# Detailed module information

Module information on this degree can be found below, separated by year of study.

The module information below applies for the current academic year. The academic year runs from August to July; the 'current year' switches over at the end of July.

Students select optional courses subject to rules specified in the Mechanical Engineering Student Handbook,  for example at most three Design and Business courses. Please note that numbers are limited on some optional courses and selection criteria will apply.

## Mathematics

### 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:  6
Contributing to Course Elements: 6 to ME3-LCTVS or ME4-LCTVS

### 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

 Summary of student timetabled hours Autumn Spring Summer Lectures — 22 — Tutorials — 10 — Total 32 hr Expected private study time 3-4 hr per week plus exam revision

### Assessments

 Written examinations: Date (approx.) Max. mark Pass mark Mathematics (3hr) This is a CLOSED BOOK Examination April/May 200 n/a