MA307-6-AU-CO:
Advanced Ordinary Differential Equations and Dynamical Systems
2023/24
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Undergraduate: Level 6
Current
Thursday 05 October 2023
Friday 15 December 2023
15
05 January 2024
Requisites for this module
MA202
(none)
(none)
(none)
(none)
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, can be described and studied using such equations.
The module builds upon the Ordinary Differential Equations module on ODEs and will introduce students to advanced topics and theories in ODEs and dynamical systems.
The aims of this module are:
- to familiarise students with advanced concepts and theories in Ordinary Differential Equations and dynamical systems.
- to equip students with the knowledge and skills to solve or study the behaviour of solutions of such equations by using advanced analytical approaches.
By the end of the module, students will be expected to:
- Know the 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.
- Have systematic understanding of the behaviour of solutions of 2nd order linear systems of ODEs near critical points, population dynamics and prey-predator systems.
- Have systematic understanding of Liapunov’s method and function.
- Have systematic understanding of periodic solutions and limit cycles.
- Have critical awareness on plane autonomous systems and linearisation.
- Have systematic understanding of autonomous systems, almost linear systems, nonlinear systems, and stability/instability.
Indicative syllabus
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)
Autonomous systems and stability
Locally linear systems, competing species, predator-prey equations
Liapunov's second method
Periodic solutions and limit cycles. The Poincaré-Bendixson theorem.
Teaching in the School 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.
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Boyce, W.E., DiPrima, R.C. and Meade, D.B. (2017)
Boyce’s elementary differential equations and boundary value problems. Eleventh edition. Hoboken, NJ: Wiley. Available at:
https://app.kortext.com/Shibboleth.sso/Login?entityID=https://idp0.essex.ac.uk/shibboleth&target=https://app.kortext.com/borrow/272233.
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Jordan, D.W. and Smith, P. (2007)
Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers. 4th ed. Oxford: Oxford University Press. Available at:
https://ebookcentral.proquest.com/lib/universityofessex-ebooks/detail.action?docID=415593.
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D. Jordan and P. Smith (2007)
Nonlinear Ordinary Differential Equations: Problems and Solutions. Oxford University Press, USA. Available at:
https://ebookcentral.proquest.com/lib/universityofessex-ebooks/detail.action?docID=415594.
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Hirsch, M.W., Smale, S. and Devaney, R.L. (2013)
Differential equations, dynamical systems, and an introduction to chaos. Third edition. Amsterdam: Academic Press. Available at:
https://www.sciencedirect.com/book/9780123820105/differential-equations-dynamical-systems-and-an-introduction-to-chaos.
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Wiandt, T., Haines, C.W. and Boyce, W.E. (2013a) ‘9.1’, in Student Solutions Manual to Accompany Elementary Differential Equations. Hoboken, NJ: Wiley, pp. 219–224.
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Wiandt, T., Haines, C.W. and Boyce, W.E. (2013b) ‘9.2’, in Student solutions manual to accompany Elementary differential equations. Hoboken, NJ: Wiley, pp. 224–228.
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Wiandt, T., Haines, C.W. and Boyce, W.E. (2013c) ‘9.3’, in Student solutions manual to accompany Elementary differential equations. Hoboken, NJ: Wiley, pp. 228–237.
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Wiandt, T., Haines, C.W. and Boyce, W.E. (2013d) ‘9.4’, in Student Solutions Manual to Accompany Elementary Differential Equations. Hoboken, NJ: Wiley, pp. 237–246.
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Wiandt, T., Haines, C.W. and Boyce, W.E. (2013e) ‘9.5’, in Student Solutions Manual to Accompany Elementary Differential Equations. Hoboken, NJ: Wiley, pp. 247–250.
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Wiandt, T., Haines, C.W. and Boyce, W.E. (2013f) ‘9.6’, in Student Solutions Manual to Accompany Elementary Differential Equations. Hoboken, NJ: Wiley, pp. 250–253.
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Wiandt, T., Haines, C.W. and Boyce, W.E. (2013g) ‘9.7’, in Student Solutions Manual to Accompany Elementary Differential Equations. Hoboken, NJ: Wiley, pp. 253–258.
The above list is indicative of the essential reading for the course.
The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.
Further reading can be obtained from this module's
reading list.
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 35 hours, 33 (94.3%) 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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