2022
15
13 April 2023
Requisites for this module
(none)
(none)
(none)
(none)
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The subject of Ordinary Differential Equations (ODEs) is a very important and fascinating branch in Mathematics. An abundance of phenomena in Physics, Biology, Engineering, Chemistry, Finance and Neuroscience to name a few, may be described and studied using such equations.
The module will introduce students to advanced topics and theories in ODEs and dynamical systems
The aim of the module is to familiarise students with advanced concepts and theories in Ordinary Differential Equations and dynamical systems. It will equip students with the knowledge and skills to solve or study the behaviour of solutions of such equations by using advanced analytical approaches.
1. Use of methods to solve 2nd order linear ODEs with variable coefficients, including power-series solutions, solutions near ordinary points, the power-series method, the Legendre equation, solutions near regular singular points, the Euler equation, the Frobenius theorem and method.
2. Systematic understanding of the behaviour of solutions of 2nd order linear systems of ODEs near critical points, population dynamics and prey-predator systems.
3. Systematic understanding of Liapunov’s method and function.
4. Systematic understanding of periodic solutions and limit cycles.
5. Critical awareness on plane autonomous systems and linearisation.
6. Systematic understanding of autonomous systems, almost linear systems, nonlinear systems, and stability/instability.
7. Systematic understanding and use of the Laplace transform to solve initial value problems for linear differential equations with constant coefficients and continuous or discontinuous functions.
The module will be based on the following syllabus:
1. Linear ODEs with variable coefficients (power-series solutions, ordinary points, solutions near ordinary points, the power series method, the Legendre equation, regular singular points, solutions near regular singular points, the Euler equation, The Frobenius theorem and method)
2. Autonomous systems and stability
3. Locally linear systems, competing species, predator-prey equations
5. Liapunov's second method
6. Periodic solutions and limit cycles. The Poincaré-Bendixson theorem.
7. The Laplace transform (definition of the Laplace transform, solution of initial value problems using the Laplace transform, step functions)
Teaching in the department will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.
This module does not appear to have a published bibliography for this year.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Test |
|
|
| Exam |
Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period)
|
| Exam |
Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period)
|
Additional coursework information
Reassessment strategy:
If a student fails the test and the exam they will only take the resit exam which will be worth 100%;
If a student fails the exam but passes the test they will only take the resit exam, the mark for which will be re-aggregated with the test mark;
If a student fails the test but passes the exam they will take the resit test, the mark for which will be re-aggregated with the exam mark.
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Chris Antonopoulos, email: canton@essex.ac.uk.
Dr Chris Antonopoulos
canton@essex.ac.uk
Yes
No
Yes
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 43 hours, 41 (95.3%) hours available to students:
0 hours not recorded due to service coverage or fault;
2 hours not recorded due to opt-out by lecturer(s), module, or event type.
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