MA30062: Analysis of nonlinear ordinary differential equations
[Page last updated: 09 August 2024]
| Academic Year: | 2024/25 |
| Owning Department/School: | Department of Mathematical Sciences |
| Credits: | 6 [equivalent to 12 CATS credits] |
| Notional Study Hours: | 120 |
| Level: | Honours (FHEQ level 6) |
| Period: |
- Semester 2
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| Assessment Summary: | EX 100% |
| Assessment Detail: | |
| Supplementary Assessment: |
- Like-for-like reassessment (where allowed by programme regulations)
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| Requisites: |
Before taking this module you must take MA20218 AND take MA20220
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| Learning Outcomes: |
After taking this unit, students should be able to:
* Demonstrate understanding of and prove basic results in the theory of ordinary differential equations including: existence and uniqueness for the initial-value problem, basic properties of flows and limit sets, Lyapunov's stability theorem and LaSalle's invariance principle.
* Apply the theory to analyse the behaviour of simple examples.
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| Aims: | To provide an accessible but rigorous treatment of initial-value problems for nonlinear systems of ordinary differential equations, including existence and uniqueness of maximal solutions and Lyapunov stability theory, illustrated by examples. Foundations will be laid for advanced studies in dynamical systems, mechanics and control. The material is also useful in mathematical biology and numerical analysis.
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| Skills: | Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets)
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| Content: | General definition of an ODE. Examples from diverse areas. Elementary methods:
separation of variables, variation of constant, estimates using Gronwall's lemma.
Existence and uniqueness for maximal solutions for ODEs with sufficiently regular coefficients (via contraction mapping theorem), continuous dependence on initial conditions.
Autonomous ODEs: introduction to local and global flows, orbits, limit sets, integral invariance principle, stability, Lyapunov functions, Lyapunov stability theorem, LaSalle's invariance principle, application to examples.
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| Course availability: |
MA30062 is Optional on the following courses:
Department of Mathematical Sciences
- USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
- USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
- USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
- USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
- USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
- USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
- USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
- USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
- USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
- USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
- USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
- USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
- USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
- USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
- USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 3)
- USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
- USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
- USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 3)
- USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 4)
- USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 4)
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Notes: - This unit catalogue is applicable for the 2024/25 academic year only. Students continuing their studies into 2025/26 and beyond should not assume that this unit will be available in future years in the format displayed here for 2024/25.
- Courses and units are subject to change in accordance with normal University procedures.
- Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
- Find out more about these and other important University terms and conditions here.
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