Module information for the next academic year is available below. Current academic year information is also available.
Mathematics A
Module aims
- To show how mathematics can be used to characterise and model the qualitative behaviour of a system without solving in detail the equations controlling the evolution of the system.
ECTS units: 5
Learning outcomes
Discuss the basic ideas of chaos
- Perform a critical point analysis
- Extract asymptotic behaviour from non-linear ordinary differential equations
- Use Fourier transforms to find particular solutions of linear ordinary differential equations.
Module syllabus
- Linear Autonomous Systems: Examples including small oscillations of simple pendulum, damped and undamped; decaying and periodic oscillations. Use of the phase plane to visualize solutions. Classification of critical points using eigenvalues. Sketching of trajectories using information including eigenvectors.
- Nonlinear Autonomous Systems: Large oscillations of simple pendulum. Use of phase plan to examine behaviour of trajectories near critical points. Trajectory laws. Role of Jacobian matrix. More complicated nonlinear examples relevant to a number of physical applications. Application to PDEs.
- Fourier Transforms: Definition and relation to Fourier series. Elementary examples. Basic properties. Convolution theorem. The delta-function and its properties.
- Finite Energy Signals: Definition. Parseval/energy theorem. Energy spectrum, autocorrelation, and the relationship between them. Autocorrelation properties. Examples including finite energy impulse train.
- Finite Power Signals: Definition. Power spectrum, autocorrelation, and the relationship between them. Autocorrelation properties. Examples including infinite energy impulse train.
- Introduction to Chaotic Systems: Pendulum with vertically oscillated pivot point: stability of critical points; numerical investigation showing period doubling phenomenon. 1D maps: fixed points and their stability; cobweb diagrams, period doubling. The Logistic Map as an example of a chaotic system: numerical experiments; analysis of period-doubling; transformation to the tent map.
Pre-requisites
ME1-hMTH, ME2-hMTH
Teaching methods
- Duration: Spring term
- Lecture/Study groups: 1 x 2hr lecture per week; 10 x 1hr tutorials; 2 extended problem sheets
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Summary of student timetabled hours | |||
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Autumn |
Spring |
Summer | |
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Lectures |
— |
22 |
— |
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Tutorials |
— |
10 |
— |
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Total |
32 hr | ||
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Expected private study time |
3-4 hr per week plus exam revision |
Assessments
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Written examinations: |
Date (approx.) |
Max. mark |
Pass mark |
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Mathematics (3hr) This is a CLOSED BOOK Examination |
April/May |
200 |
n/a |
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